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quantum electrodynamics, photons and electrons, renormalization group, quantum resonances, spectral theory, Schroedinger operators, ground state, quantum dynamics, non-relativistic theory
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\newcommand{\DATUM}{February 26, 2011} % %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{Renormalization Group and Problem of Radiation\\ %Non-relativistic Quantum Electrodynamics\\
Les Houches, August, 2010}
\author{Israel Michael Sigal
\thanks{Dept.~of Math.,
Univ. of Toronto, Toronto, Canada; Supported by NSERC Grant No. NA7901} \\
%\small{Dept.~of Math.; Univ. of Toronto; Toronto; Canada;}\\[-1ex]
%\small{Canada (www.math.toronto.edu/sigal)}
}
\begin{document}
\date{\DATUM}
\maketitle
\begin{abstract} The standard model of non-relativistic quantum electrodynamics describes non-relativistic quantum matter, such as atoms and molecules, coupled to the quantized electromagnetic field. Within this model, we review basic notions, results and techniques in theory radiation. We describe the key technique in this area - the spectral renormalization group. Our review is based on joint works with Volker Bach and J\"urg Fr\"ohlich and with Walid Abou Salem, Thomas Chen, J\'er\'emy Faupin and Marcel Griesemer. Brief discussion of related contributions is given at the end of these lectures. This review will appear in "Quantum Theory from Small to Large Scales", Lecture Notes of the Les Houches Summer Schools, volume 95, Oxford University Press, 2011.
\end{abstract}
\textit{Key words}: quantum electrodynamics, photons and electrons, renormalization group, quantum resonances, spectral theory, Schr\"odinger operators, ground state, quantum dynamics, non-relativistic theory.
%\section{Renormalization Group and Non-relativistic QED}
\section{Overview}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We will describe some key results in theory of radiation for the standard model of non-relativistic electrodynamics (QED). % mathematical theory of the non-relativistic QED. This theory
The non-relativistic QED %is based on the Pauli-Fierz Hamiltonian, proposed in 1939
was proposed in early days of Quantum Mechanics\footnote{It was used, as already known, by Fermi (\cite{fermi}) in 1932 in his review of theory of radiation.} and it describes quantum-mechanical particle systems coupled to quantized electromagnetic field. It arises from a standard quantization of the corresponding classical systems %particle system interacting with electromagnetic field
(with possible addition of internal - spin - degrees of freedom)\footnote{In fact, it is the most general quantum theory obtained as a quantization of a classical system.} and it gives a complete and consistent account of electrons and nuclei interacting with electro-magnetic radiation at low energies. %(at the energies $\ll mc^2$, the rest energy of electron).
In fact, it accounts for all physical phenomena in QED, apart from vacuum polarization.
%It deals with quantum-mechanical particle systems coupled to quantized electromagnetic field at the energies $\ll mc^2$ (the rest energy of electron).
Sample of issues it addresses are
\begin{itemize}
\item
Stability; %(existence of the \textit{ground state});
\item Radiation; %(instability of the excited states of particle systems); %is \textit{stable} as the coupling is turned on, while
%The \textit{resonance} structure.%excited states, generically, are not. They turn into
\item Renormalization of mass;
\item Anomalous magnetic moment;
\item One-particle states;
\item Scattering theory.
\end{itemize}
There was a remarkable progress in the last 10 or so years in rigorous understanding of the corresponding phenomena. In this brief review we will deal with results concerning the first two items. We translate %some of the physical notions above
them into mathematical terms:
\begin{itemize}
\item
Stability $ \Longleftrightarrow$ Existence of the \textit{ground state};
\item Radiation $ \Longleftrightarrow$ Formation of %Turning of the excited states of particle systems into
\textit{resonances} out of the excited states of particle systems, scattering theory.
\end{itemize}
One of the key notions here is that of the \textit{resonance}. It gives a clear-cut mathematical description of processes of emission and absorption of the electro-magnetic radiation.
\medskip
The key and unifying technique we will concentrate on is the spectral \textit{renormalization group}. It is easily combined with other techniques, e.g. complex deformations (for resonances), the Mourre estimate (for dynamics), analyticity, fiber integral decompositions and Ward identities (used so far for translationally invariant systems). %it is augmented by other techniques
%The techniques used here
It was also extended to analysis of existence and stability of thermal states.
\medskip
\noindent \textbf{Acknowledgements} \\
The author is grateful to Walid Abou Salem, Thomas Chen, J\'er\'emy Faupin, Marcel Griesemer and especially Volker Bach and J\"urg Fr\"ohlich for fruitful collaboration and for all they have taught him in the course of joint work. %He is grateful to J\"urg Fr\"ohlich for hospitality at ETH Z{\"u}rich and acknowledges the support of the Oberwolfach Institute.
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Non-relativistic QED}\label{sec:nrqed}
\subsection{Schr\"odinger equation}\label{sec:model}
We consider a system consisting of $n$ charged particles
interacting between themselves and with external fields, which are coupled to quantized electromagnetic field. The %mathematical framework
starting point of the non-relativistic QED is the state Hilbert space $\cH=\cH_{p}\otimes\cH_{f}$,
which is the tensor product of the state spaces of the particles, $\cH_\at$, say, $\cH_\at=L^2(\R^{3n})$,
and of Bosonic Fock space $\cH_{f}$ of the quantized electromagnetic field, and the standard quantum Hamiltonian $\hsm\equiv H_{g\chi}$ on $\cH=\cH_{p}\otimes\cH_{f}$, given (in the units in which the Planck constant divided by $2\pi$ and the speed of light % and the electron mass
are equal to $1:\ \hbar=1$ and
$c=1$) %and $m_\el =1$) %(the ratio of the mass of the $j$-th particle to the mass of an electron) and
by %(after rescaling)
\begin{equation} \label{Hsm}
\hsm=\sum\limits_{j=1}^n{1\over 2m_j}
(i\nabla_{x_j}-g\Af(x_j))^2+V(x)+H_f
\end{equation}
%given in terms of
%Here $\cH_{p}=$ the particle state space of the ($\cH_p$, say, $\cH_\at=L^2(R^{3n})$)and $\quad \cH_{f}=$ the state space of the quantized electromagnetic field (Bosonic Fock space).
%It acts on the Hilbert space $\cH=\cH_{p}\otimes\cH_{f}$, which is
%the tensor product of the state spaces of the matter system
%$\cH_{p}$ and the quantized electromagnetic field $\cH_{f}$. Here,
%as was mentioned above, the superindex $SM$ stands for 'standard
%model' and, as explained below, $g$ is related to the particle
%charge, or, more precisely, to the fine-structure constant. As
%mentioned above, for the sake of simplicity, we do not include the interaction of the
%spin with magnetic field - a term proportional $\sum\limits_{j=1}^n
%\sigma_j \cdot B(x_j),\ B(y)=curl A(y)$.
%
%It is given by the state space $\cH=\cH_{p}\otimes\cH_{f}$, which is
%the tensor product of the state spaces of the matter system (say,
%consisting of $n$ particles) and the quantized electromagnetic field
%and the evolution on this space described by the time-dependent
%Schr\"odinger equation, $i\partial_t\psi= H^{SM}_g\psi$, where
%$H^{SM}_g$ is the standard quantum Hamiltonian
%
%The mathematical framework of the theory of non-relativistic matter
%interacting with the quantized electro-magnetic field
%(non-relativistic quantum electrodynamics) is well established. In
%this framework the state space is the tensor product,
%$\cH=\cH_{p}\otimes\cH_{f}$, of the state spaces of the matter
%system (say, consisting of $n$ particles) and the quantized
%electromagnetic field.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Standard Hamiltonian}\label{sec:ham}The standard quantum Hamiltonian for non-relativistic QED is an operator on $\cH=\cH_{p}\otimes\cH_{f}$, given (after rescaling) by
%Thus the Hilbert space of the total system is ${\mathcal H}:= {\mathcal H}_p \otimes
%{\mathcal F}$. %We consider a particle system (electrons in an external field of static nuclei) interacting with quantized electromagnetic field.
(see \cite{fermi} and \cite{PauliFierz}). Here, $m_j$ and $x_j$, $j=1, ..., n$, are the ('bare') particle masses and the particle positions,
$x=(x_1,\dots,x_n)$, $V(x)$ is the total potential affecting particles and $g>0$ is a coupling constant related to the particle charges, %(for the theory describing electron and static nuclei, $g:= \alpha^{3/2}$, where $\alpha =\frac{e^2}{4\pi \hbar c}\approx {1\over 137}$, the fine-structure constant),
%$A(x)$ is the electromagnetic vector potential and $H_f$ is the photon Hamiltonian. To have a self-adjoint and bounded from below quantum Hamiltonian we have to subject $A(x)$ to an ultraviolet (UV) cut-off.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Quantized electromagnetic field}\label{sec:qemf}
$\Af:=\check\chi *A$, where $A(y)$ is the \textit{quantized vector potential}, in the Coulomb gauge ($\textrm{div} A(y)=0$), describing the quantized electromagnetic field, %which, subject to
and $\chi$ is an \textit{ultraviolet cut-off}, %$\chi$, %and is given by
\begin{equation}\label{A}
\Af(y)=\int(e^{iky}a(k)+e^{-iky}a^*(k))\chi(k){d^3k\over \sqrt{|k|}}
\end{equation}
($a(k)$ and $a^*(k)$ are annihilation and creation operators
acting on the Fock space $\cH_{f}\equiv \cF$, see Supplement \ref{sec:crannihoprs} for the definitions), and $\hf$ is the quantum Hamiltonian of the quantized electromagnetic field, describing the dynamics of the latter, it is given by
%
\begin{equation} \label{Hf}
\hf \ = \ \int d^3 k \; \om(k) a^*(k) \cdot a(k),
\end{equation}
%
where $\om(k) \ = \ |k|$ is the dispersion law connecting the energy
of the field quantum with its wave vector $k$.
For simplicity we omitted the interaction of the spin with magnetic field.
%$$\sum\limits_{j=1}^n \frac{g}{2m_j}\sigma_j \cdot \textrm{curl} A(x_j).$$
(For a discussion of this Hamiltonian including units, the removal of the center-of-mass motion of the particle system and taking into account the spin of the particles, see Appendix \ref{sec:pot}. Note that our units are not dimensionless. We use this units since we want to keep track of the particle masses. To pass to the dimensionless units we would have to set $m_\el=1$ also.)
The Hamiltonian $H$ determines the dynamics via the time-dependent Schr\"odinger equation
$$i\partial_t\psi= H\psi,$$
where $\psi$ is a differentiable path in $\cH=\cH_{p}\otimes\cH_{f}$.
The ultraviolet cut-off, $\chi$, satisfies
$\chi(k)= 1 $ in a neighborhood of $k=0$ and is decaying at infinity on the scale $\kappa$ and sufficiently fast. %say $|\chi(k)| \lesssim \langle k\rangle^{-3} $. %(here we absorbed the relevant constant into $\chi$).
We assume that $V(x)$ is a generalized $n$-body potential, i.e. it satisfies the assumptions:
\begin{itemize}
\item [(V)] $V(x) = \sum_i W_i(\pi_i x)$, where $\pi_i$ are a linear maps from
$\mathbb{R}^{3n}$ to $\mathbb{R}^{m_i},\ m_i \le 3n $ and $ W_i$ are
Kato-Rellich potentials (i.e. $W_i(\pi_i x) \in
L^{p_i}(\mathbb{R}^{m_i}) + (L^\infty (\mathbb{R}^{3n}))_\eps$ with
$p_i=2$ for $m_i \le 3,\ p_i>2 $ for $m_i =4$ and $p_i \ge m_i/2$
for $m_i > 4$).
%general $\Delta-$compact (\textbf{Kato} ??) potentials $V(x)$.
\end{itemize}
%For a large class of potentials $V(x)$ and for an ultra-violet cut-off in $A(x)$,
Under the assumption (V), the operator $H$ is self-adjoint and bounded below.
%In \eqref{Hsm} the vector potential $A(x)$ is subjected to an ultraviolet (UV) %cut-off. For a given quantum model the UV cut-off is defined by an energy %scale on which this model is applicable. In our case, the relevant energy scale %is the characteristic energies of the particle motion (see a subsection below %for a detailed discussion). Thus,
We assume for simplicity that our matter consists of electrons and the nuclei and that the nuclei are infinitely heavy and therefore are manifested through the interactions only (put differently, the molecules are treated in the Born - Oppenheimer approximation). In this case, the coupling constant $g$ is related to the electron charge $-e$ as $g:= \alpha^{3/2}$, where $\alpha =\frac{e^2}{4\pi \hbar c}\approx {1\over 137}$, the fine-structure constant, and $m_j =m$.
It is shown (see Section \ref{sec:liter} and a review in \cite{bach}) %the contribution \cite{bach} of Volker Bach in this volume
for references and discussion) that the physical electron mass, $m_\el $, is not the same as the parameter $m\equiv m_j$ (the 'bare' electron mass) entering \eqref{Hsm}, but depends on $m$ and $\kappa$. Inverting this relation, we can think of $m$ as a function of $m_{\el}$ and $\kappa$. %If we fix $m_{\el}$ and $e$ %Thus the ultraviolet cut-off scale, $\kappa$, is determined by the electron's physical mass.
If we fix the particle potential $V(x)$ (e.g. taking it to be the total Coulomb potential), and $m_{\el}$ and $e$, then the Hamiltonian \eqref{Hsm} depends on one free parameter, the bare electron mass $m$ (or the ultraviolet cut-off scale, $\kappa$). %electron charge $-e$, which plays role of the coupling constant, and the electron's physical mass. %, which, as was mentioned below, is related to the %particle (electron) charges
%the ultraviolet cut-off, $\chi$, not displayed here. %As was mentioned above, $\vep$ is related to the particle (electron) charges and the ultraviolet cut-off, to the particle (electron) renormalized mass (see \cite{HS, HiroshimaSpohn2, LiebLoss2, LiebLoss3, BCFS2, Chen}, and \cite{Spohn} for a recent review).
%
\DETAILS{If we fix the particle potential $V(x)$, %(e.g. taking it to be the total Coulomb potential),
then the Hamiltonian $\hsm\equiv H_{g\chi}$ depends on \textit{two free parameters}:
\begin{itemize}
\item The coupling constant $g$ (related to the electron charges);
\item The ultraviolet cut-off $\kappa$ (related to the electron renormalized mass. %Appendix \ref{sec:mass-renorm}).
\end{itemize}}
%
%the present paper omits the interaction of the spin with
%magnetic field %- $\sum\limits_{j=1}^n \frac{g}{2m_j}\sigma_j \cdot B(x_j)$ -
%in the Hamiltonian.
%One can think about the particles as electrons in an external field of static nuclei.
%We assume some smoothness conditions on $G_j(k)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The state spaces of the quantized electro-magnetic field, denoted by
%$\cF$ is the Fock space of a massless quantum field (photons) (see
%Appendix).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%We are interested in the dynamics of a particle
%system coupled to a massless quantum field (photons or phonons). The
%former is described by a Schr\"odinger operator, $H_\at$, acting on
%an $L^2$ Hilbert space, $\cH_\at$, say, $\cH_\at=L^2(R^{3n})$. The
%latter is determined by It is described in terms of creation- and
%annihilation operators by the quantum Hamiltonian $\hf$ is given by
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%**Charged particles coupled to the quantized electromagnetic
%field can be satisfactorily described within the
%standard model in which the matter is treated non-relativistically
%while the radiation is quantized.
%The quantum Hamiltonian of the standard model of the
%non-relativistic QED which is given by the operator In the
%expression above $\hf$ is the Hamiltonian of the quantized
%electromagnetic field (the photon Hamiltonian) defined above
%\begin{equation}\label{eq2}
%H_f = \int|k| a^*(k)a(k){d^3k}
%\end{equation}
%\begin{equation}\label{eq4}
%\chi(k)=1{\rm\quad for\quad} |k|\leq \kappa, {\rm\quad and \quad}
%=0{\rm\quad for\quad} |k|\geq \kappa+1,
%\end{equation}
%for some positive $\kappa$.
%
%Furthermore, $\cH_{\prt}$ and
%$\cH_{\rad}$ are the Hilbert spaces of the particle system and of
%the quantized electro-magnetic field respectively, say
%$\cH_\prt=L^2(R^{3n}$) (see \cite{BFS3} for a careful
%description).
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Stability and radiation}\label{sec:stabrad}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Particle system}\label{sec:partsyst}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The matter system considered consists of $n$ charged particles interacting between themselves and with external fields. %and with a quantized electromagnetic field. %The particles have masses $m_j$
%%(the ratio of the mass of the $j$-th particle to the mass of an electron)
%and positions $x_j$, where $j=1, ..., n$. We write
%$x=(x_1,\dots,x_n)$. The total potential of the particle system is
%denoted by $V(x)$.
We begin with considering the matter system alone. As was mentioned above, its state space, $\cH_{p}$, is either $L^2(\mathbb{R}^{3n})$ or a subspace of
this space determined by a symmetry group of the particle system, and its Hamiltonian operator, $H_p$, acting on %a Hilbert space of the particle system,
$\cH_{p}$, is given by
\begin{equation} \label{Hp}
H_p:=\sum\limits_{j=1}^n {-1\over 2m_j} \Delta_{x_j}+V(x),
\end{equation}
where $\Delta_{x_j}$ is the Laplacian in the variable $x_j$ and, recall, $V(x)$ is the total potential of the particle system. Under the conditions on the potentials $V(x)$, given above, the operator $H_p$ is self-adjoint and bounded below. Typically, according to the HVZ theorem, its spectrum consists of isolated eigenvalues, $\epsilon^{(p)}_0 < \epsilon^{(p)}_1 < ... <\Sigma^{(p)}$, and continuum $[\Sigma^{(p)}, \infty)$, starting at the ionization threshold $\Sigma^{(p)}$, as shown in the figure below. %We assume that $V(x)$ is real and s.t. the operator $H_p$ is self-adjoint on the domain of $\sum\limits_{j=1}^n {1\over 2m_j}\Delta_{x_j}$.
$$ \textbf{Hidden Picture} $$ \DETAILS{
\bigskip
\bigskip
%----------------------- 8a -------------------
%\begin{center}
%
\includegraphics[height=4cm]{lh2.pdf}
%$$PICTURE$$
%\end{center}
\bigskip
\bigskip }
%Let $\epsilon^{(p)}_0 < \epsilon^{(p)}_1 < ...$ be the isolated eigenvalues of the particle Hamiltonian $H_p$.
%For a large class of potentials $V(x)$, including Coulomb potentials, %and under an ultra-violet cut-off,
%the operator $H$ is self-adjoint and bounded below. %(Bach-Fr\"ohlich-Sigal, Hiroshima).
The eigenfunctions corresponding to the isolated eigenvalues are exponentially localized. Thus left on its own the particle system, either in its ground state or in one of the excited states, is stable and well localized in space. We expect that this picture changes dramatically when the total system (the universe) also includes the electromagnetic field, which at this level must be considered to be quantum. As was already indicated above what we expect is the following
\begin{itemize}
\item The stability of the system under consideration is
equivalent to the statement of existence of the ground state of
$H$, i.e. an eigenfunction with the smallest possible energy.
\bigskip
\item The physical phenomenon of radiation is expressed mathematically as
emergence of resonances out of
%formalized mathematically by the fact that the
excited states of a particle system
%turn into the (quantum) resonances when the system is coupled
due to coupling of this system to the quantum electro-magnetic
field.
\end{itemize}
Our goal is to develop the spectral theory of the Hamiltonian $H$ and relate to the properties of the relevant evolution. %to understanding the evolution prove existence of the ground states and resonances in the non-relativistic QED.
Namely, %for a quantum-mechanical system of particles coupled to quantized electromagnetic field
we would like to show that
\bigskip
1) The \textit{ground state} of the particle system is \textit{stable} when the coupling is turned on, while
\bigskip
2) The excited states, generically, are not. They turn into \textit{resonances}.
%The resonances are responsible for processes of emission and absorption of the electro-magnetic radiation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Ultra-violet cut-off} \label{subsec:UV}
We reintroduce the Planck constant, $\hbar$, speed of light, $c$, and electron mass, $m_\el$, for a moment. Assuming the ultra-violet cut-off $\chi(k)$
decays on the scale $\kappa$, in order to correctly describe the
phenomena of interest, such as emission and absorption of
electromagnetic radiation, i.e. for optical and rf modes, we have
to assume that the cut-off energy, $$\hbar c \kappa\ \gg\ \alpha^2 m_\el c^2,\ \mbox{ionization energy, characteristic energy of the particle motion}.$$
%is much greater
%than the characteristic energies of the particle motion. %(We
%reintroduced the Planck constant, $\hbar$, and speed of light, $c$,
%for a moment.)
%The latter motion takes place on the energy scale of
%the order of the ionization energy, i.e. of the order
%$\alpha^2 mc^2$ $\rightarrow$ assume $$\alpha^2 \ll \kappa\ (\alpha^2 mc^2 \ll \hbar c \kappa).$$
%or $\alpha^2 \ll \kappa$ in our units.
On the other hand, we should exclude the energies where the relativistic effects, such as
electron-positron pair creation, vacuum polarization and
relativistic recoil, take place, and therefore we assume
$$ \hbar c \kappa \ll m_\el c^2,\ \mbox{the rest energy of the
the electron}.$$
%for energies higher than the rest energy of the the electron ($mc^2$)
%$\rightarrow$ assume
%$ \kappa \ll 1\,\,\ (\hbar c \kappa \ll mc^2)$.
Combining the last two conditions we arrive
at $\alpha^2 m_\el c/\hbar \ll \kappa \ll m_\el c/\hbar$, or in our units, %the restriction $\alpha^2 mc^2 \ll \hbar c \kappa \ll mc^2$ or %$\alpha^2 mc/\hbar \ll \kappa \ll mc/\hbar.$
$$\alpha^2m_\el \ll \kappa \ll m_\el \,\,\ .$$ The Hamiltonian \eqref{Hsm} is obtained by the
rescaling $x \rightarrow \alpha^{-1} x$ and $k \rightarrow \alpha^2
k$ of the original QED Hamiltonian (see Appendix \ref{sec:pot}). After this rescaling, the new cut-off momentum scale, $\kappa'= \alpha^{-2} \kappa$, satisfies
$$m_\el \ll \kappa' \ll \alpha^{-2}m_\el,$$
which is easily accommodated by our estimates (e.g. we can have $\kappa'
=O(\alpha^{-1/3}m_\el)).$ %Thus we can assume for simplicity that $\chi$
%is fixed.
%The relevant energy scale in our case is the characteristic energies of the particle motion.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Resonances}
As was mentioned above, the mathematical language which describes the physical phenomenon of radiation is that of \textit{quantum resonances}. We expect that the latter emerge out of
%formalized mathematically by the fact that the
excited states of a particle system
%turn into the (quantum) resonances when the system is coupled
due to coupling of this system to the quantum electro-magnetic
field.
Quantum resonances manifest themselves in three different ways:
1) Eigenvalues of complexly deformed Hamiltonian;
2) Poles of the meromorphic continuation of the resolvent across the continuous spectrum;
3) Metastable states.
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Complex deformation}
To define resonances we use complex deformation method. In order to be able to apply this method %tackle the resonances
we choose the ultraviolet cut-off,
$\chi(k)$, so that
\begin{itemize}
\item [] The function $\theta \rightarrow \chi(e^{-\theta} k)$ has an
analytic continuation from the real axis, $\mathbb{R}$, to the strip
$\{\theta \in \mathbb{C} | |\rIm\ \theta | < \pi/4 \}$ as a $L^2
\bigcap L^\infty (\mathbb{R}^3)$ function, \end{itemize} e.g.
$\chi(k)= e^{-|k|^2/\kappa^2}$. For the same purpose, we assume that the potential, $V(x)$,
satisfies the condition:
%(\textbf{we need a stronger condition}!)
\begin{itemize}
\item [(DA)] The
%ultraviolet cut-off $\chi(k)$ and
the particle potential $V(x)$ is dilation analytic in the sense that
the operator-function
%$\theta \rightarrow \chi(e^{-\theta} k)$ and
$\theta \rightarrow V(e^{\theta} x)$ $(-\Delta +1)^{-1}$ has an
analytic continuation from the real axis, $\mathbb{R}$, to the strip
$\{\theta \in \mathbb{C} | |\rIm\ \theta | < \theta_0 \}$ for some
$\theta_0 > 0$.
\end{itemize}
%(Note that our approach can also handle the perturbations quadratic
%in creation and annihilation operators, $a$ and $a^*$.)
%We define the resonances of $H^{SM}_g$ as complex eigenvalues of the
%complex deformation, $H^{SM}_{g, \theta},\ \rIm \theta > 0,$ of
%$H^{SM}_g$ defined in Section \ref{sec-III}. We discuss their
%definition and physical manifestations below.
To define the resonances for the Hamiltonian $H$
%as complex eigenvalues of the complex deformation, $H^{SM}_{g, \theta},\ \rIm \theta > 0,$ of $H^{SM}_g$,
we pass to the one-parameter (deformation) family
\begin{equation}\label{Htheta} %{I.5}
H_{ \theta} := U_{\theta} H U_{\theta}^{-1},
\end{equation}
where $\theta$ is a real parameter and $U_{\theta},$ on
the total Hilbert space ${\mathcal H}:= {\mathcal H}_p \otimes
{\mathcal F}$, is the one-parameter group of unitary operators, whose action is rescaling particle positions and of
photon momenta:
$$x_j\rightarrow e^\theta x_j\ \mbox{and}\ k\rightarrow e^{-\theta} k.$$
%$\theta \in \mathbb{R}$.
One can show show that:
1) Under a certain analyticity condition on coupling functions, the family $H_{\theta}$ has an analytic
continuation in $\theta$ to the disc $D(0, \theta_0)$, as a type A
family in the sense of Kato;
%A standard argument shows that
2) The real eigenvalues of $H_{ \theta},\ \rIm \theta
>0,$ coincide with eigenvalues of $H$ and that complex
eigenvalues of $H_{ \theta},\ \rIm \theta >0,$ lie in the
complex half-plane $\mathbb{C}^-$;
3) The complex
eigenvalues of $H_{ \theta},\ \rIm \theta
>0,$ are locally independent of $\theta$.
The typical spectrum of $[H_{ \theta}\equiv H^{SM}_{ \theta}]|_{g=0},\ \rIm \theta
>0$ (here the superindex SM stands for the standard model) is shown in the figure below.
$$ \textbf{Hidden Picture}$$ \DETAILS{
%\end{center}
\bigskip
\bigskip
\bigskip
%----------------------- 8a -------------------
%\begin{center}
%\center{
\includegraphics[height=4cm]{picture-b.pdf}
%\end{center}
\bigskip
\bigskip }
We call complex eigenvalues of $H_{ \theta},\ \rIm \theta
>0$ the \textit{resonances} of $H$.
%Clearly under a small perturbation an embedded eigenvalue turns
%generally into a resonance.
%When a quantum-mechanical system interacts with the quantized
%electro-magnetic field, or, more generally, with an environment, one
%expects that all eigenvalues of this system, save for the smallest
%one, disappear and re-emerge as resonances.
%\textbf{Exercise.*} Find
As an example of the above procedure we consider the complex deformation of the hydrogen atom and photon Hamiltonians $H_{hydr}:=-\frac{1}{2m}\Delta-\frac{\alpha}{|x|}$ and $H_f$: $$H_{hydr \theta}=e^{-2\theta}(-\frac{1}{2m}\Delta)-e^{-\theta}\frac{\alpha}{|x|},\ H_{f \theta}=e^{-\theta}H_f.$$ Let $e^{hydr}_j$ be the eigenvalues of the hydrogen atom. Then the spectra of these deformations are
$$\s(H_{hydr\theta})=\{e^{hydr}_j\}\cup e^{-2\Im\theta}[0,\infty),\ \s(H_{f\theta})=\{0\}\cup e^{-\Im\theta}[0,\infty).$$
%where $e^{hydr}_j$ are the eigenvalues of the hydrogen atom.
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Resonances as poles}
%the resonances share two 'physical' manifestations of eigenvalues, as poles of the resolvent and frequencies of time-periodic and spatially localized solutions of the time-dependent Schr\"odinger equation, but with a caveat. To explain the first property, we use the Combes argument which goes as follows.
%The relation between the analytic continuation of
%Consider the matrix elements $F(z, \Psi, \Phi)$ $:=\langle \Psi,(H^{SM}_g-z)^{-1}\Phi\rangle$.
%To explain how the resonances arise as poles of the resolvent we
Similarly to eigenvalues, we would like to characterize the resonances in terms of poles of matrix elements of the resolvent $(H -z)^{-1}$ of the Hamiltonian $H$. To this end we have to go beyond the spectral analysis of $H$. Let $\Psi_{\theta}=U_{\theta}\Psi$, etc., for $\theta\in
\mathbb{R}$ and $z\in \mathbb{C}^+$. Use the unitarity of
$U_{\theta}$ %e^{-igF}$ %$X_{\theta}:= U_{\theta}e^{-igF}$
for real $\theta$, to obtain (the Combes argument)
\begin{equation} \label{meromcont}
\langle\Psi, \ (H -z)^{-1}\Phi\rangle = \langle
\Psi_{\bar\theta} , ( H_{ \theta} -z)^{-1}\Phi_{\theta}
\rangle.
\end{equation}
Assume now that for a dense set of $\Psi$'s and $\Phi$'s (say, $\cD$, defined below), $\Psi_{\theta}$ and $\Phi_{\theta}$ have analytic continuations into a complex
neighbourhood of $\theta=0$ and continue the r.h.s of \eqref{meromcont} analytically first in $\theta$ into the upper half-plane and then in $z$ across the continuous spectrum. %Then the r.h.s. of \eqref{I.7} has an analytic continuation in $\theta$ into a complex neighbourhood of $\theta=0$. Since \eqref{I.7} holds for real $\theta$, it also holds in the above neighbourhood. Fix $\theta$ on the r.h.s. of \eqref{I.7}, with ${\rm Im}\theta>0$. The r.h.s.~of \eqref{I.7} can be analytically extended across the real axis into the part of the resolvent set of $H_{g \theta}$ which lies in $\overline{\mathbb{C}^-}$ and which is connected to $\mathbb{C}^+$. This yields an analytic continuation of the l.h.s. of \eqref{I.7}.
This meromorphic continuation has the following properties:
\begin{itemize}
\item The real eigenvalues of $H_{ \theta}$ give real poles of the
r.h.s. of \eqref{meromcont} and therefore they are the eigenvalues of
$H$.
\item The complex eigenvalues of $H_{\theta}$ %, which are at the resonances of $H_{g}$, yield complex poles of the r.h.s. of \eqref{I.7}. They
are poles of the meromorphic continuation of the l.h.s. of \eqref{meromcont} across the
spectrum of $H$ onto the second Riemann sheet.
\end{itemize}
The poles manifest themselves physically as bumps in the scattering cross-section or poles in the scattering matrix. %There are some subtleties involved which we explain below.
%There are some subtleties involved which we explain below.
%Consider the analytic continuation of $\langle \psi, (H_{g} -z)^{-1}
%\phi \rangle$, for $\psi,\phi\in\cD$, from $\mathbb{C}_+$ (the upper
%half-plane) across the real axis into the lower half-plane, provided
%such a continuation exists. We define the {\it resonance
%eigenvalues} (or simply, {\it resonances}) as positions of poles on
%the second sheet, i.e., in the lower half-plane $\mathbb{C}_-$, of
%the analytic continuation described above.
%
%
%\textbf{Exercise.*}
\DETAILS{A simple example illustrating the use of complex deformation in order to continue analytically matrix elements of resolvents is an analytic continuation of the integral $\int_0^\infty \frac{f(\omega)}{\om-z}d\om$. Namely, we would like to continue analytically this integral across the semi-axis $(0, \infty)$ from the upper semi-plane $\C^+$ to the second Riemann sheet. %Formulate conditions on $f(\om)$ so that such a continuation is possible. (Answer to the first part:
Assuming that $f(e^{-\theta}\omega)$ is analytic in a neighbourhood of $\theta=0$, we can write such a continuation as $\int_0^\infty \frac{f(e^{-\theta}\omega)}{\om-e^{\theta}z}d\om,\ \Im\theta>0,\ \Im z<0.$}
%
%
%\noindent\textbf{Remark.}
The r.h.s. of \eqref{meromcont} has an analytic continuation into a complex neighbourhood of $\theta=0$, if $\Psi, \Phi\in \cD$, where
\begin{equation} \label{D}
\cD :=\bigcup_{n >0,a >0} \Ran \big(\chi_{N \le n}\chi_{|T| \le
a}\big). \end{equation}
Here $N = \int d^3 k a^*(k) a(k)$ be the
photon number operator and $T$ be the self-adjoint
generator of the one-parameter group $U_{\theta},\ \theta \in
\mathbb{R}$. (It is dense, since $N$ and $T$ commute.)
%Consider the analytic continuation of $\langle \psi, (H_{g} -z)^{-1}
%\phi \rangle$, for $\psi,\phi\in\cD$, from $\mathbb{C}_+$ (the upper
%half-plane) across the real axis into the lower half-plane, provided
%such a continuation exists. We define the {\it resonance
%eigenvalues} (or simply, {\it resonances}) as positions of poles on
%the second sheet, i.e., in the lower half-plane $\mathbb{C}_-$, of
%the analytic continuation described above.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Resonance states as metastable states}\label{sec:meta}
%The second manifestation of resonances alluded to above is as metastable states (metastable attractors of system's dynamics). Namely, one expects that the ground state is asymptotically stable and the resonance states are (asymptotically) metastable, i.e. attractive for very long time intervals.
%In other words one would like to investigate the
%effect of the ground state and resonances on the
%long-time behaviour of solutions of the time-dependent Schr\"odinger
%equation, $i\partial_t\psi= H^{SM}_g\psi$, for initial conditions
%localized in energy intervals around ground state and resonance energies.
While bound states are stationary solutions, one expects that resonances to lead to almost stationary, long-living solutions. Let $z_*,\ \rIm z_* \le0,$ be the ground state or
resonance eigenvalue. One expects that for an initial condition,
$\psi_0$, localized in a small energy interval around the ground state
or resonance energy, $\rRe z_*$, the solution, $\psi=e^{-i H t}\psi_0$, of the
time-dependent Schr\"odinger equation, $i\partial_t\psi=
H\psi$, is of the form
%above theorem has the following consequence. (In what follows
%functions of self-adjoint operators are defined by functional calculus.)\\
%\begin{corollary}
%For $\Delta$ as above and for any function $f(\lambda)$ with $\supp
%g\subseteq\Delta$ and for $\nu< \theta-\frac{1}{2}$, we have
\begin{equation} \label{ResonDecay}
\psi = e^{-i z_* t}\phi_* +
O_{\textrm{loc}}(t^{-\alpha})+ O_{\textrm{res}}(g^{\beta}),
%O(g ^{-\epsilon}) ,
\end{equation}
for some $\alpha,\ \beta > 0$ (depending on $\psi_0$),
%and for $t \ll 1/|\rIm z_*|$. %$z$ is either the ground state or resonance eigenvalue and
where
\begin{itemize}
\item $\phi_*$ is either the ground state or an excited state of the unperturbed system, depending on whether $z_*$ is the ground
state energy or a resonance eigenvalue;
\item The error term $O_{\textrm{loc}}(t^{-\alpha})$ satisfies $\|(\one+|T|)^{-\nu} O_{\textrm{loc}}(t^{-\alpha})\| \le C t^{-\alpha}$, where $T$ is the generator of the group $U_\theta$, with an appropriate $\nu>0$.
%\item The error term $O_{\textrm{res}}(g^{\beta})$ is absent in the ground state case. %The reason for the latter is that, unlike bound states, there is no 'canonical' notion of the resonance state.
\end{itemize}
%$$****$$ The dynamical picture of the resonance described above implies that the imaginary part of the resonance eigenvalue, called the resonance width, can be interpreted as the decay rate probability, and its reciprocal, as the life-time, of the resonance.
For the \textit{ground state}, \eqref{ResonDecay}, without the error term $O_{\textrm{res}}(g^{\beta})$, is called the local decay property (see Section \ref{sec:relat-res}). %{sec:loc-dec})
One way to prove it is to
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Relation between poles and time asymptotics}\label{sec:relat} \begin{itemize}
%\item
%To prove the equation \eqref{ResonDecay}, i.e. to determine the asymptotic behaviour of solutions $e^{-i H t}\psi_0$, we
use the formula connecting the propagator and the resolvent:
\begin{align}\label{prop-intrepr}
& e^{-i H t}f(H)= \frac{1}{\pi} \int_{-\infty}^\infty d\lambda
f(\lambda)e^{-i\lambda t}\rIm(H-\lambda-i0)^{-1}.
\end{align}
%\item
%For the \textit{ground state} %the absolute continuity of the spectrum outside the ground state energy, or a stronger property of
Then one controls the boundary values of the resolvent on the spectrum (the corresponding result is called the limiting absorption principle, see Appendix \ref{sec:locdectransf}) and uses properties of the Fourier transform. %, suffices to establish the result above (again without the error term $O_{\textrm{res}}(g^{\beta})$).
%\item
\medskip
For the\textit{ resonances}, \eqref{ResonDecay} implies that $-\rIm z_*$ has the meaning of the decay probability per unit time, and $(-\rIm z_*)^{-1}$, as the life-time of the resonance.
To prove it,
%For the\textit{ resonances}, %observes that the meromorphic continuation of matrix elements of the resolvent (on an appropriate dense set of vectors) to the second Riemann sheet has poles at resonances
one uses \eqref{meromcont} and the analyticity of its r.h.s. in $z$ and performs in \eqref{prop-intrepr} %the integral on the r.h.s.
a suitable deformation of the contour of integration to the second Riemann sheet to pick up the contribution of poles there. This works %in the usual quantum mechanical situation since there
when the resonances are isolated. In the present case, they are not. This is a consequence
of the infrared problem.
%\end{itemize}
Hence, determining the long-time behaviour of $e^{-i H t}\psi_0$ is a subtle problem in this case.
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Comparison with Quantum Mechanics}\label{sec:qm}
This situation is quite different from the one in Quantum Mechanics
(e.g. Stark effect or tunneling decay) where the resonances are
isolated eigenvalues of complexly deformed Hamiltonians. This makes
the proof of their existence and establishing their properties, e.g.
independence of $\theta$ (and, in fact, of the transformation group
$U_{\theta}$), relatively easy. In the non-relativistic QED (and
other massless theories),
%where the eigenvalues of $H^{SM}_{g \theta}$ are not isolated,
giving meaning of the resonance poles and proving independence of
their location of $\theta$ is a rather involved matter. %We discuss it briefly below. The point above can be illustrated on the proof of the statement \eqref{ResonDecay}.
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Infrared problem}\label{sec:IRprob}
The resonances arise from the eigenvalues of the non-interacting
Hamiltonian $H_{g=0}$. The latter is of the form
\begin{equation}\label{H0} %{H0EM} %{eq:2.1}
H_{0} = H_\part \otimes \bfone_f + \bfone_\part \otimes H_f\, .
\end{equation}
%To find the spectrum of $H^{SM}_0$ one
%observes that $\hf$ defines a positive operator, with purely absolutely continuous spectrum, except for a %simple eigenvalue $0$ corresponding to the vacuum eigenvector $\Om$. Thus, For $g=0$
The low energy spectrum of the
operator $H_0$ consists of branches $[\epsilon^{(p)}_i,
\infty)$ of absolutely continuous spectrum and of the eigenvalues
$\epsilon^{(p)}_i$'s, sitting at the continuous spectrum
'thresholds' $\epsilon^{(p)}_i$'s.
%Thus the points $\epsilon^{(p)}_i$'s are 'thresholds' of the
%continuous spectrum and the eigenvalues of the operator $H^{SM}_0$.
Here, recall, $\epsilon^{(p)}_0 < \epsilon^{(p)}_1 < ... <\Sigma^{(p)}$ are the isolated eigenvalues of the particle Hamiltonian $H_p$.
Let $\phi^{(p)}_i$ be the eigenfunctions of the particle system, while $\Om$ be the photon vacuum. The eigenvalues $\epsilon^{(p)}_i$'s correspond to the eigenfunctions $\phi^{(p)}_i\otimes \Om$ of $H_0$. The branches $[\epsilon^{(p)}_i, \infty)$ of absolutely continuous spectrum are associated with generalized eigenfunctions of the form $\phi^{(p)}_i\otimes g_\lam$, where $g_\lam$ are the generalized eigenfunctions of $H_f:\ H_fg_\lam=\lam g_\lam,\ 0<\lam<\infty$.
The absence of gaps between the eigenvalues and thresholds is a
consequence of the fact that the photons are massless. %This leads
%to hard problems in perturbation theory, known collectively as the
%\textit{infrared problem}.
To address this problem we use the spectral renormalization group (RG). %(Bach-Fr\"ohlich-Sigal, Sigal). %In the terminology of the RG approach
The problem here is that the leading part of the perturbation in $H$ is marginal. %This leads to the presence of the
%second zero eigenvalue in the spectrum of the linearized RG flow.
%which indicates that either the fixed point manifold is embedded in
%a larger central manifold or part of the stable manifold .
%(note that there is no spectral gap in the linearized RG flow).
% (Bach-Fr\"ohlich-Sigal remove it by additional assumptions.)
\DETAILS{either assuming the non-physical infrared behaviour of the
vector potential by replacing $|k|^{-1/2}$ in the vector potential
\eqref{A} by $|k|^{-1/2 +\varepsilon}$, with $\varepsilon
>0$, or by assuming presence of a strong confining external potential
so that $V(x) \ge c|x|^2$ for $x$ large.}
\bigskip
%\begin{center}
%\includegraphics[height=4.5cm]{picture-b.pdf}
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Results}\label{sec:results}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Existence of the Ground and Resonance States}\label{sec:existgrres}
\subsection{Bifurcation of eigenvalues and resonances}\label{subsec:existgrres}
%In this paper we prove the existence of the ground states and
%resonances for the Hamiltonians $H^{SM}_g$ and $H_g^{N}$.
%, labeled with their multiplicities.
%Fix an energy $ \nu \in (\epsilon^{(p)}_0 , \inf\sigma_{ess} (H_p))$ and %below the ionization threshold $\inf\sigma_{ess} (H_p)$
Stated informally what we show is
\begin{itemize}
\item The ground state of $H |_{g=0}$ $\Rightarrow$ the ground state $H$ ($\epsilon_{0} = \epsilon^{(p)}_{0} +O(g^2)$ and $\epsilon_{0} < \epsilon^{(p)}_{0}$); %, originating from
%\item %$\forall j$, $\epsilon_{j,k} = \epsilon^{(p)}_{j} +O(g^2)$ and the total multiplicity of $\epsilon_{j, k}\ \forall k$ equals the multiplicity of $\epsilon^{(p)}_{j}$; %($\epsilon_{0}=\epsilon_{0,0}$);
\item The excited states of $H |_{g=0}$ $\Rightarrow$ (generically) the resonances of $H$ %Eigenvalues, $\epsilon^{(p)}_{j} < \nu$, of $H |_{g=0}$ turn into %, which
%is less than $\nu$,
%resonance and/or bound state eigenvalues, $\epsilon_{j, k}$, of $H$,
($\epsilon_{j,k} = \epsilon^{(p)}_{j} +O(g^2)$);
%The ground state of $H^{SM}_g$ for $g=0$ turns into the ground state of
%$H^{SM}_g$ and the excited states below the energy level $\nu$ turn
%into resonance and/or bound states;
%(iv) The eigenvalues/resonances, $\epsilon_{j,k}$, of $H^{SM}_g$ are
%related to the unperturbed eigenvalues of $H^{SM}_0$ as
%(here we ignore the
%multiplicities);
\item There is $ \Sigma>\inf\s(H)$ (the ionization threshold, $\Sigma+\Sigma^{(p)}+O(g^2)$) s.t. for energies $<\Sigma$ that particles are exponentially localized around the common center of mass.
\end{itemize}
For energies $>\Sigma$ the system either sheds off locally the excess of energy and descends into a localized state or breaks apart with some of the particles flying off to infinity.
To formulate this result more precisely, denote
$$\epsilon^{(p)}_{gap}(\nu):=\min \{|\epsilon^{(p)}_i
-\epsilon^{(p)}_j |\ |\ i\neq j,\ \epsilon^{(p)}_i, \epsilon^{(p)}_j
\le \nu \}.$$ %and $j(\nu):= \max\{j: \epsilon^{(p)}_{j} \le \nu\}$.
%, where $z^{(p)}_i$ are eigenvalues and resonances of the
%particle Hamiltonian $H_p$ ??.
%
%, where $e^{(p)}_0$ and $e^{(p)}_1$ are the ground state and the first
%excited state energies of the particle system.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf{Theorem 4.1}} \textit{(Fate of particle bound states}).
%\begin{theorem} %\label{thm-main} %Assume Condition (V).
Fix $e^{(p)}_0 < \nu < \inf \sigma_{ess} (H_p)$ and let
%$H$ be either $H^{SM}_g$ or $H_g^{N}$, the two Hamiltonians defined above,
%and let
$g\ll \epsilon^{(p)}_{gap}(\nu)$. %\sqrt{\epsilon^{(p)}_{gap}(\nu)\tan (\theta_0/2)}\ )$.
Then for $ g \ne 0,$
\begin{itemize}
%(i) $H^{SM}_g$ has a ground state, whose eigenvalue we denote
%$\epsilon_{0}$; %eigenvalue of comes out of $\epsilon^{(p)}_{0}$;
\item $H$ has a ground state, originating from a ground state
of $H |_{g=0}$ ($\epsilon_{0} = \epsilon^{(p)}_{0} +O(g^2),\ \epsilon_{0} < \epsilon^{(p)}_{0}$);
\item Generically, $H$ has no other bound state (besides the ground state);
\item Eigenvalues, $\epsilon^{(p)}_{j} < \nu,\ j\ne 0$, of $H |_{g=0}\ \Longrightarrow$ %, which
%is less than $\nu$,
%turns into
resonance eigenvalues, $\epsilon_{j, k}$, of $H$;
\item $\epsilon_{j,k} = \epsilon^{(p)}_{j} +O(g^2)$ and the total multiplicity of $\epsilon_{j, k}$ equals the multiplicity of
$\epsilon^{(p)}_{j}$; %($\epsilon_{0}=\epsilon_{0,0}$);
%The ground state of $H^{SM}_g$ for $g=0$ turns into the ground state of
%$H^{SM}_g$ and the excited states below the energy level $\nu$ turn
%into resonance and/or bound states;
%(iv) The eigenvalues/resonances, $\epsilon_{j,k}$, of $H^{SM}_g$ are
%related to the unperturbed eigenvalues of $H^{SM}_0$ as
%(here we ignore the
%multiplicities);
%\item $\epsilon_{j, k}$'s are independent of $\theta$. %provided $\rIm
%\theta \ge \theta_0/2$.
\item The ground and resonance states are exponentially localized in the physical space: $$\|e^{\delta|x|}\psi\|<\infty,\ \forall \psi\in\Ran E_\Delta(H),\ \delta<\Sigma^{(p)}-\sup\Delta.$$
%$\epsilon_{j, k}$'s are independent of $\theta$. %provided $\rIm \theta \ge \theta_0/2$.
\end{itemize}
%(iii) The total multiplicity of the eigenvalues and resonances
%emerging from a given eigenvalue of $H_p$ is equal to the
%multiplicity of this eigenvalue;
%(iv) The ground and resonance states and the corresponding
%eigenvalues can be computed explicitly in terms of fast convergent
%expressions in the coupling constant $g$.
%
%
$$ \textbf{Hidden Picture}$$ \DETAILS{
\bigskip
\bigskip
\includegraphics[height=5cm]{lh3.pdf}
\bigskip
\bigskip }
%The statements concerning the excited states are proven under
%additional Condition (DA). \end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%In order not to complicate the formulation of the theorem we assumed
%tacitely that the ground state eigenvalue $\epsilon^{(p)}_{0}$ of
%$H^{SM}_{g=0}$ is simple. If not then it turns into ground state and
%resonance eigenvalues of $H^{SM}_g$ of the total multiplicity equal
%to the multiplicity of $\epsilon^{(p)}_{0}$ , with at least one of
%them being the ground state eigenvalue of $H^{SM}_g$.
%
%With our labeling the eigenvalues counting their multiplicities
%result (i) of Theorem \ref{thm-main} implies that the total
%multiplicity of the eigenvalues and resonances emerging from a given
%eigenvalue of $H_p$ is equal to the multiplicity of this eigenvalue;
%
%Let $\epsilon_{j}$ be the eigenvalues and resonances of the
%Hamiltonian $H^{SM}_g$ established in the above theorem (in
%In what follows we omit the subindex $k$ in $\epsilon_{j,k}$ and
%write $\epsilon_{j}$.
\noindent\textbf{Remark.} The relation $\epsilon_{0} < \epsilon^{(p)}_{0}$ is due to the fact that the electron surrounded by clouds of photons become heavier.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Meromorphic continuation across spectrum}\label{subsec:meromcont}
%By statement (ii), we have
%Information about meromorphic continuation of the matrix elements of the resolvent and position of the resonances is given in the next theorem.
%which uses the notion of dilation entire vectors defined in the Section \ref{sec-III}.
%\begin{theorem} %\label{thm-main2}
\textbf{Theorem 4.2}. (\textit{Meromorphic continuation of the matrix elements of
the resolvent}) Assume $g\ll \epsilon^{(p)}_{gap}(\nu)$ and let $\epsilon_{0}:=\inf \sigma (H)$ be the ground state energy of $H$.
%and Conditions (V) and (DA).
Then
\begin{itemize}
\item For a dense set (defined in \eqref{D} below) %of dilation entire
of vectors $\Psi$ and $\Phi$, the matrix elements
$$F(z, \Psi, \Phi):=\langle \Psi, (H-z)^{-1}\Phi\rangle$$
have meromorphic continuations from $\mathbb{C}^+$
across the interval $(\epsilon_{0}, \nu)\subset \s_{ess}(H)$ %of the essential spectrum of $H_g$
into %the domain
$$\{z \in \mathbb{C}^- |\ \epsilon_{0}
< \rRe z < \nu\}/\bigcup_{0 \le j \le j(\nu)}S_{j, k},$$
%lower complex semi-plane, $\mathbb{C}^-$,
%with the wedges $S_{j, k},\ 0 \le j \le j(\nu)$, deleted;
where $S_{j, k}$ are the wedges starting at the resonances %The bifurcation result implies that $\epsilon_{j} \rightarrow
%\epsilon^{(p)}_{j}$ as $g \rightarrow 0$.
\begin{equation} \label{Sj}
S_{j, k}:=\{ z \in \mathbb{C} %e^{-\theta}Q_j
\mid \frac{1}{2}\rRe (e^{\theta} (z - \epsilon_{j, k})) \ge %0\ \hbox{and}\
| \rIm (e^{\theta} (z - \epsilon_{j, k})) |\};
%\le \frac{1}{2} | \rRe (e^{\theta} (z - \epsilon_{j, k})) |
\end{equation}
\item This
continuation has poles at $\epsilon_{j, k}$: $\lim_{z
\rightarrow \epsilon_{j, k}}(\epsilon_{j, k} -z) F(z, \Psi, \Phi)$ is finite
and $\ne 0$. %, for a finite-dimensional subspace of $\Psi$'s and $\Phi$'s, nonzero.
\end{itemize}
%A similar statement holds also for the Hamiltonian $H_g^{N}$.
%\end{theorem}
%\begin{remark} \label{remI-2}
%\begin{center}
%\includegraphics[height=6.5cm]{lh3.pdf}
%$$PICTURE$$ \end{center}
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Discussion}\label{sec:disc}
%(i) Condition (DA) could be weakened considerably so that it is satisfied by the potential of a molecule with fixed nuclei
%and correspondingly the set of the dilation analytic vectors could
%be replaced by the set $ C_0^\infty(\mathbb{R}^3 )\otimes \cF (C_0)$
%where $\cF(C_0)$ be the bosonic Fock space over $C_0(\mathbb{R}^3)$,
%i.e., $\cF(C_0)=\bigoplus\limits_{n=0}^\infty C_0
%(\mathbb{R}^3)^{\otimes_s n}$, where $\otimes_s$ stands for the
%symmetric tensor product. Here $C_0(\mathbb{R}^3)$ is the one-photon
%space of continuous and compactly supported transverse vector fields
%on $\mathbb{R}^3$, i.e., vector fields, $f$, obeying $k\cdot
%f(k)=0$.
%
%, in the vector model case and the one-phonon space of
%continuous and compactly supported functions on $\mathbb{R}^3$, in
%the case of the Nelson model.
%
%(In the case of the Nelson model considered below,
%$C_0(\mathbb{R}^3)$ is the one-phonon space of continuous and
%compactly supported functions on $\mathbb{R}^3$.) This space is a
%core for $H_f$, while $ C_0^\infty(\mathbb{R}^3 )\otimes \cF (C_0)$
%is a core for $H^{SM}_{g}$.
\begin{itemize}
\item Generically, excited states turn into the resonances, not
bound states. %A condition which guarantees that this happens is the Fermi Golden Rule (FGR) (BachFroehlichSigal), which %It expresses the fact that the coupling of unperturbed embedded eigenvalues of $H_{0}$ to the continuous spectrum is effective in the second order of the perturbation theory. It is generically satisfied.
%(iii) With a little more work one can establish an explicit restriction on the coupling constant $g$ in terms of the particle energy difference $e^{(p)}_{gap}(\nu)$ and appropriate norms of the coupling functions.
\item The second theorem implies the absolute continuity of the
spectrum and its proof gives also the limiting absorption principle for $H$ (see Appendix \ref{sec:locdectransf} for the definitions). %below the ionization threshold, $\inf \sigma_{ess} (H_p)$,
%in the interval $(\epsilon_{0}, \nu)$. (These results can be also proven by the spectral deformation and commutator techniques.) %BachFroehlichSigal1999,BachFroehlichSigalSoffer1999,FroehlichGriesemerSigal2008a.
\item The proof of first theorem gives fast convergent
expressions in the coupling constant $g$ for the ground state energy
and resonances.
%states and the corresponding eigenvalues can be computed explicitly in terms of.
\item One can show analyticity of $\epsilon_{j,k}$ in the coupling constant $g$ (see Appendix \ref{sec:relat-res} for a result on the ground state energies).
%(v) Representation \eqref{poles} for resonances is proven in
%\cite{AFFS} based on results of this work.
%\end{remark}
%As was already eluded to above, the resonances are descendants of
%eigenvalues; they replace the latter
%\item The dense set mentioned in the second theorem is %defined as
%follows. $\cD := \{\Psi \in \cH |\ U_{\theta} \Psi$ has an entire
%continuation into the complex plane $\mathbb{C}\}$.
\item The meromorphic continuation in question is constructed in terms
of matrix elements of the resolvent of a complex deformation,
$H_{ \theta},\ \rIm \theta > 0,$ of the Hamiltonian
$H$.
%The dense set mentioned in the last theorem is defined below.
\item A description of resonance poles is given in Section \ref{sec:relat-res}.
\end{itemize}
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Analysis of QED Hamiltonian}\label{sec:Idea}
\subsection{Approach}\label{sec:Idea}
%We want to understand the spectral structure of the quantum Hamiltonian \begin{equation*} %\label{Hsm} H=\sum\limits_{j=1}^n{1\over 2m_j}(i\nabla_{x_j}-g\Af(x_j))^2+V(x)+H_f. \end{equation*}
%\textit{Problem.}
The main steps in our analysis of the spectral structure of the quantum Hamiltonian $H$ are:
%Our approach contains two new ingredients:
\begin{itemize}
\item Perform a new canonical transformation (a generalized Pauli-Fierz transform)
\begin{equation*}
H \rightarrow H^{PF}:= e^{-ig F}H e^{ig F},
\end{equation*}
in order to bring $H$ to a more convenient for our analysis form;
\item Apply the spectral renormalization group (RG) on new -- momentum anisotropic -- Banach spaces. %The latter allow us to control the RG flow for more singular coupling functions.
\end{itemize}
\bigskip
%To find the spectral structure of $H^{PF}_\theta$ we use
The main ideas of the spectral RG are as follows:
\begin{itemize}
\item Pass from a single operator $H^{PF}_\theta$ to a Banach space $\cal B$ of Hamiltonian-type
operators;
\item Construct a map, $\cR_{\rho}$, (RG transformation) on $\cB$,
%whose domain contains the neighborhood $D$. The transformation, $\cR_{\rho}$, has
with the following properties:
%\begin{itemize}
(a) $\cR_{\rho}$ is 'isospectral'; %and 'preserves' the limiting absorption principle;
(b) $\cR_{\rho}$ removes the photon degrees of freedom related to
energies $\ge \rho$.
%\item \verb"Semigroup property of" $\cR_{\rho}$?
%\end{itemize}
%We then consider the discrete
\item Relate the dynamics of semi-flow, $\cR_{\rho}^n, n \ge 1$, %generated by the
%renormalization transformation, $\cR_{\rho}$
(called renormalization group) to spectral properties of individual
operators in $\cB$.
\end{itemize}
\bigskip
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Generalized Pauli-Fierz transformation} \label{sec-PF-transf}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
We perform a canonical transformation (generalized Pauli-Fierz transform) of $H$ in order to bring it to a form which is accessible to spectral renormalization group (it removes the marginal operators). For simplicity, consider one particle of mass $1$. We define the generalized Pauli-Fierz
transformation as:
\begin{equation} \label{HPF}
H^{PF}: = e^{-ig F} H e^{ig F},
%e^{-ig x \cdot \bar A(0)} H^{SM}_g e^{ig x\cdot \bar A(0)},
\end{equation}
where %$F(x)$ introduced below. We call the resulting Hamiltonian
%on the l.h.s. of this expression
%\noindent which we call
%the generalized Pauli-Fierz Hamiltonian. Now,
$F(x)$ is the self-adjoint operator %on the state space $\cH$
given by
%$\eps=\alpha^{3/2}$ and
%
%$\bar A(x) = (\check\chi_1 * A)(x)$, with $\chi_1$ a function of $|k|$ s.t.
%$\chi_1 \equiv 1$ for $ |k| \leq \nu$ and $\equiv 0$ for $|k|\geq 2\nu$.
\begin{equation}\label{F}
F(x)=\sum_\lambda
\int(\bar{f}_{x,\lambda}(k)a_{\lambda}(k)+f_{x,\lambda}(k)a_{\lambda}^*(k))
\frac{\chi(k)d^3k}{\sqrt{|k|}},
\end{equation}
with the coupling function $f_{x,\lambda}(k)$ chosen as
\begin{equation}\label{f}
f_{x,\lambda}(k):=
e^{-ikx}\frac{\varphi(|k|^{\frac{1}{2}}e_\lambda(k)
\cdot x)}{\sqrt{|k|}}, %\ \varphi \in C^2,\ \varphi'(0)=1.
\end{equation}
with $\varphi \in C^2\ \mbox{bounded, with bounded derivatives and satisfying}\ \varphi'(0)=1.$ %is assumed to be $C^2$, bounded, with bounded second derivative, and satisfying $\varphi'(0)=1.$ %We assume also that $\varphi$ has a
%bounded analytic continuation into the wedge $\{z \in \mathbb{C} |\
%|\arg(z)| < \theta_0\}$.
For the \textit{standard} Pauli-Fierz transformation, we have $\varphi(s)=s$.
\bigskip
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Generalized Pauli-Fierz Hamiltonian} \label{sec-genHpf}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The Hamiltonian $H^{PF}$ is of the same form as $H$. %, but with better infrared behaviour for $|x|$ bounded.
Indeed, using the commutator expansion\\
$e^{-ig F(x)} H_f e^{ig F(x)}= - i g [F,H_f] - g^2 [F, [F,
H_f]],$ we compute
\begin{equation} \label{H^PF}
%{align}\mbox{}\,\,\,\,\,\,\ \hspace{20mm}
H^{PF} = \frac{1}{2m} (p + g A_{\chi\varphi}(x))^2 + V_g(x) + H_f + gG(x), %\,\,\,\,\,\,\,\,\,\,\,\,\ \hspace{25mm} %\\ %\notag &- g x \cdot E + g^{2}\frac{2}{3} |x|^2 \int \chi^2 \cdot %\chi_1,\
\end{equation}
where
\begin{equation*}
A_{\chi\varphi}(x)=\sum_\lambda
\int(\bar{\varphi}_{x,\lambda}(k)a_{\lambda}(k)
+\varphi_{x,\lambda}(k)a_{\lambda}^*(k)){\chi(k)d^3k\over \sqrt{|k|}},
\end{equation*}
%Here the coupling function $\chi_{\lambda, x}(k)$ is defined as follows
with the new coupling function $\varphi_{\lambda, x}(k):=
e_{\lambda}(k)e^{-ikx}- \nabla_x f_{x,\lambda}(k)$ and
$$%A_1(x) = A(x) - \nabla F(x),\
V_g(x):= V(x) + 2
g^2\sum_\lambda \int |k| |f_{x,\lambda}(k)|^2d^3k, $$
\begin{equation*}
G(x):=- i\sum_\lambda
\int|k|(\bar{f}_{x,\lambda}(k)a_{\lambda}(k)-f_{x,\lambda}(k)a_{\lambda}^*(k))
\frac{\chi(k) d^3k}{\sqrt{|k|}}.
\end{equation*}
The potential $V_g(x)$ is a small perturbation of $V(x)$ and the operator $G(x)$ is easy to control. The new coupling function has better infrared behaviour for bounded $|x|$: %satisfying the estimates
%\begin{equation}\label{chi-estim2}$$\int \frac{d^3 k}{|k| } \: |\chi_{\lambda, x}(k)|^{2} \ < \ \infty,$$
%\end{equation}and
\begin{equation}\label{chi-estim1}
|\varphi_{\lambda, x}(k)|
%+ (\sqrt{\omega}|x|)^{-1}|\chi_1(k)|
\le \const \min (1, \sqrt{|k|}\la x\ra).
\end{equation}
To prove the results above we first establish the spectral properties of the generalized Pauli-Fierz Hamiltonian $H^{PF}$ and then transfer the obtained information to the original Hamiltonian $H$.
%\textbf{Remark.*} The formula \eqref{H^PF} can be obtained by
%with $\la x\ra := (1 +|x|^2)^{1/2}$, and
%
%Note that the exponential $x$-localization of $f(H)$ implies the
%exponential $x$-localization of $f(H_1)$. Using the latter and
%
%\begin{equation}
%$E = \int(ia - ia^*) \sqrt{\omega} \chi \cdot \chi_1$.
%\end{equation}
%
\DETAILS{(The terms $gG$ and $V_g - V$ come from the commutator expansion
$e^{-ig F} H_f e^{ig F}$ $= - i g [F,H_f] - g^2 [F, [F,
H_f]]$.) Observe that the operator-family $A_1(x)$ is of the form
The fact that the operators $A_1$ and $G$ have better infra-red
behavior than the original vector potential $A$, is used in proving,
with a help of a renormalization group, the existence of the ground
state and resonances for the Hamiltonian $H_g^{SM}$.
%For the standard Pauli-Fierz transformation: $f_{x,\lambda}(k)=\chi(k)e_\lambda(k)\cdot x\ \Longrightarrow$ $G=E\cdot\ x$. %($E$ is proportional to the electric field at $x=0$). %growing as $x$.
We mention for further references that the operator \eqref{H^PFa} can be
written as
\begin{equation} \label{Hpf}
H_g^{PF} \ = \ H_{0}^{PF} \, + \, I_{g}^{PF} ,
\end{equation}
where $H^{PF}_{0}=H_0 + 2 g^2\sum_\lambda \int
|k||f_{x,\lambda}(k)|^2d^3k + g^2\sum_\lambda
\int{|\chi_\lambda(k)|^2\over |k|}d^3k$, with $H_0 := H_p+H_f$
%defined in (\ref{H_0}),
and $I_{g}^{PF}$ is defined by this relation. Note that the operator
$I_{g}^{PF}$ contains linear and quadratic terms in the creation and
annihilation operators and that the operator $H^{PF}_{0}$ is of the
form $H^{PF}_{0}= H^{PF}_{p}+H_{f}$ where
\begin{equation} \label{Hpg}
H^{PF}_{p}:=H_{p}+2 g^2\sum_\lambda \int |k|
|f_{x,\lambda}(k)|^2d^3k + g^2\sum_\lambda
\int{|\chi_\lambda(k)|^2\over |k|}d^3k
\end{equation}
with $H_p$ given in \eqref{Hp}.
%In what follows we will study the operator $H_g^{PF}$ (in fact, we
%will study more general operators).
%
%and we will use the following theorem in order to translate the properties
%of the operator $H_g^{PF}$ into properties of the operator $H_g^{SM}$:
%\begin{theorem} \label{thm-II.1}
%The Hamiltonians $H^{PF} _{g }$ and $H^{SM} _{g }$ have the same
%eigenvalues and resonances and their resolvents have meromorphic
%continuations across the continuous spectra into the same domain.
%\end{theorem}
Since the operator $F(x)$ in \eqref{HPF} is self-adjoint, the
operators $H_g^{SM}$ and $H_g^{PF}$ have the same eigenvalues with
closely related eigenfunctions and the same essential spectra.}
%
%
%In the next section we prove the remaining part of the theorem.
%
%, i.e. show that they have
%the same resonances and closely related the resolvent matrix
%elements. Below we will study a general class of operators which
%includes the Hamiltonians $H^{PF} _{g }$ and $H^{N} _{g }$.
%In order to get an information about the resonances and meromorphic
%continuation of the resolvent of $H^{SM} _{g }$ we
%with the coupling functions (form-factors) in the linear terms
%satisfying estimate \eqref{I.7} with any $\mu< 1/2$ and with the
%coupling functions in the quadratic terms satisfying a similar estimate.
%In fact, we consider a general class of Hamiltonians of the form
%\eqref{H^PF}, denoted by $H_g$, of which $H_g^{PF}$ is a special
%case.
\bigskip
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\section{Renormalization Group Map}\label{sec:RG}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{RG map}\label{sec:RGmap}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The \textit{renormalization map} is defined on Hamiltonians acting on $\cH_f$ which as follows
\begin{equation}
\cR_{\rho }=\rho^{-1} S_{\rho}\circ F_{ \rho}, \label{RGmap}
\end{equation}
where $ \rho >0$, $S_\rho:
\cB[\cH] \to \cB[\cH]$ is the \textit{scaling transformation}:
%
%\begin{equation} \label{eq-III-2-1}
%S_\rho(1) \ := \ \frac{1}{\rho}1 \;
%S_\rho(A) \ := \ \frac{1}{\rho} \; \Gamma_\rho \; A \;
%\Gamma_\rho^* \comma
%\end{equation}
%
%where $\Gamma_\rho$ is the unitary dilatation on $\cF$ defined by
%$\Gamma_\rho \Om := \Om \comma$ and
%
\begin{equation} \label{Srho}
S_\rho(\one) \ := \ \one , \hspace{5mm} S_\rho (a^\#(k)) := \
\rho^{-3/2} \, a^\#( \rho^{-1} k) ,
\end{equation}
and $F_{ \rho}$ is
the \textit{ Feshbach-Schur map}, or decimation, map,
%
\begin{equation} \label{Fesh}
F_{\rho} (H ) \ := \ \chi_{\rho} (H -
H \bchi_{\rho} (\bchi_{\rho} H \bchi_{\rho})^{-1} \bchi_{\rho} H) \chi_{\rho} ,
\end{equation}
where $\chi_{\rho}$ and $ \overline{\chi}_{\rho}$ is a pair of orthogonal projections, defined as
$$\chi_{\rho} =\chi_{H_{p\theta}=e_j}\otimes\chi_{H_f\le\rho}\ \quad \mbox{and}\ \quad \overline{\chi}_{\rho}:=\mathbf{1}- \chi_{\rho}.$$
%\begin{equation} \label{pi}
%\tau_0(H_g-\lambda)=H_{0g}-\lambda\ \mbox{and}\
%$\chi_{\rho} =\chi_{H_f\le\rho}\ \mbox{and}\ \overline{\chi}_{\rho} =\chi_{H_f\ge\rho}$
%\end{equation}
%$\pi_0$
\noindent\textbf{Remark.} For simplicity we defined the decimation map as the Feshbach-Schur map. Technically it is more convenient to use the \textit{smooth Feshbach-Schur map}, which we defined and discussed in Appendix \ref{sec:sfm}. The smooth Feshbach-Schur map %construction of the map $F_{\rho}$ can be generalized to
uses 'smooth' projections which form a partition of unity
$\chi_{\rho}^{2}+ \overline{\chi}_{\rho}^{2} = \mathbf{1}$, instead of true projections as defined above.
%
%\begin{equation}
%\label{eq-II-1-1}
%W_\chi \ \; := & \chi \, W \, \chi
%\comma \hspace{15mm}
%W_\bchi \ \; := & \bchi \, W \, \bchi \comma
%\\ \label{V.5}
%H_{\tau,\chi} \ \; := \tau(H) \: + \: \chi\overline{\tau}(H)\chi
%\comma \hspace{15mm}
%H_{\bpi} \ \; := H_{0g}\: + \: \bpi I_{g}\bpi .
%\end{equation}
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Isospectrality of $F_{ \rho}$}\label{sec:IsospRG}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
The map $F_{ \rho}$ is \textit{isospectral} in the sense of the following theorem:
\begin{theorem}\label{thm:isospF}
{\em \begin{itemize}
\item[(i)] $ \lambda \in \rho(H ) \Leftrightarrow 0 \in \rho(F_{\rho}
(H - \lambda))$; % i.e. $H_{g} - \lambda$ is bounded invertible on
%$\cH$ if and only if $F_{\rho} (H_{g} - \lambda)$ is bounded
%invertible on $\Ran\, \chi$;
\item[(ii)] %$\psi \in \cH \setminus \{0\}$ solves
$H\psi = \lambda \psi\ \Longleftrightarrow$
%$\vphi := \pi \psi \in \Ran\, \pi \setminus \{0\}$ solves
$F_{\rho} (H - \lambda) \, \vphi = 0;$
%
\DETAILS{\item[(iii)] If $\vphi
\in \Ran\, \pi \setminus \{0\}$ solves $F_{\rho} (H - \lambda)
\, \vphi = 0$ then $\psi := Q_{\pi} (H - \lambda) \vphi \in \cH
\setminus \{0\}$ solves $H\psi = \lambda \psi$;}
%
\item[(iii)] %The multiplicity of the spectral value $\{0\}$ is conserved in the sense that
$\dim \cern (H - \lambda) = \dim \cern F_{\rho} (H -
\lambda)$;
\item[(iv)] %If one of the inverses,
$(H - \lambda)^{-1}$ exists $ \Longleftrightarrow$ $F_{\rho} (H - \lambda)^{-1}$
exists.
%\hspace{8mm}
%\\ \label{eq-II-7}
%and $$ F_{\rho} (H_{g}- \lambda)^{-1} = \pi \, (H-
%\lambda)^{-1} \, \pi \; + \; \bpi \, (H- \lambda)^{-1} \bpi .
%$$
% \hspace{8mm}
%\end{eqnarray}
%
\end{itemize}}
\end{theorem}
For the proof of this theorem as well as for the relation between $\psi$ and $\varphi$ in (ii) and between $(H - \lambda)^{-1}$ and $F_{\rho} (H - \lambda)^{-1}$ in (iv) see Appendix \ref{sec:isosprel}.
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A Banach Space of Hamiltonians} \label{sec:Bansp}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\DETAILS{We construct a Banach space of Hamiltonians on which the
renormalization transformation will be defined. In order not to
complicate notation unnecessarily we will think about the creation-
and annihilation operators used below as scalar operators,
neglecting the helicity of photons.
%rather than operator-valued transverse vector functions.
We explain at the end of Supplement A how to reinterpret the
corresponding expression for the photon creation- and annihilation
operators.
%Let
%
%\begin{equation} \label{eq-III-1-1}
%\cH_\red \ := \ \Ran\, \one[ \hf < 1]
%\ \equiv \ \one[ \hf < 1] \cF \ \subseteq \ \cF
%\end{equation}
%
%be the spectral subspace corresponding to photon energies less than $1$.
and $m,n \ge 0$. Given functions $w_{m,n}:
I\times B_1^{m+n} \rightarrow \mathbb{C}, m+n > 0$, we consider
monomials, $W_{m,n} \equiv W_{m,n}[w_{m,n}]$, in the creation and
annihilation operators defined as follows:}
%
%introduce Hamiltonians
%
%$H \in \cB[\cH_\red]$
%of the form
%
%\begin{equation} \label{eq-III-1-2}
%H \ = \sum_{m+n \geq 0}
%W_{m,n} \comma
%\end{equation}
We will study operators on the subspace $\rRan\chi_1$ of the Fock space $\cF$. Such operators are said to be
in the \textit{generalized normal form} if they can be written as:
\begin{eqnarray} \label{H} %\nonumber
H&=&\sum_{m+n\ge 0} W_{m+n},\\
W_{m+n}&=&\int_{B_1^{m+n}} \prod_{i=1}^{m+n} d^3 k_i \; %\prod_{i=1}^n d^3 \tk_i
%\frac{ dk_{(m,n)} }{ |k_{(m,n)}|^{1/2} } \;
\prod_{i=1}^m a^*(k_i ) \, w_{m,n} \big( \hf ; k_{(m+n)} \big) \, \prod_{i=m+1}^{m+n} a(k_i ), \nonumber
\end{eqnarray}
where $B_1^r$ denotes the Cartesian product of $r$ unit balls in
$\RR^{3}$, $ k_{(m)} \: := \: (k_1, \ldots, k_m)$ and $w_{m,n}: I\times B_1^{m+n}\to \C,\ I:=[0,1]$.
We %write $H=\sum_{m+n\ge 0} H_{m,n}$ and
sometimes we display the dependence of $H$ and $W_{m,n}$ on the coupling functions $\uw:= (w_{m,n},\ m+n\ge 0)$ by writing $H[\uw ]$ and $W_{m,n}[\uw ]$.
We assume that %, for every $m$ and $n$ with $m+n>0$ and for $s \ge %1$,
the functions $w_{m,n}( r, , k_{(m+n)})$ are %$s$ times
continuously differentiable in $r \in I$, symmetric w.~r.~t.\ the
variables $ (k_1, \ldots, k_m)$ and $ (k_{m+1}, \ldots, k_{m+n})$
and obey $\| w_{m,n} \|_{\mu,1} \ :=
%\| w_{m,n} \|_{\mu} \; + \;
\sum_{n=0}^{1} \| \partial_r^n w_{m,n} \|_{\mu} \ < \ \infty ,
%(r\partial_r)^p
$
where $\mu \ge 0$ and
\begin{equation} \label{wmn}
\| w_{m,n} \|_{\mu } \ := %\sum_{n=0}^{s}
\max_j \sup_{r \in I, k_{(m+n)} \in B_1^{m+n}} \big| | k_j|^{-\mu}\prod_{i=1}^{m+n} | k_i|^{1/2} %\partial_r^n
w_{m,n}(r ; k_{(m+n)}) \big|.
\end{equation}
%where $\mu \ge -1/2$ and $I:=[0,1]$.
Here $k_j \in \mathbb{R}^3$ is the $j-$th $3-$%dimensional components of the $3(m+n)-$dimensional $k-$
vector in $k_{(m,n)}$ over which we take the supremum.
Note that these norms are anisotropic in the total momentum space.
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For $\mu \ge0$ and $0 < \xi < 1$ %and $s \ge 1$
we define the Banach space %$\cW^{\mu}$, % \ \equiv \cW^{\mu,s}_{\xi}$, := \ \bigoplus_{m+n \geq 0}\cW_{m,n}^{\mu,s}$, with the norm
\begin{equation} \label{Bmuxi}
\cB^{\mu\xi}:=\{H\ :\ \big\| H \big\|_{\mu,\xi} \ := \ \sum_{m+n \geq 0}
\xi^{-(m+n)} \; \| w_{m,n} \|_{\mu, 1} \ < \ \infty \}.
\end{equation}
We mention some properties of these spaces %(see \cite{bcfs1}): %BachChenFroehlichSigal2003}):
\begin{itemize} %\label{prop:sa}
\item For any $\mu \ge 0$ and $0 < \xi < 1$, the map $H : \uw \to H[\uw ]$,
given in \eqref{H}, is one-to-one.
\item If $ H$ is self-adjoint, then so are $W_{0,0}$ and $ \sum_{m+n \geq 1} \chi_1 W_{m,n}\chi_1$ (see \eqref{H}). %(Follows from \cite{BachChenFroehlichSigal2003}, Eq. (3.33).)
%\item If $ H(\lam),\ \lam\in \Lam,$ is an analytic family, then so are $W_{0,0}(\lam)$ and $ \sum_{m+n \geq 1} \chi_1 W_{m,n}(\lam)\chi_1$.
\end{itemize}
\textbf{Remarks.} 1) Unlike the Banach spaces defined in \cite{bfs1, bfs2, bcfs1}, the Banach spaces are anisotropic in the momentum space. This is needed to overcome the problem of marginal operators which arise in the renormalization group approach. %which are defined in Appendix \ref{sec:anisotr-B-sp}.
2) The self-adjointness statement follows from \cite{bcfs1}, Eq. (3.33).
%3) The analyticity statement was proven in \cite{GriesemerHasler2}, see Appendix \ref{sec:relat-res} for a precise statement.
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Basic bound} \label{sec:bnd}
The following bound shows that our Banach space norm control the operator norm and the terms with higher numbers of creation and annihilation operators make progressively smaller contributions:
\begin{theorem} %\label{thm;basicineq}
{\em Let
$\chi_\rho\equiv\chi_{H_f\le\rho}$. Then for all $\rho >0$ and $m+n \geq 1$}
\begin{equation}
%\big\| W_{m,n} \big\|_\op \ \leq \
\big\| \chi_\rho \, H_{m,n} \, \chi_\rho
% (P_\Om^\perp \hf)^{-n/2}
\big\|
\ \leq \ \frac{\rho^{m+n+\mu}}{\sqrt{m! \, n!} } \, \| w_{m,n}
\|_{\mu }.
\end{equation}
%where $\| \, \cdot \, \|$ denotes the operator norm on $\cB[\cF]$.
% $P_\Om := |\Om \ra \la \Om |$ is the orthogonal projection onto
%the vacuum vector in $\cF$, and $P_\Om^\perp :=\one - P_\Om$.
\end{theorem}
%This shows that the terms with higher numbers of creation and annihilation operators are easier to control.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Sketch of Proof of Basic Bound*} \label{sec:bndpf}
\noindent \textit{Sketch of proof.}
For simplicity we prove this inequality for $m=n=1$. Let $\phi\in \cF$ and $\Phi_k=a(k ) \chi_\rho\phi$. We have
\begin{eqnarray} \nonumber
\la \chi_\rho\phi, H_{1,1} \, \chi_\rho\phi\ra &=& \int_{B_1^{2}} \prod_{i=1}^{2} d^3 k_i \; %\prod_{i=1}^n d^3 \tk_i
%\frac{ dk_{(m,n)} }{ |k_{(m,n)}|^{1/2} } \;
\la \Phi_{k_1}, \, w_{1,1} \big( \hf ; k_1, k_2 \big) \, \Phi_{k_2}\ra.
\end{eqnarray}
Now we write $\chi_\rho=\chi_\rho\chi_{2\rho}$ and pull $\chi_{2\rho}$ toward $w_{1,1} ( \hf ; k_1, k_2)$ using the pull-trough formulae
$$a(k) \, f(\hf) \ = \ f( \hf + |k| ) \, a(k),\
%\hspace{4mm} \mbox{and} \hspace{3mm}
f(\hf) \, a^*(k) \ = \ a^*(k) \, f( \hf +|k|)$$
(see Appendix \ref{sec:pullthrough}). This gives
\begin{eqnarray*}
&&|\la \phi, \chi_\rho \, H_{1,1} \, \chi_\rho\phi\ra|\\
&=& |\int_{|k_i|\le 2\rho, i=1,2} \prod_{i=1}^{2} d^3 k_i \; \la \Phi_{k_1}, \chi_{2\rho-|k_1|} w_{1,1} \big( \hf ; k_1, k_2 \big) \chi_{2\rho-|k_2|} \Phi_{k_2}\ra|\\
&\le& \int_{|k_i|\le 2\rho, i=1,2} \prod_{i=1}^{2} d^3 k_i \; \| \Phi_{k_1}\|\ \| w_{1,1} \big( \hf ; k_1, k_2 \big) \|\ \| \Phi_{k_2}\| \\
&\le& \bigg(\int_{|k_i|\le 2\rho, i=1,2} \prod_{i=1}^{2} d^3 k_i \frac{\| w_{1,1} \big( \hf ; k_1, k_2 \big) \|^2}{|k_1||k_2|}\bigg)^{1/2}\int d^3 k\|\sqrt{|k|} \Phi_{k}\|^2.
\end{eqnarray*}
Now, using $\| w_{1,1} \big( \hf ; k_1, k_2 \big) \|\le \| w_{1,1} \|_\mu\frac{|k_1|^{\mu}+|k_2|^{\mu}}{|k_1|^\frac{1}{2}|k_2|^\frac{1}{2}}$ and $\int d^3 k\|\sqrt{|k|} \Phi_{k}\|^2=\|\sqrt{H_f} \chi_\rho\phi\|^2$, we find
$$|\la \phi, \chi_\rho \, H_{1,1} \, \chi_\rho\phi\ra|\lesssim \rho^{2+\mu} \| w_{1,1}\|_{\mu }.$$
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Unstable, neutral and stable components} \label{sec:comp}
%The operators in $\cW^{\mu,s}_{op}$ will decomposed as
We decompose $H\in \cB^{\mu\xi}$ into the components %\begin{equation*}
$E:=\la\Om, H\Om\ra,\ T:=H_{0,0}-\la\Om, H\Om\ra,\ %\sim H_f,\ %w_{0,0}(0),\ T:=w_{0,0}(\hf) - w_{0,0}(0),\
W:=\sum_{m+n \geq 1} W_{m,n},$ %(\uw). \boxed{\chi_1W_{m,n}[\uw]\chi_1}.
%\end{equation*}
so that
\begin{equation}\label{Hsplit}
H=E\one + T + W.
\end{equation}
%where
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Scaling Properties} \label{sec:comp}
%Different
If we assume $\sup_{r \in [0,\infty)}| T'(r) - 1 | \ll 1$, then we have $T \sim H_f$. These Hamiltonian components scale as follows
\begin{itemize}
\item % $S_\rho (\hf) = \rho \hf$, and hence $S_\rho ( \chi_\rho) = \ \chi_1,$ %\hspace{5mm} \mbox{and} \hspace{6mm}
$\rho^{-1} S_\rho \big( \hf \big) \ = \ \hf $ %, i.e. the operator
($\hf$ is a \emph{fixed point} of $\rho^{-1} S_\rho$);
\item $\rho^{-1} S_\rho(E \cdot \one) =
\rho^{-1} E \cdot \one$ ($E \cdot \one$ \emph{
expand} under $\rho^{-1} S_\rho$ at a rate $\rho^{-1}$);
\item $
\| S_\rho(W_{m,n}) \|_{\mu} \le \rho^{\alpha} \, \| w_{m,n} \|_{\mu},\ \alpha:=m+n- 1 +\mu\del_{m+n=1}$ (%either $\mu >0, m+n\ge 1$ or $\mu \ge 0, m+n> 1$ the interactions
$W_{mn}$ \textit{contract} under $\rho^{-1} S_\rho$, if $\mu> 0$).
\end{itemize}
Thus for $\mu >0$, $E,\ T,\ W$ behave, in the terminology of the renormalization group approach, as relevant, marginal, and irrelevant operators, respectively. For $\mu =0$, the operators $W_{mn},\ m+n=1,$ become marginal.
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Action of Renormalization Map} \label{sec:actionRG}
To control the components $E, T, W$ of $H$ we introduce, for $\alpha, \beta, \gamma >0$, the following polydisc:
\begin{eqnarray} %\label{disc}
\cD^{\mu}(\alpha,\beta,\gamma) && := \Big\{ H=E+T+W \in
\cB^{\mu\xi} \ | \ |E|\leq\alpha, \nonumber\\
%\text{and} \ T'-\delta\ \text{has compact support for some}\ \delta\in\RR
&&\sup_{r \in [0,\infty)}| T'(r) - 1 | \leq
\beta,\ \| W \|_{\mu, \xi}\leq\gamma \Big\}.\nonumber
\end{eqnarray}
(Strictly speaking we should write %$\chi_{H_{p\theta}=e_j}\otimes \cB^{\mu\xi}$ in the definition above, or
$\chi_{H_{p\theta}=e_j}\otimes \cD^{\mu}(\alpha,\beta,\gamma)$ instead of $\cD^{\mu}(\alpha,\beta,\gamma)$ (for various sets of parameters $\alpha,\beta,\gamma$) in the statement below.)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \label{thm:act-RGmap}
{\em Let $ 0<\rho<1/2,\ \alpha,\ \beta,\ \gamma \le \rho/8$ and $\mu_0=1/2$. Then there is $c>0$, s.t.
%Let %$\epsilon_0:H\rightarrow \la H\ra_\Omega$ and $\mu>0$. Then for an absolute constant $c$ and for any $ 0<\rho<1/2,\ \alpha,\beta \le \frac{\rho}{8},$ $\gamma \le \frac{\rho}{2c}$, we have
%$\epsilon_0:H\rightarrow \la H\ra_\Omega$ and $\mu>0$ and %for an absolute constant $c$ and $g \ll 1$. Then, for any $ 0<\rho<1/2$,
\begin{itemize}
\medskip
\item $\cR_\rho(H^{PF}_\theta)\in \cD^{\mu_0}(\al_0, \beta_0, \gamma_0),\ \al_0=c g^2 \rho^{\mu_0-2}, \beta_0=c g^2\rho^{\mu_0 -1}$, $\gamma_0 =c g\rho^\mu_0$,
provided $g \ll 1$;
\item $ \cD^{\mu}(\alpha,\beta,\gamma)\subset D(\cR_\rho)$, provided $\mu>0$;
\item
$\cR_\rho :\cD^{\mu}(\alpha,\beta,\gamma)\rightarrow \cD^{\mu}(\alpha',\beta',\gamma'),$
continuously, with %$\forall 0<\rho<1/2,\ \alpha,\beta, \gamma \le \rho/8,$ and
$\alpha'=\rho^{-1}\alpha+c\lb\gamma^2/2\rho\rb, \beta'=\beta+c\lb\gamma^2/2\rho\rb,\ \gamma'=c\rho^\mu\gamma.$
%$\cR_\rho$ has fixed points $wH_f,\ w\in \C$
%$\forall \zeta\in \C,\ \cR_{\rho}(\zeta H_f)=\zeta H_f$ and $\cR_{\rho}(\zeta\id)=\rho^{-1}\zeta\id$;
\medskip
%\item $\|\cR_{\rho}(H)-\cR_{\rho}(E+T)\|_\mu\le c\rho^\mu\|W\|_\mu$.%$\cR_\rho$ contracts in the co-dimension $2$:
%\item $\cR_\rho$ expands along a subspace of complex dimension $1$; $\cR_{\rho}(w\id)=\rho^{-1}w\id$.
\end{itemize}}
%$$\cR_\rho :\cD^{\mu}(\alpha,\beta,\gamma)\rightarrow \cD^{\mu}(\alpha',\beta',\gamma'),$$ continuously, with %$\xi:=\frac{\sqrt{\rho}}{c}$ and \begin{equation*}\alpha'=\rho^{-1}\alpha+c\lb\gamma^2/2\rho\rb, \beta'=\beta+c\lb\gamma^2/2\rho\rb,\ \gamma'=c\rho^\mu\gamma.\end{equation*}
\end{theorem}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\DETAILS{\begin{theorem} {\em Let %$\epsilon_0:H\rightarrow \la H\ra_\Omega$ and
$\mu>0$. Then for an absolute constant $c$ and
for any $ 0<\rho<1/2,\ \alpha,\beta \le \frac{\rho}{8},$
$\gamma \le \frac{\rho}{2c}$, we have that
\begin{equation*}
\cR_\rho %-\rho^{-1}\epsilon_0
:\cD^{\mu,s}(\alpha,\beta,\gamma)\rightarrow
\cD^{\mu,s}(\alpha',\beta',\gamma'),
\end{equation*}
continuously, with %$\xi:=\frac{\sqrt{\rho}}{c}$ and
\begin{equation*}
\alpha'=\alpha+c\lb\gamma^2/2\rho\rb,
\beta'=\beta+c\lb\gamma^2/2\rho\rb,\ \gamma'=30
c^2\rho^\mu\gamma.
\end{equation*}}
\end{theorem}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Idea of Proof of $\cD^{\mu}(\rho/8, 1/8, \rho/8)\subset D(\cR_\rho)$*} \label{sec:RGMpf1}
%\begin{proof} (1) Show $\cD^{\mu,s}(\rho/8, 1/8, \rho/8)\subset D(\cR_\rho).$
%\begin{lemma} If $0 < \rho < 1$, $\mu > 0, s \geq 1$, and $0 < \xi < 1$. Then
%is in the domain of the Feshbach map $F_\rho$. \end{lemma}
%
%Let $H(\uw) \in \cD^{\mu,s}(\rho/8, 1/8, \rho/8)$. We remark that
\noindent \textit{Sketch of proof of the second and third properties.} 1) %Idea of Proof of
$\cD^{\mu}(\rho/8, 1/8, \rho/8)\subset D(\cR_\rho)$. Since $W:= H-E-T$ defines a bounded operator on $\cF$, we
only need to check the invertibility of $H_{\tau \chi_\rho}$ on
$\Ran \,\bchi_\rho$. The operator $E+T$ is
invertible on $\Ran \,\bchi_\rho$:
for all $r \in [3\rho/4, \infty)$
\begin{eqnarray*} %\label{IV.4}
\Re\ T(r) + \Re\ E & \geq & r \, - \, | T(r) - r | \, - \, |E|
\nonumber \\ & \geq & r \big( 1 \, - \, \sup_{r} | T'(r) - 1 | \big) \: - \: |E| \nonumber \\
& \geq & \frac{3 \, \rho}{4} ( 1 - 1/8 ) \: - \: \frac{\rho}{8} \
\geq \ \frac{ \rho}{2} \nonumber \\
&\Rightarrow & E+T\ \mbox{is invertible and}\ \|(E+T)^{-1}\|
\le 2/\rho.
\end{eqnarray*}
%Eqn \eqref{IV.4} implies also that $\|(E+T)^{-1}\|\le 2/\rho$.
Now, by the basic estimate, $\big\| W \|\leq
\rho/8$ and therefore,
\begin{eqnarray*} && \big\|\bchi_\rho W \bchi_\rho (E +
T)^{-1}\|\leq 1/4 \\
&\Rightarrow & %$\qquad H(\uw)_{\tau, \bchi_\rho}= [1+\bchi_\rho W \bchi_\rho (E +T)^{-1}](E + T)$
E + T+\bchi_\rho W \bchi_\rho\ \mbox{is invertible on}\ \Ran \,\bchi_\rho \\
&\Rightarrow & \cD^{\mu}(\rho/8, 1/8, \rho/8)\subset D(F_\rho)=D(\cR_\rho).
\end{eqnarray*}
%\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Sketch of Proof of $\cR_\rho %-\rho^{-1}\epsilon_0 :\cD^{\mu,s}(\alpha,\beta,\gamma)\rightarrow \cD^{\mu,s}(\alpha',\beta',\gamma')$*} \label{sec:RGMpf2}
2) $\cR_\rho :\cD^{\mu}(\alpha,\beta,\gamma)\rightarrow \cD^{\mu}(\alpha',\beta',\gamma')$ (normal form of $\cR_{\rho}(H)$). Recall that $\chi_\rho \equiv \chi_{H_f \le
\rho}$ and $\bchi_\rho:= 1 -\chi_\rho$. Let $H_0:=E+T$, so that $H=H_0+W$. We have shown above
\begin{equation*}
\big\| H_0^{-1} \bchi_\rho \big\| \ \leq \ \frac{2}{\rho}
\hspace{5mm} \mbox{and} \hspace{5mm} \| W \| \ \leq \ \frac{\rho}{8}.
\end{equation*}
In the Feshbach-Schur map, $F_{\rho}$,
%
%\begin{eqnarray*} \lefteqn{ F_{\rho} \big( H \big) \ = \ H_0 \; + \; \chi_\rho W \chi_\rho }\\[1mm] & & \hspace{-3mm}\; - \; \chi_\rho \, W \, \bchi_\rho \big(H_0 + \, \bchi_\rho W\bchi_\rho \big)^{-1} \bchi_\rho \, W \, \chi_\rho .\end{eqnarray*}
\begin{eqnarray*}
F_{\rho} \big( H \big) \ = \ \chi_\rho \big(H_0+W
- W \, \bchi_\rho \big( \bchi_\rho (H_0+W)
\bchi_\rho \big)^{-1} \bchi_\rho \, W \,\big) \chi_\rho ,
\end{eqnarray*}
we expand the resolvent $( \bchi_\rho (H_0+W)
\bchi_\rho \big)^{-1}$ %in $F_{\rho} \big( H \big)$
in the norm convergent Neumann series
\begin{equation*}
F_{\rho} \big( H \big) \ = \ \chi_\rho\big[H_0 + \sum_{s=0}^\infty
(-1)^{s} \, W \big( H_0^{-1}\bchi_\rho^2 \; W
\big)^{s}\big] \chi_\rho .
\end{equation*}
Next, we transform the right side to the generalized normal form using generalized Wick's theorem.
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Generalized Wick's Theorem*}\label{sec:wick}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{Generalized Wick's theorem.} To write the product $W \big( H_0^{-1}\bchi_\rho^2 \; W\big)^{s}$
in the generalized normal form we pull the annihilation operators, $a$, to the right and the creation operators, $a^*$, to the left, apart from those which enter $H_f$. We use the rules (see Appendix \ref{sec:pullthrough}): %The creation and annihilation operators interchange positions according to the formula
$$a(k)a^*(k') = a^*(k')a(k) +\delta(k-k'),$$
%Thus they either pass through each other without a change or produce the $\delta-$function (contract with each other). They pass through functions of the photon Hamiltonian operator $H_f$ according to the Pull-Through formulae
\begin{equation*}
a(k) \, f(\hf) \ = \ f( \hf + |k| ) \, a(k),\
%\hspace{4mm} \mbox{and} \hspace{3mm}
f(\hf) \, a^*(k) \ = \ a^*(k) \, f( \hf +|k|) .
\end{equation*}
Some of the creation and annihilation operators reach the extreme
left and right positions, while the remaining ones contract (see the figure below). The
terms with $m$ creation operators on the left
and $n$ annihilation operators on the right
contribute to the $(m,n)-$ formfactor, $w^{(s)}_{m,n}$, of the
operator $W \big( H_0^{-1}\bchi_\rho^2 \; W\big)^{s}$. As the result we obtain the generalized normal form of $F_\rho(H)$:
\begin{equation*}
F_\rho(H) = \ \sum_{m+n\ge0}
W^{'}_{m,n}.
\end{equation*}
The term $W^{'}_{0,0}=\la W^{(s)}_{0,0}\ra_\Om +(W^{(s)}_{0,0}-\la W^{(s)}_{0,0}\ra_\Om)$ contributes the corrections to $E+T$.
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%
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\DETAILS{
\begin{center}
\psset{unit=1cm} \pspicture(-4,-5)(8,3.5)
% stable manifold and stable orbit
\pscustom[linestyle=none,fillstyle=solid,fillcolor=lightgray]{
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\rput(0.5,-3.5){$\mathcal{M}_s$} \rput(0.5,0.8){$w H_f$}
% manifold of fixed points
\psset{linewidth=0.5pt} \psline(-3,-1.5)(3,1.5)
\rput(3.5,1.5){$\mathcal{M}_{fp}$}
% vertical axis
\psline(0,-1)(0,2)\rput(0,2.3){$\mathcal{M}_u$}
% dashed vertical line and unstable orbit
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\rput(3,2.2){$\mathcal{R}^{n}_{\rho}(H_\theta-\lambda)$}
% H and H-lambda with dashed connection
\psline[linewidth=0.5pt,linestyle=dashed](3,0.2)(3,-0.7)
\qdisk(3,0.2){2pt}\qdisk(3,-0.7){2pt}\rput(3.4,0.2){$H_\theta$}
\psline[linewidth=0.3pt]{<-}(3.1,-0.7)(4.5,-0.5)\rput(5.2,-0.5){$H_\theta-\lambda$}
\endpspicture
Stable and unstable manifolds.
\end{center} }
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$$ \textbf{Hidden Picture}$$ \DETAILS{
\bigskip
%\begin{center}
\includegraphics[height=4cm]{lh4.pdf} }
%$$PICTURE$$ %\end{center}
\bigskip
This is the standard way for proving the Wick theorem, taking into account the presence of $H_f-$ dependent factors. (See \cite{bfs1} for a different, more formal proof.)
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Estimating Formfactors*}\label{sec:wick}
\textbf{Estimating formfactors.} The problem here is that the number of terms generated by various contractions is %, which is the number of pairs which can be formed by creation and annihilation operators, is, roughly,
$O(s!)$. %for a product of $s$ terms.
Therefore a simple majoration of the series for the $(m,n)-$ formfactor, $w^{(s)}_{m,n}$, of the operator $W \big( H_0^{-1}\bchi_\rho^2 \; W\big)^{s}$ will diverge badly.
To overcome this we re-sum the series %in order to take advantage of possible cancelations. The latter is done
by, roughly, representing the sum over all contractions, for a given $m$ and $n$, as
$$w^{(s)}_{m,n}\sim \la\Omega, [W \big( H_0^{-1}\bchi_\rho^2 \; W\big)^{s}]_{m,n} \mbox{}\Omega\ra,$$
where $[W \big( H_0^{-1}\bchi_\rho^2 \; W\big)^{s}]_{m,n}$ is $W \big( H_0^{-1}\bchi_\rho^2 \; W\big)^{s}$, with $m$ escaping creation operators and $n$ escaping annihilation operators deleted. % the vacuum expectation effects only the 'contracting' creation and annihilation operators and does not apply to those 'escaping' to the left or right.
Now the estimate of $w^{(s)}_{m,n}$ is straightforward and can be written (symbolically) as
$$\|w^{(s)}_{m,n}\|\lesssim \|\chi_{\rho'} \; W'\chi_{\rho'} \|^{s+1},$$
( for the operator norm, and similarly for $\cB^{\mu,\xi}-$norm.
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Renormalization Group}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Iteration of $\cR_\rho$}
To analyze spectral properties of individual
operators in $\cB^{\mu\xi}$ we use the discrete semi-flow, $\cR_{\rho}^n, n \ge 1$
(called renormalization group) , generated by the renormalization transformation, $\cR_{\rho}$.
By Theorem \ref{thm:act-RGmap}, in order to iterate $\cR_\rho$ we have to control the expanding direction: $\cR_{\rho}(\z\id)=\rho^{-1}\z\id.$
To control this direction, we adjust, inductively, at each step the constant component $\la H\ra_\Omega :=\la \Omega, H\Omega\ra$ of the initial Hamiltonian, $H$: %the parameter $\la H\ra_\Omega :=\la \Omega, H\Omega\ra$ inductively,
$$|\la H\ra_\Omega - e_{n-1}| \leq \frac{1}{12} \rho^{n+1},$$
\begin{equation*}
e_{n-1}\ \mbox{is the unique zero of the function}\ \lam\to \big\la\cR_\rho^{n-1}(H-\la H\ra_\Omega+\lambda)\big\ra_\Omega,
%\mbox{in}\ D(e_{(n-1)}(H_s),\ 2\alpha_n \rho^n).\
\end{equation*}
%(i.e. $H $ is in a $\rho^n-$neighborhood of the stable manifold $\cM_s$)
so that $$H \in\ \quad \mbox{the domain of}\ \quad \cR_{\rho}^n.$$
This way one adjusts the initial conditions closer and closer to the stable manifold $\mathcal{M}_s=\cap_n D(\cR_{\rho}^n)$.
%
\DETAILS{We have shown that $\cR_{\rho}$
\begin{itemize}
\item contracts, for $\mu>0$, in the co-dimension $2$: $\|\cR_{\rho}(H)-\cR_{\rho}(E+T)\|_\mu\le c\rho^\mu\|W\|_\mu$;
\item has fixed points $\z H_f,\ \z \in \C$: $\cR_{\rho}(\z H_f)=\z H_f$; %manifold of complex dimension $1$,
\item expands along a subspace of complex dimension $1$; $\cR_{\rho}(\z\id)=\rho^{-1}\z\id$.
\end{itemize}
To control the expanding direction, we adjust the parameter $\la H\ra_\Omega$ inductively,
$$|\la H\ra_\Omega - e_{n-1}| \leq \frac{1}{12} \rho^{n+1},$$
\begin{equation*}
e_{n}\ \mbox{is the unique zero of the function}\ \big\la\cR_\rho^{n}(H-\la H\ra_\Omega+\lambda)\big\ra_\Omega,
%\mbox{in}\ D(e_{(n-1)}(H_s),\ 2\alpha_n \rho^n).\
\end{equation*}
%(i.e. $H $ is in a $\rho^n-$neighborhood of the stable manifold $\cM_s$)
so that $H \in$ the domain of $\cR_{\rho}^n$.}
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$$ \textbf{Hidden Picture}$$ \DETAILS{
\begin{center}
\includegraphics[height=6.0cm]{rglh-ps-picture1.pdf}
\end{center}
}
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\psset{unit=1cm}
\begin{pspicture}(-1,-3)(8,3.5)
\psset{linewidth=0.5pt} \psline(0,-2)(0,3)
\psline(0,0)(6,0) % axes of coordinates
\psbezier[linewidth=1pt](0,0)(2,0)(4,-0.5)(5.5,-2) % trajectory
\psbezier[linewidth=0.5pt,linestyle=dashed](4.5,-0.5)(2.5,0.5)(0.5,1)(0.5,2.8)
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\qdisk(4.5,-0.5){2pt} \qdisk(4.5,-1.19){2pt}
\psline[linewidth=0.5pt,linestyle=dashed](4.5,0)(4.5,-1.2) % dashed line
\psline[linewidth=0.5pt,linestyle=dashed](0,-0.5)(4.5,-0.5)
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\rput(-0.3,-0.5){$H_u$} \rput(4.8,-0.5){$H$}\rput(4.6,0.25){$H_s$}
%\rput(4.8,-0.3){$H_s$}\qdisk(4.5,0){2pt}
\rput(6,-2){$\mathcal{M}_s$} \rput(0,3.2){$\mathcal{M}_u$}
\rput(6.6,0){$V_s$} %\mathcal{W}_{}$}
\end{pspicture}
$\mathcal{R}^{n}_{\rho}(H)$ converges to the unstable-fixed point manifold $\mathcal{M}_{ufp}:=\{\z_1\id+\z_2H_f\}$.
Here $V_s:=\{\sum_{m+n\ge 1}W_{mn}\},\ H_u:=\la H\ra_\Omega$ and $H_s:=H-\la H\ra_\Omega$.
\end{center}}
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\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{RG dynamics}\label{sec:RGDyn}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%We show that the flow,
The procedure above leads to the following results:
\begin{itemize}
\item $\cR_{\rho}^n$ has the fixed-point manifold $\cM_{fp}:=\mathbb{C}H_f$;
\item $\cR_{\rho}^n$ has an unstable manifold $\cM_{u}:=\mathbb{C}\one$;
\item $\cR_{\rho}^n$ has a (complex) co-dimension $1$ stable
manifold $\cM_s$ for $\cM_{fp}$;
\item $\cM_s$ is foliated by (complex) co-dimension
$2$ stable manifolds for each fixed point.
\end{itemize}
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Stable and unstable manifolds.
\end{center}}
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$$ \textbf{Hidden Picture}$$ \DETAILS{
\begin{center}
\includegraphics[height=6.0cm]{rglh-ps-picture2.pdf}
\end{center}
}
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\DETAILS{
\begin{center}
\psset{unit=1cm} \pspicture(-4,-5)(8,3.5)
% stable manifold and stable orbit
\pscustom[linestyle=none,fillstyle=solid,fillcolor=lightgray]{
\psbezier(2,1)(3,0.5)(4,0)(4.5,-1.5)
\psbezier[liftpen=1](0,-4.5)(-0.25,-3)(-1,-2)(-2,-1)}
\psbezier[linewidth=0.5pt,linestyle=dashed](1,0.5)(2.2,-0.1)(3,-0.5)(3.5,-2.25)
\rput(0.5,-3.5){$\mathcal{M}_s$} \rput(0.5,0.8){$w H_f$}
% manifold of fixed points
\psset{linewidth=0.5pt} \psline(-3,-1.5)(3,1.5)
\rput(3.5,1.5){$\mathcal{M}_{fp}$}
% vertical axis
\psline(0,-1)(0,2)\rput(0,2.3){$\mathcal{M}_u$}
% dashed vertical line and unstable orbit
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\rput(3,2.2){$\mathcal{R}^{n}_{\rho}(H-\lambda)$}
% H and H-lambda with dashed connection
\psline[linewidth=0.5pt,linestyle=dashed](3,0.2)(3,-0.7)
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\psline[linewidth=0.3pt]{<-}(3.1,-0.7)(4.5,-0.5)\rput(5.2,-0.5){$H-\lambda$}
\endpspicture
Stable and unstable manifolds.
\end{center}}
%
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%\includegraphics[height=1.5cm]{picture1.pdf} Stable and unstable manifolds.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Following the Eigenvalues} To control the expanding direction, on each step of application of $\cR_\rho$, we restrict the unstable component $\la H\ra_\Omega$ of $H$ to a smaller set, so that the image of $\cR_\rho$ on this step still belongs to $D(\cR_\rho)$. On the $n-$th step we assume that $$|\la H\ra_\Omega+e_{n-1}| \leq \frac{1}{12} \rho^{n+1}$$ \begin{equation}e_{n}\ \mbox{is the unique zero of the function}\ \big\la\cR_\rho^{n}(H_s-\lambda)\big\ra_\Omega\\end{equation} $%H_u:=\la H\ra_\Omega\ \mbox{and}\H_s:=H - \la H\ra_\Omega\\mathbf{1}$ (%the unstable- and stable-central-space component of $H$).
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{RG and spectral properties}\label{sec:RGSp}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%We show that $L_{\theta}-\lambda $ is in the domain of
%$\cR_{\rho}^n$, provided the parameter $\lambda$ is adjusted
%appropriately, so that $L_{\theta}-\lambda $ is, roughly, in a
%$\rho^n-$neighborhood of the stable manifold $\cM_s$.
%Adjust the parameter $\la \Omega, H\Omega\ra$ iteratively, so that
Define the polydisc $\cD_s:=\cD^{\mu}(0, \beta_0, \gamma_0)$, with $\mu>0,\ \beta_0=c g^2\rho^{\mu_0 -1}$ and $\gamma_0 =c g\rho^\mu_0$, the same as in Theorem \ref{thm:act-RGmap}. Using the above results, we draw the following conclusions about the spectrum of an operator $H\in \cD_s$. Find $\lam\in$ a neighbourhood of $0$, s.t. $H(\lambda):=H -\lam \in \cD(\cR_{\rho}^n)$ %is in a $\rho^n-$neighborhood of the stable manifold $\cM_s$ %and therefore
%$\Longrightarrow$ $H_{\theta}-\lambda $ is in the domain of $\cR_{\rho}^n$
%Then, for $n$ sufficiently large,
%which we apply iteratively to the family of
%operators $H^{(0)}(\lambda)$ . After sufficiently many iterations we
%obtain an operator, $\Longrightarrow$
and define $H^{(n)}(\lambda):=\cR_{\rho}^n(H(\lambda) )$. We illustrate this as putting $H(\lambda)$ through the RG (black) box: %are close to the operator
%\approx \z H_f$, for some $\z \in \mathbb{C},\ \rRe\ \z >0$, and $n$ sufficiently large.
$$H(\lambda) \Longrightarrow\ \textbf{RG BOX}\ \Longrightarrow\ H^{(n)}(\lambda).$$
Use that for $n$ sufficiently large, $H^{(n)}(\lambda)\approx \z H_f$, for some $\z \in \mathbb{C},\ \rRe\ \z >0$, to find wanted spectral information about $H^{(n)}(\lambda)$ in a neighbourhood of $0$. Then, use the isospectrality of $\cR_{\rho}$ to pass this information up the chain. %as illustrated in
These steps are described schematically as follows:
%In particular we show that
%$H^{(n)}_{\lambda }$ has the ground state.
%
%Our approach is rather robust. After the first application of the decimation
%map we reduce the problem to that of analyzing a family of
%Since the renormalization map is 'isospectral', we can pass the
%spectral information about $L^{(n)}(\lambda)$ %, which is easy to obtain to
% 'preserves' the spectral information, this
%implies the existence of the ground state (or the resonance) for
%to the operator
$\,$
Spectral information about $H^{(n)}(\lambda)$ $\Longrightarrow$ (by 'isospectrality' of $\cR_{\rho}$)
Spectral information about $H^{(n-1)}(\lambda)$ %(by 'isospectrality' of $\cR_{\rho}$)
%and so forth,
$...$
%until we obtain
%
%existence of the ground state (or the resonance)
%
%the desired spectral information for the initial operator
$\Longrightarrow$ Spectral information about $H(\lambda)$.
\noindent We sum up this as
$$\textbf{Spec info}\ (H)\ \Longleftarrow\ \textbf{RG BOX}\ \Longleftarrow\ \textbf{Spec info}\ (H^{(n)}(\lambda)).$$
%, so the desired result follows.
%
\DETAILS{Choose the parameter $\lambda$ inductively, so that $H_{\theta}-\lambda \in D(\cR_{\rho}^n)$, %$$|\lam-e_{n-1}| \leq \frac{1}{12} \rho^{n+1},$$ \begin{equation}e_{n}\ \mbox{is the unique zero of the function}\ \big\la\cR_\rho^{n}(H_\theta-\lambda)\big\ra_\Omega\
%\mbox{in}\ D(e_{(n-1)}(H_s),\ 2\alpha_n \rho^n).\ \end{equation}
i.e. $H_{\theta}-\lambda $ is in a $\rho^n-$neighborhood of the stable manifold $\cM_s$. Then
%$\Longrightarrow$
$H^{(n)}(\lambda):=\cR_{\rho}^n(H_{\theta}-\lambda )$ are close to the fixed point manifold $\cM_{fp}$
i.e. $ H^{(n)}(\lambda) \approx w H_f$, for some $\z\in \mathbb{C},\ \rRe\ \z >0$ and for $n$ sufficiently large,
%\mathbb{R}^+$
$\Longrightarrow$ Spectral information about $H^{(n)}(\lambda)$.
%In particular we show that
%$H^{(n)}_{\lambda }$ has the ground state.
%
%Our approach is rather robust. After the first application of the decimation
%map we reduce the problem to that of analyzing a family of
%Since the renormalization map is 'isospectral', we can pass the
%spectral information about $L^{(n)}(\lambda)$ %, which is easy to obtain to
% 'preserves' the spectral information, this
%implies the existence of the ground state (or the resonance) for
%to the operator
$\Longrightarrow$ Spectral information about $H^{(n-1)}({\lambda })$ (by isospectrality of $\cR_{\rho}$)
%and so forth,
$...$}
%
%until we obtain
%
%existence of the ground state (or the resonance)
%
%the desired spectral information for the initial operator $\Longrightarrow$
If we start with the Hamiltonian $H_{\theta}$, then we are interested in, we derive this way the required spectral information about it.
%, so the desired result follows.
%Note that, in this approach, a specific form of the interaction and
%the coupling functions becomes irrelevant.
%In particular, as was mentioned above, our approach can handle the
%perturbations quadratic in creation and annihilation operators, $a$
%and $a^*$.
%$$PICTURE$$
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\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Related Results}\label{sec:relat-res}
%
\textbf{Existence of the ionization threshold}. There is $ \Sigma= \Sigma^{(p)}+O(g^2)>\inf\s(H)$ (the ionization threshold) s.t. for any energy interval in $\Delta\subset (\inf\s(H), \Sigma)$,
$$\|e^{\delta|x|}\psi\|<\infty,\ \forall \psi\in\Ran E_\Delta(H),\ \delta<\Sigma-\sup\Delta.$$
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Analyticity}\label{sec:anal}
\noindent\textbf{Analyticity.}
%\begin{theorem}%[Griesemer-Hasler-Herbst, \cite{GriesemerHasler2, HaslerHerbst1, HaslerHerbst2,HaslerHerbst3}]
%{\em
If the ground state energy $\e_g$ of $H$ is non-degenerate, then it is analytic in $g$ and has the following expansion
\begin{equation}\label{GSE-exp}
\e_g=\sum\limits_{j=1}^\infty \e^{2j}_\al \al^{3j},
\end{equation}
where $\e^{2j}_\al$ are smooth function of $\al$ (recall that $g:= \alpha^{3/2}$). %} %eigenvector $\Phsig(P)$ of $\Kns(P)$ converges strongly in $\Fo$: $\Phi(P):=\lim_{\sigma\rightarrow0}\Phsig(P)$ exists in $\Fo$.
%\end{theorem}
The proof relies on the renormalization group analysis together with the analyticity and the form-factor rotation symmetry transfer by the Feshbach-Schur map (see Appendix \ref{sec:anal-transf}) and on treating two sources of the dependence of $H$ on the coupling constant $g$ - in the prefactor of $\Af(y)$ and in its argument %, $A(y) = A_{\chi}(\alpha y)$
(see the line after \eqref{Hsm'}), differently, i.e. by considering the following family of Hamiltonians %Recall that the quantized vector potential in \eqref{Hsm'} is given by $A(y) = A_{\chi}(\alpha y)$.
%So far we suppressed the dependence of $A$ on $g$ or $\al$ (recall that $g$ was defined as $g:= \alpha^{3/2}$). We restore it now writing
\begin{equation}\label{Hsm''}
\hsm:=\sum\limits_{j=1}^n{1\over 2m_j}
(i\nabla_{x_j}-gA_{\chi}(\al x_j))^2+ V(x)+H_f,
\end{equation}
and consider $g$ and $\al$ as independent variables. The powers of $g$ in \eqref{GSE-exp} come from $g$ in \eqref{Hsm''}.
%$V(x)$ of the order $O(1)$ %$A(x)$ replaced by $A'(x)$, where.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{ Analyticity of all Parts of $H$}\label{sec:analytic}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\textbf{Analyticity of all parts of $H$.} %In this appendix we state a useful result, due to \cite{GriesemerHasler2}, about families Hamiltonians $H_\lam\equiv H(\underline{w^\lam})$ of the form \eqref{H}. The result says that if $H_\lam$ is analytic, then so are every component, $E_\lam, T_\lam, W_\lam$, of it. Here, recall, that $E_\lam:=w_{0,0}^{\lambda}(0), T_\lam :=w_{0,0}^{\lambda}(H_f)-w_{0,0}^{\lambda}(0), W_\lam:=H_{\lambda}-E_\lam-T_\lam$ (see \eqref{Hsplit}).
%\begin{proposition}[\cite{GriesemerHasler2}] \label{analytic00} {\em
Suppose that $\lambda\mapsto
H_\lam\equiv H(\underline{w}^{\lambda})$
is of the form \eqref{H} and is analytic in $\lambda\in S\subset\CC$ and that $H(\underline{w}^{\lambda})$ belongs to some polydisc
$\cD(\alpha,\beta,\gamma)$ for all $\lambda\in S$. Then
$\lambda\mapsto E_\lam:=w_{0,0}^{\lambda}(0),\ T_\lam :=w_{0,0}^{\lambda}(H_f)-w_{0,0}^{\lambda}(0),\ W_\lam:=H_{\lambda}-E_\lam-T_\lam $ %+T_\lam:=w_{0,0}^{\lambda}(H_f),\quad \lambda\mapsto W_\lam:=H_\lam-W(\uw^{\lambda})
are analytic in $\lambda\in S$. %} \end{proposition}
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Resonance poles}\label{sec:reson-poles}
%We say To characterize the resonances as the resolvent poles
%As was mentioned above, the resonances in Quantum Mechanics lead to
%isolated poles of meromorphic continuations of the matrix elements
%of the resolvent on a dense set.
%of dilation entire vectors.
%The same property allows one to show that they lead to complex poles
%in the scattering matrix and are responsible for the bumps in the
%scattering cross-section.
\noindent\textbf{Resonance poles.} Can we make sense of the resonance poles in the present context? %Abou Salem-Faupin-Fr\"ohlich-Sigal:
%Thus one would like to show that if a Hamiltonian $H^{SM}_g$ has a
%resonance at $z_0 \in \mathbb{C}^-$, then,
Let $$ Q:= \{z \in \mathbb{C}^- |\ \epsilon_{0} < \rRe z < \nu \} / \bigcup_{j \le j(\nu), k} S_{j, k} .$$
%\textbf{Theorem.}
\begin{theorem} {\em For each $\Psi$ and $\Phi$ from a dense set of
%dilation-entire
vectors (say, \eqref{D}), the meromorphic continuation, $F(z, \Psi, \Phi)$, of the
matrix element $\langle \Psi, (H-z)^{-1}\Phi\rangle$ is of the following form near the resonance
$\epsilon_j$ of $H$:
%of the resolvent of $H^{SM}_g$ has a meromorphic continuation from $\mathbb{C}^+$
%across the essential spectrum of $H^{SM}_g$ into a domain, $Q$, in
%the lower complex semi-plane, s.t. $z_0 \in \overline{Q}$, and this
%continuation is of the form
%
%has a pole at $z_0$.
%
%A subtle point here is that, unlike in Quantum Mechanics, in our
%context the poles of the meromorphic continuations in question are
%not isolated but occur on the boundary
%
%(more precisely at tips of cusps of the complement)
%
%of the domain, $Q$, of the meromorphic continuation. (This is due to
%the fact that the photon mass is zero.) We characterize such a
%(semisimple) pole,
%
%as follows. For a (semisimple) pole,
%
%$z_0$, by requiring
%
%at boundary $\partial Q$ we require
%
%that for a dense subset of the set of dilation holomorphic vectors,
%
\begin{equation} \label{poles}
%\langle \Psi, (H-z)^{-1}\Phi\rangle =
F(z, \Psi, \Phi)=(\epsilon_{j, k} -z)^{-1} p(\Psi, \Phi) + r(z, \Psi,
\Phi) .
\end{equation}
Here $p$ and $r(z)$ are sesquilinear forms in $\Psi$ and $\Phi$, s.t.
\begin{itemize}
\item $r(z)$ is analytic in $Q$
and bounded on the intersection of a neighbourhood of
$\epsilon_{j, k}$ with $Q$ as $$|r(z, \Psi, \Phi)| \le C_{\Psi,
\Phi}|\epsilon_{j, k}-z|^{-\gamma}\ \mbox{for some}\ \gamma <1;$$
\item $p \ne 0$ at least for one pair of vectors $\Psi$ and
$\Phi$ and $p = 0$
%and $\langle \Psi, (H-z)^{-1}\Phi\rangle \le |z_0-z|^{-\gamma}$
for a dense set of vectors $\Psi$ and $\Phi$ in a finite
co-dimension subspace.
%The multiplicity of a resonance is the rank of the residue at the pole.
\end{itemize}}
\end{theorem}
%A simpler but still important open problem is to show that (assuming
%\textbf{asymptotic completeness holds}??)
\noindent\textbf{Local decay.} For any compactly supported function $f(\lambda)$ with $\supp
f \subseteq (\inf H, \infty)/$(a neighbourhood of $\Sigma$), and for $\theta> \frac{1}{2},\ \nu< \theta-\frac{1}{2}$, we have that
\begin{equation}\label{loc-dec}
\|\la Y\ra^{-\theta}e^{-i H t}f(H)\la Y\ra^{-\theta}\|\le C
t^{-\nu}.
\end{equation}
Here
$\la Y\ra := (\one +Y^2)^{1/2}$, and $Y$ denotes the self-adjoint operator on Fock space $\cF$ of photon co-ordinate,
%
\begin{equation} \label{Y}
Y\ : = \ \; \int d^3k \; a^*(k) \: i\nabla_k \: a(k),
\end{equation}
extended to the Hilbert space $\cH=\cH_{\at}\otimes\cF$.
(A self-adjoint operator $H$ obeying \eqref{loc-dec} is said to have the ($\Delta, \nu, Y, \theta$) - \textit{local decay} (LD) property.) \eqref{loc-dec} shows that for well-prepared initial conditions $\psi_0$, the probability of
finding photons within a ball of an arbitrary radius $R < {\infty} $,
centered say at the center-of-mass of the particle system, tends to 0, as time $t$
tends to ${\infty}$.
Note that one can also show that as $t\to \infty$, the photon coordinate and wave vectors in the support of the solution $e^{-iHt}\psi_0$ of the Schr\"odinger equation become more and more parallel. This follows from the local decay for the self-adjoint generator of dilatations on Fock space $\cF$,
\begin{equation} \label{B}
B \ : = \ \frac{i}{2}\; \int d^3k \; a^*(k) \: \big\{ k \cdot
\nabla_k + \nabla_k \cdot k \big\} \: a(k) .
\end{equation}
(In fact, one first proves the local decay property for $B$ and then transfers it to the photon co-ordinate operator $Y$.) %The local decay for $B$ shows that the solution $e^{-iHt}\psi_0$ of the Schr\"odinger equation moves out of the bounded spectral domain for the operator $B$, and in particular the photon coordinate and wave vectors become more and more parallel.)
\bigskip
\section{Conclusion}
Apart from the vacuum polarization, the non-relativistic QED provides a good qualitative description of the physical phenomena related to the interaction of quantized electrons and nuclei and the electro-magnetic field. (Though construction of the scattering theory is not yet completed and the correction to the gyromagnetic ratio is not established, it is fair to conjecture that while both are difficult problems, they should go through without a hitch.)
The quantitative results though are still missing. Does the free parameter, $m$ (or $\kappa$), suffice to give a good fit with the experimental data say on the radiative corrections? Another important open question is the behaviour of the theory in the ultra-violet cut off. % In particular, with an appropriate dependence, $m=m (m_\el, \kappa)$, of the bare mass on the physical one and on the ultra-violet cut-off (as say given by the mass renormalization) in the quantum Hamiltonian, do the physical quantities still depend on $\kappa$?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{End of Proof}\label{sec:endproof}
\DETAILS{
\section{Open Problems}
%Connection between the ground state and resonance eigenvalues to poles of the scattering matrix;
Compute With $m=m(m_\el, \kappa)$ in the quantum Hamiltonian
$$$$
Minimal and maximal velocity of photons;
$$$$
Asymptotic completeness; %($\|N\psi_t \| \le C$);
$$$$
Bohr photon frequency laws. }
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Comments on Literature}\label{sec:liter}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
These lectures follow the papers \cite{sig, fgs2, AFFS}, which in turn extend \cite{bfs1, bfs2, bcfs1}.
The papers \cite{sig, fgs2, AFFS} use the smooth Feshbah-Schur map (\cite{bcfs1, GriesemerHasler1}), which is much more powerful (see Appendix \ref{sec:sfm}), while in these lectures we use, for simplicity, the the original, Feshbach-Schur map (see \cite{bfs1, bfs2}), which is simpler to formulate.
The self-adjointness of $H$ is not difficult and was proven in \cite{ bfs3} for sufficiently small coupling constant ($g$) and in \cite{Hiroshima4} (see also \cite{Hiroshima2a}), for an arbitrary one.
Theorems 4.1 and 4.2 were proven in \cite{bfs1, bfs2, bfs3} for 'confined particles' (the exact conditions are somewhat technical) and in the present form in \cite{sig}.
%The binding results are given in \cite{bfs3, Griesemer}.
The results of \cite{ bfs3} on existence %(and uniqueness)
of the ground state were considerably improved in \cite{ Hiroshima1,
Hiroshima2, Hiroshima3, HiroshimaSpohn, AraiHirokawa, LMS2} (by compactness techniques) and
\cite{BachFroehlichPizzo1} (by multiscale techniques), with the sharpest result given in \cite{GriesemerLiebLoss, LiebLoss3}. %\footnote{
(The papers
\cite{bfs3, Hiroshima1, GriesemerLiebLoss,
LiebLoss1, LiebLoss2, LiebLoss3, HiroshimaSpohn, LMS2} include the interaction of the spin with magnetic field %- $\sum\limits_{j=1}^n \frac{g}{2m_j}\sigma_j \cdot B(x_j)$ -
in the Hamiltonian.)
%The existence of the resonances was proven so far only for confined potentials (see \cite{BachFroehlichSigal1998a,BachFroehlichSigal1998b} and, for a book exposition, \cite{GustafsonSigal}). Lieb-Loss, Bach-Fr\"ohlich- Sigal, Hiroshima, Hiroshima-Spohn, Arai-Hirokawa (by compactness techniques) and Bach-Fr\"ohlich-Pizzo (in a constructive way), with the sharpest one due to Griesemer, Lieb and Loss (\cite{gll}).
%The existence of the resonances, radiative corrections and life-times:Bach-Fr\"ohlich-Sigal ( for confined potentials or molified infra-red behaviour).
Related results:
\begin{enumerate}
\item[\nonumber] The asymptotic stability of the ground state (local decay, see Section \ref{sec:relat-res}):
\cite{bfss, fgs1, fgs3}.
%Fr\"ohlich- Griesemer-Sigal, Bach-Fr\"ohlich-Sigal,Bach-Fr\"ohlich-Sigal-Soffer. $\bullet$
\item[\nonumber] The survival probabilities of excited states (see \eqref{ResonDecay}): %which is related to the metastability of the resonances: Hasler-Herbst-Huber, Abou Salem-Faupin-Fr\"ohlich-Sigal
\cite{bfs3, HaslerHerbstHuber, AFFS}.
%(see for related results and Bach-Fr\"ohlich-Sigal, M\"uck, King.
\item[\nonumber] Atoms with dynamic nuclei: %Faupin, Amour- Gr\'ebert - Guillot, Loss-Miyao-Spohn (
\cite{faupin, AGG, LMS}.
\item[\nonumber] Analyticity of the ground state eigenvalues in parameters and asymptotic expansions (see Section \ref{sec:relat-res}): \cite{bfs3, BachFroehlichPizzo1, BachFroehlichPizzo3, GriesemerHasler2, HaslerHerbst1, HaslerHerbst2,HaslerHerbst3}. %Griesemer-Hasler.
%(see Hainzl - Seiringer, Hiroshima-Spohn, Lieb-Loss, Bach-Chen-Fr\"ohlich-Sigal, Chen).
\item[\nonumber] Existence of the ionization threshold (see Section \ref{sec:relat-res}): \cite{Griesemer}.
\item[\nonumber] Resonance poles (see Section \ref{sec:relat-res}): \cite{
%, HaslerHerbstHuber,
AFFS} (see also \cite{bfs3}).
\item[\nonumber] Self-energy and binding energy: \cite{HiroshimaSpohn1, LiebLoss1, LiebLoss2, LiebLoss3, H1, bcv, cvv, HVV, H2, HS, HHS, CEH, bv, bcvv}.
%\item[\nonumber] Local decay: \cite{fgs1, fgs2, fgs3}.
\item[\nonumber] Electron mass renormalization: %(see Appendix \ref{sec:mass-renorm}):
\cite{HS, HiroshimaSpohn2, bcfs2, ch, FroehlichPizzo}. %LiebLoss3, LiebLoss1,
\item[\nonumber] One particle states: \cite{fr1, fr2, cfp1, cfp2, FroehlichPizzo}.
\item[\nonumber] Scattering amplitudes: \cite{BachFroehlichPizzo2}. %Bach-Fr\"ohlich-Pizzo.
\item[\nonumber] Semi-relativistic Hamiltonians: \cite{BDG, MiyaoSpohn, MatteStockmeyer, KMS, Stockmeyer, HiroshimaSasaki}.
\end{enumerate}
%\bigskip \noindent Among other aspects of non-relativistic QED, we mention the photo-electric effect (\cite{bkz, GriesemerZenk}), scattering theory (\cite{FroehlichGriesemerSchlein1,FroehlichGriesemerSchlein2, FroehlichGriesemerSchlein3}) and stability of matter (\cite{BugliaroFroehlichGraff, FeffermanFroehlichGraff, LiebLoss1}).
%\DETAILS{
Other aspects of non-relativistic QED:
\begin{enumerate}
\item[\nonumber] Photo-electric effect: \cite{bkz, GriesemerZenk}.
%\item[\nonumber] Anomalous magnetic moment of electron: \cite{pst}. %Renormalization of the electron gyromagnetic ratio:
\item[\nonumber] Scattering theory:
\cite{FroehlichGriesemerSchlein1,FroehlichGriesemerSchlein2, FroehlichGriesemerSchlein3}.
\item[\nonumber] Stability of matter:
\cite{BugliaroFroehlichGraff, FeffermanFroehlichGraff, LiebLoss1}.
%\item[\nonumber] Positive temperatures: \cite{bfs4, MMS1, MMS2}.
\end{enumerate} %}
%
\bigskip
\noindent There is an extensive literature on related models, which we do not mention here: Nelson model describing a particle linearly coupled to a free massless scalar field (phonons), semi-relativistic models, based on Dirac equation, and quantum statistics (open systems, positive temperature) models. (In the latter case, %translated to the language of
one deals with Liouvillians, rather than Hamiltonians, on positive temperature Hilbert spaces. %, but the techniques sketched above still apply.
The main results above were proved simultaneously for the QED and Nelson models and extended, at least partially, to positive temperatures.) %\cite{ghps}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%This model describes, in particular, the problem of radiation --- emission and
%absorption of radiation by systems of matter, such as atoms and
%molecules --- as well as other processes of interaction of quantum
%radiation with matter (e.g. photoeffect). It has been extensively studied in the last decade, see the papers \cite{Arai1999,Arai2,AraiHirokawa,
%BachFroehlichSigal1998a,BachFroehlichSigal1998b,
%BachFroehlichSigal1999,BachFroehlichSigalSoffer1999,
%BarbarouxDimassiGuillot1,BarbarouxDimassiGuillot2,Betz,Chen2001,
%DerezinskiJaksic2001,DerezinskiGerard,DimassiGuillot,FroehlichGriesemerSchlein1,
%BachChenFroehlichSigal2003, BachChenFroehlichSigal2006,
%FroehlichGriesemerSchlein2, FroehlichGriesemerSchlein3,
%GergescuGerardMoeller1,GergescuGerardMoeller2,HirokawaHiroshimaSpohn,Hirokawa1,
%Hirokawa2,Hiroshima1,Hiroshima2,Hiroshima3,
%Hiroshima4,Hiroshima5,Hiroshima6,Hiroshima7,HiroshimaSpohn,HuebnerSpohn1,
%HuebnerSpohn2,Skibsted,Spohn, GriesemerLiebLoss,
%BachFroehlichPizzo1, Pizzo2003, Pizzo2005,
%FroehlichGriesemerSigal2007a, Faupin2007} and references therein for
%a partial list of contributions.
%The existence of the ground state for full vector model was proven
%by a compactness technique in \cite{BachFroehlichSigal1999,
%AraiHirokawa, Hiroshima1, Hiroshima2, Hiroshima3, GriesemerLiebLoss,
%Hiroshima7} and in a constructive way, in
%\cite{BachFroehlichPizzo1}. A computational algorithm for the ground
%state energy was designed in \cite{BachFroehlichPizzo1}. Thus the
%The main new result of this work is the existence of resonances and an algorithm for their computation.
%We also give a simple but effective extension of the spectral
%renormalization group technique and prove estimates of the
%meromorphic continuation of the resolvent near the resonances needed
%in the study of the resonance dynamics, see below.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Hamiltonian of the Standard Model} %Rescaling and Conditions on Potentials}
\label{sec:pot}
In this appendix we demonstrate the origin the quantum Hamiltonian $H_g$ given in \eqref{Hsm}. To be specific consider %a system of charged particles First, we consider the Hamiltonian $H_g$ for
an atom or molecule with $n$ electrons interacting with radiation field. %For simplicity we assume the nuclei to be infinitely heavy (the general case we be considered below).
In this case the Hamiltonian of the system in our units is given by
\begin{equation}\label{Hsm'}
H(\al)=\sum\limits_{j=1}^n{1\over 2m}
(i\nabla_{x_j}-\sqrt{\alpha}A_{\chi'}(x_j))^2+\al V(x)+H_f,
\end{equation}
%the Bohr radius $r_{bohr}=(m\alpha)^{-1}$, and the energy, in the units of
%$m\alpha^2=2(Rydberg)$, where $m$ is the electron mass. FIND THE
%RIGHT DISTANCE AND ENERGY SCALES, SOMETHING LIKE $\hbar/mc$ and
%$mc^2$ BUT WITHOUT $c$???
%In the physical units (??) the quantum Hamiltonian of the total
%system
%
%of the non-relativistic quantum electrodynamics
%
%is of the form \eqref{eq1} but with $g=e/c$ and with
%Then, in the natural units, $g= \sqrt{\alpha}$ and
where $\al V(x)$ is the total Coulomb potential of the particle system, $m$ is the electron bare mass, $\alpha =\frac{e^2}{4\pi \hbar c}\approx {1\over 137}$ (the
fine-structure constant) %, is proportional to $\alpha$.
and $A_{\chi'}(y)$ is the original vector potential with the ultraviolet cut-off $\chi'$.
%(Note that in our units, the electron charge is equal to $-\sqrt{\alpha}$, %\(e=-\sqrt{\alpha})$,
%and the distance, time and energy are measured in units of $\hbar/m_\el c =3.86 \cdot 10^{-11}cm,\ \hbar/m_\el c^2 =1.29 \cdot 10^{-21} sec$ and $m_\el c^2 = 0.511 MeV$, respectively.)
%(natural units).
Rescaling $x \rightarrow \alpha^{-1} x$ and $k \rightarrow
\alpha^2 k$, we arrive at the Hamiltonian \eqref{Hsm},
% \begin{equation}\label{Hsm'} H_g=\sum\limits_{j=1}^n{1\over 2m_j} (i\nabla_{x_j}-gA(x_j))^2+ V(x)+H_f, \end{equation}
where $g:= \alpha^{3/2}$ %, $V(x)$ of the order $O(1)$ %$A(x)$ replaced by $A'(x)$, where
and $A(y) = A_{\chi}(\alpha y)$, %|_{\chi(k) \rightarrow \chi'(k)}$,
with $\chi( k):=\chi'(\alpha^2 k)$. %\footnote{} %(see Bach-Fr\"ohlich-Sigal). %Hence the new cut-off momentum scale satisfies $$1 \ll k_c \ll \alpha^{-2},$$
%which is easily accommodated by our estimates (e.g. we can have $k_c
%=O(\alpha^{-1/3})).$ %Dropping the prime in the vector potential and
%assuming for simplicity chi fixed we arrived at the \eqref{Hsm}
%with ... .
%
%(and is in particular a positive-homogeneous function of the degree $-1$)
%To pass to dimensionless units we rescale this original Hamiltonian
%appropriately. The result is the expression \eqref{Hsm} with $g:=
%\alpha^{3/2}$ (see \cite{BachFroehlichSigal1999}). Then
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Units}
%We now describe the standard model of
%non-relativistic QED. %and the corresponding quantum Hamiltonian.
%We now describe the notions involved in Eqn \eqref{Hsm}.
%We use the units in which the Planck constant divided by $2\pi$, the speed of light and the electron mass are equal to $1\ (\ \hbar=1$, $c=1$ and $m=1$).
After that %we drop the prime in the vector potential $A'(x)$ and the ultraviolet cut-off $\chi'(x)$ (see a discussion of the latter in Subsection \ref{subsec:UV}). %assume for simplicity $\chi$ fixed. Finally,
we relax the restriction on $V(x)$
by allowing it to be a standard generalized $n$-body potential (see Subsection \ref{sec:model}). Note that though this is not displayed, $A(x)$ does depend on $g$. This however does not effect the analysis of the Hamiltonian $\hsm$. (If anything, this makes certain parts of it simpler, as derivatives of $A(x)$ bring down $g$.)
$$****$$
In order not to deal with the problem of center-of-mass motion
which is not essential in the present context, we assume that either
some of the particles (nuclei) are infinitely heavy (a molecule in the
Born-Oppenheimer approximation), or the system is
placed in a binding, external potential field. (In the case of the
Born-Oppenheimer molecule, the resulting $V(x)$ also depends on
the rescaled coordinates of the nuclei, but this does not effect our analysis except of making the complex deformation of the particle system more complicated (see \cite{HunzikerSigal}).) This means that the operator $H_p$ has isolated eigenvalues below its essential
spectrum. The general case is considered below. %However, we expect that the techniques we discuss here can be extended to translationally invariant particle systems. %(see AGG, LMS, Faupin2007).
In order to take into account the particle spin we change the state space the particle system to $\cH_\at=\otimes_1^nL^2(\R^{3}, \C^2)$ (or the antisymmetric in identical particles subspace thereof),
and the standard quantum Hamiltonian on $\cH=\cH_{p}\otimes\cH_{f}$, is taken to be (see e.g. \cite{Cohen-TannoudjiDupont-RocGrynberg1, Cohen-TannoudjiDupont-RocGrynberg2, sakurai})%(after rescaling)
\begin{equation} \label{Hsm-spin}
H_{spin}=\sum\limits_{j=1}^n{1\over 2m}
[\sigma_j\cdot(i\nabla_{x_j}-g\Af(x_j))]^2+V(x)+H_f,
\end{equation}
where $\s_j:=(\s_{j1}, \s_{j2}, \s_{j3}),\ \s_{ji}$ are the Pauli matrices of the $j-$th particle and the identity operator on $\C^{2n}$ is omitted in the last two terms. It is easy to show that
\begin{equation} \label{spin}
[\sigma\cdot(i\nabla_{x}-g\Af(x))]^2=(i\nabla_{x}-g\Af(x))^2+ g\sigma \cdot B(x).
\end{equation}
where $B(x):=\textrm{curl} \Af(x)$ is the magnetic field. As a result the operator \eqref{Hsm-spin} can be rewritten as
\begin{equation} \label{Hsm-spin}
H_{spin}=H\otimes \one+\one\otimes g\sum\limits_{j=1}^n{1\over 2m}
\sigma_j\cdot B(x_j).
\end{equation}
For the semi-relativistic Hamiltonian, the non-relativistic kinetic energy $\frac{1}{2m}|p|^2$ is replaced by the relativistic one, $\sqrt{|p|^2+m^2}$ or $\sqrt{(\sigma\cdot p)^2+m^2}$.
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Translationally Invariant Hamiltonians} \label{sec:transl-Ham}
If we do not assume that the nuclei are infinitely and there are no external forces acting on the system, then the Hamiltonian \eqref{Hsm} is translationally symmetric. This %Typically symmetries of a physical system lead to conservation laws (Noether theorem).
%In our case, of a particular importance is space-time
%We consider the space translational symmetry, which
leads to conservation %laws of the energy, $H_{e\chi}$, and
of the total momentum (a quantum version of the classical Noether theorem). Indeed,
%\[P=\sum_ip_i+P_f,\ \quad \mbox{where}\ \quad P_f=\int :E\wedge B:\ .\]
\index{translation ! invariance}
%A crucial role in what follows will be played by the fact that
the system of particles interacting with the quantized electromagnetic fields %we consider here
is invariant under translations of the particle coordinates, $\ux\to \ux+\uy$, where $\uy=(y, \dots, y)$ ($n-$ tuple) and the fields, $A(x)\to A(x-y)$, i.e. $\hsm$ commutes with the translations
%
\DETAILS{\begin{equation}\label{Ty}
T_{y}: \Psi(\ux, A)\to \Psi(\ux+\uy, t_{y}A), %\Psi(x)\ra e^{ iy\cdot P_\re} \Psi(x+y); \oplus_n\Psi_n(x, k_1, \dots, k_n)\ra \oplus_n e^{ iy\cdot (k_1+\dots k_n)} \Psi_n(x+y, k_1, \dots, k_n).
\end{equation}
where %$\uy=(y, \dots, y)$ ($n-$ tuple) and
$(t_{y}A)(x, A)=A(x-y)$. (We say that $\hsm$ is \textit{translation invariant}.) Indeed,
we use \eqref{Ty} and the definitions of the operators $A(x)$ and $E(x)$, to obtain $T_y A(x) = (t_{y}A)(x) T_y$ and $T_y E(x) = (t_{y}E)(x) T_y$, which, due to the definition of $H_{e\chi}$, give %We also note that $T_y$ commutes with $H$,
\begin{equation*} T_y \hsm = \hsm T_y, \end{equation*}
Note that in the Fock space representation this gives: $T_{y}: \oplus_n\Psi_n(\ux, k_1, \dots, k_n)$\\ $\to \oplus_n e^{ iy\cdot (k_1+\dots k_n)} \Psi_n(\ux+\uy, k_1, \dots, k_n)$ and therefore can be rewritten as}
%
%
$T_{y}: \Psi(\ux)\to e^{ iy\cdot P_\re} \Psi(\ux+\uy)$, where $ P_\re$ is the momentum
operator associated to the quantized radiation field,
\[\Pf=\sum_\lambda\int dk \, k \, a_\lambda^*(k)a_\lambda(k).\] %$P_\re := \int_{\mathbb{R}^3} k a^*(k) a(k) \d k.$
%Indeed, we check this for the spatial translations.
It is straightforward to show that $T_y$ are unitary operators and that they satisfy the relations
$ T_{x+y} = T_xT_y,$
and therefore $y\to T_{y}$ is a unitary Abelian representation of $\R^3$.
Finally, we observe that the group $T_y$ is generated by the total momentum operator, $\Ptot$, of the electrons and the photon field: $T_y= e^{ iy\cdot \Ptot}$. Here $\Ptot$ is the selfadjoint operator on $\cH$, given by %$ P_\el + P_\re$,
\eqn
\Ptot \, := \, %P_\el
\sum_ip_i\otimes\1_f \, + \, \1_{el}\otimes P_f
\eeqn
where, as above, $p_j:=-i\nabla_{x_j}$, the momentum of the $j-$th electron and $ P_f$ is the field momentum given above. %associated to the total momentum of the system containing the electron and the photon field
%\begin{equation} P_\re := \int_{\mathbb{R}^3} k a^*(k) a(k) \d k.\end{equation}
Hence $[\hsm, \Ptot]=0$.
%%%%%%%%%%%%%%%%
\medskip
%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Fiber decomposition with respect to total momentum}\label{subsec:fiber}
%\subsection{Direct Integral Representation}
%For each $P \in \R^3$, We define the Hilbert space
Let $\cH$ be the direct integral
%\begin{equation*}
$\cH = \int_{\R^3}^\oplus \cH_P dP, $ %\end{equation*}
with the fibers $\cH_P := L^2(X)\otimes \cF$, where $X:=\{x\in {\mathbb R}^{3n}\ |\ \sum_i m_ix_i=0\}\simeq{\mathbb R}^{3(n-1)}$, (this means that $\cH = L^2(\R^3, dP; L^2(X)\otimes \cF)$) and %\begin{enumerate}[(a)] \item \end{enumerate}
define $U : \cH_\el \otimes \cH_{\re} \to \cH$ on smooth functions with compact domain by the formula
\begin{equation}\label{U}
(U \Psi)(\ux', P) = \int_{\mathbb{R}^3}e^{i(P-P_\re)\cdot x_{cm}}\Psi(\ux'+\ux_{cm}) \d y, %\int_{y \in \R^3} \chi_k^{-1}(y) T_y \Psi(x) dy,
\end{equation}
%where $\chi_k$, $k \in \R^3$, are the characters of given explicitly by %\begin{equation*}
% $\chi_k(y) = e^{ik\cdot y}.$ %\end{equation*}
where $\ux'$ are the coordinates of the $N$ particles in the center-of-mass frame and $\ux_{cm}=(x_{cm}, \dots, x_{cm})$ ($n-$ tuple), with $ x_{cm}=\frac{1}{\sum_im_i}\sum_im_ix_i$, the center-of-mass coordinate, so that $\ux=\ux'+\ux_{cm}$. Then $U$ extends uniquely to a unitary operator (see below). Its converse is written, for $\Phi (\ux', P)\ \in L^2(X)\otimes \mathcal{F}$, as
\begin{equation}\label{Uinv} %{Psidecomp}
(U^{-1}\Phi)(\ux)=\int_{\mathbb{R}^3} e^{-\i x_{cm}\cdot(P-P_\re)}\Phi (\ux', P) \d P.
\end{equation}
The functions $\Phi (\ux', P)\ =(U \Psi)(\ux', P)$ %\int_{\mathbb{R}^3} dye^{i(P-P_\re)\cdot y}\Psi(y)$
are called fibers of $\Psi$. One can easiely prove the following
%
%
\DETAILS{ For $k \in \R^3$, let $H_{ k}$ be the operator $H$ acting on $\cH_k$ with domain consisting of those $v \in \cH_k \cap H^2$ such that $\Psi$
satisfies the boundary conditions $T_y \Psi(x) = \chi_k(t) \Psi(x)$. Then
\begin{equation}
\label{K-decomposition}
UH U^{-1} = \int_{\R^3}^\oplus H_{ k} dk.
\end{equation}
and
\begin{equation}
\label{KKkspecrelat}
\sigma(H) = \bigcup_{k\in\R^3} \sigma(H_{ k}).
\end{equation}}
%
%
%
%
\DETAILS{\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Spectral decomposition with respect to total momentum}
We consider the direct integral decomposition
\eqn
\cH \, = \, \int^\oplus dp \, \cH_p \,,
\eeqn
with respect to $\Ptot$, where each fiber $\cH_p\cong {\rm Ker}(\Ptot-p)$ is isomorphic to $\Fo$.
It corresponds to the natural isomorphism
\eqn
\cH \, \cong \, L^2(\R^3,\Fo) \,
\eeqn
which we will use in the sequel.
%obtained from
%\eqn
% (x,\Psi) \, \mapsto \, \iota_{x}\Psi \,\in \Fo
%\eeqn
%where $\iota_x$ is the evaluation map, for $x\in\R^3$.
For $\psi\in\cH$, we define the Fourier transform with respect to the electron variable,
\eqn
\widehat\psi(p) \, = \, \int dx \, e^{-i(p-\Pf)x} \psi(x)
\eeqn
with inverse transform
\eqn
\check\phi%^\vee(x)
\, = \, \int dp \, e^{i(p-\Pf)x}\phi(p) \,.
\eeqn}
%
%
\begin{lemma}
{\em The operations \eqref{U} %{eq-FT-def-1}
and \eqref{Uinv} %{eq-FTinv-def-1}
define unitary maps $L^2({\mathbb R}^{3n})\otimes \cF\rightarrow
\cH$ and $\cH\rightarrow L^2({\mathbb R}^{3n})\otimes \cF$, and are mutual inverses.}
\end{lemma}
\DETAILS{\prf %We denote $\widehat\Psi(P) \, = \, (U \Psi)(P)$ and $ \check\Phi (x)%^\vee(x) \, = \, (U^{-1} \Phi)(x)$.
By density, we may assume that $\Psi$ is a $C_0^\infty$ in $\ux$. %generalized Schwartz class function, \eqn {\mathcal S}_{\Fo} \, :=\, \big\{ \, \Psi\in L^2({\mathbb R}^{3})\otimes \cF \, \big| \, \sup_x\|x^\alpha\partial_x^\beta\Psi\|_{ \Fo} \, < \, \infty \; \; \; , \; \; \; \forall \alpha,\beta\in\N_0^3 \, \big\} \,. \eeqn
Then, it follows from standard arguments in Fourier analysis that
\eqn\label{eq-FT-def-1}
%(\widehat\psi)^\vee(x)
(U^{-1}U\Psi)(\ux) &=&
\int \d P \, e^{-i(P-\Pf)x_{cm}} \int dy \, e^{i(P-\Pf)\cdot y} \, \Psi(\ux' +y)
\nonumber\\
&=&
\int dy \, \int \d P \, e^{-iP\cdot(x_{cm}-y)}e^{i\Pf\cdot(x_{cm}-y)} \, \Psi(\ux'+y)
\nonumber\\
&=&
\int dy \, \delta(x_{cm}-y) \, e^{i\Pf\cdot(x_{cm}-y)} \, \Psi(\ux'+y)
\nonumber\\
&=&\Psi(\ux) \,.
\eeqn
On the other hand, for $\psi\in {\mathcal S}_{\Fo} $,
\eqn\label{eq-FTinv-def-1}
(UU^{-1}\Phi)(\ux', P)
&=&
\int dy \, e^{i(P-\Pf)y} \int dq \, e^{i(q-\Pf)y} \, \Phi(\ux', q)
\nonumber\\
&=&
\int dq \, \int dy \, e^{i(p-q)y} \, \Phi(\ux', q)
\nonumber\\
&=&
\int dq \, \delta(P-q) \, \Phi(\ux', q)
\nonumber\\
&=&\Phi(\ux', P) \,.
\eeqn
From the density of %${\mathcal S}_{\Fo}$ in $L^2({\mathbb R}^{3})\otimes \cF$,
$C_0^\infty$ in $\ux$ functions, we infer that \eqref{eq-FT-def-1}
and \eqref{eq-FTinv-def-1} define bounded maps %$\cH\rightarrow\cH$, and
which are mutual inverses. Unitarity can be checked easily.
\endprf}
%
\DETAILS{We observe that
\eqn
(\Ptot\psi)^{\widehat{\;}}(p) & = & \int dx \, e^{i(p-\Pf)x} (-i\nabla_x+\Pf)\psi(x)
\nonumber\\
&=& \int dx \, \big( \, (-i\nabla_x+\Pf)e^{i(p-\Pf)x}\,\big) \, \psi(x)
\nonumber\\
&=&p \, \widehat\psi(p) \,.
\eeqn
Hence, $\widehat\psi(p)\in\cH_p\cong {\rm Ker}(\Ptot-p)$, for any $p\in\R^3$.}
%In particular, \eqn \psi(x) \, = \, \int dP_{\Ptot}(p) \, e^{i(p-\Pf)x}\widehat\psi(p)\eeqn where $dP_{\Ptot}$ denotes the spectral measure associated to $\Ptot$.
%
Since $\hsm$ commutes with $\Ptot$, it follows that it admits the fiber decomposition
\begin{equation}
U \hsm U^{-1} = \int_{\mathbb{R}^3}^{\oplus} \hsm(P) \d P,
\end{equation}
where the fiber operators $\Hn(P)$, $P \in \mathbb{R}^3$, are self-adjoint operators on $\mathcal{F}$. Using $a(k)e^{-\i y\cdot P_\re}=e^{-\i y\cdot (P_\re+k)}a(k)$ and $a^*(k)e^{-i y\cdot P_\re}=e^{-i y\cdot (P_\re-k)}a^*(k)$, we find $\nabla_y e^{i y\cdot(P-P_\re)}\Af(x'+y) e^{\i y\cdot(P-P_\re)} =0$ and therefore
\begin{equation}
\Af(x) e^{i y\cdot(P-P_\re)} = e^{i y\cdot(P-P_\re)}\Af(x-y).
\end{equation}
Using this and \eqref{Uinv}, we compute $\hsm (U^{-1}\Phi)(x)=\int_{\mathbb{R}^3} e^{i x\cdot(P-P_\re)}\hsm(P)\Phi (P) d P$, where $\hsm(P)$ are Hamiltonians on the space fibers $\cH_P := \cF$ given explicitly by
\begin{equation}
\hsm(P) = \sum_j\frac{1}{2m_i}\big ( P - P_\re -i\nabla_{x_j'} -e_i\Af(x_j')\big)^2 + V_{\rm coul}(\ux')+H_\re
\end{equation}
where $x_i'=x_i - x_{cm}$ is the coordinate of the $i-$th particle in the center-of-mass frame. Now, this hamiltonian can be investigated similarly to the one in \eqref{Hsm}. %$\Af:=\Af(0)$. Explicitly, $\Af$ is given by \begin{equation}\label{A0} \Af \, = \, \sum_{\lambda}\int \d k \, \frac{\kappa(|k|)}{|k|^{1/2}} \, \e_\lambda(k) \,\{ \, a_\lambda(k) \, + \, a^*_\lambda(k) \, \}. %\frac{ \kappa(k) }{ |k|^{\frac{1}{2}}}\end{equation}
%We define $E(P) := \inf \sigma( H(P) )$. If $g=0$, and if $|P|$ is less than the bare electron mass (equal to 1 in the units used in this paper), then $E(P) = P^2 /2$ is an \emph{eigenvalue} of $H(P)$. If $|P| > 1$, then $E(P) = |P| - 1/2$, and $E(P)$ is \emph{not} an eigenvalue of $H(P)$.
%
%
\DETAILS{For an observable $A$ on $ \mathcal{H}_{\mathrm{el}} \otimes \mathcal{H}_\re$, we consider the operator $UA U^{-1}$, acting on the space $\cH = \int_{\R^3}^\oplus \cH_P dP$. It can be written as
$$(UA U^{-1}\Phi)(P)=\int_{\mathbb{R}^3} A_{PQ}\Phi(Q)dQ,$$
for some operator-function $A_{PQ}$ on $\cF$. If $A\in \cA$, then $A_{PQ}\in {\frak W}(L^{2}_0)$.
We write $UA U^{-1}=\int A_{PQ} dPdQ.$ Note that the evolution, $\al^t(A)$, of $A$ is then given by
\begin{equation}
\al^t(A)= \int_{\mathbb{R}^3\times \R^3}dPdQ\alpha_{PQ}^t (A_{PQ}),
\end{equation}
where $\alpha_{PQ}^t (A_{PQ}):=e^{itH_P}A_{PQ} e^{-itH_Q}$. If an observable $A$ is translation invariant, then we can write it as a fiber integral, $A = \int_{\mathbb{R}^3}^{\oplus} A_P \d P.$ If $A$ is translation invariant, then so is the observable $\al^t(A)$ and
\begin{equation}
\al^t(A)=\int_{\mathbb{R}^3} \al^t_{P}(A_{P})dP,\ \mbox{where}\ \al^t_{P}(A_{P}):=e^{itH_P}A_{P} e^{-itH_P}.
\end{equation}
We write $\al^t=\int \al^t_{P}dP$. An example of a translation invariant observable is the particle momentum $P_\el=-i\nabla=i[H, x]$. In this case we have $P_\el = \int_{\mathbb{R}^3}^{\oplus} (P-P_\re) \d P = \int_{\mathbb{R}^3}^{\oplus} P \d P-P_\re.$}
%
%
%
%
\DETAILS{\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Next, we consider the Hamiltonian \eqref{eq-Hn-def-1}.
Clearly, $[\Hn,\Ptot]=0$, by translation invariance.
Then,
\eqn
(\Hn \psi)^{\widehat{\;}}(p) \, = \, \Hn(p) \widehat\psi(p)
\eeqn
where $\Hn(p) = \Hn\big|_{\cH_p}$ is the fiber Hamiltonian associated to total momentum $p$. Let $\Af:=\Af(0) $. We compute
\eqn
\Hn(p) =\frac12( \, p \, - \, \Pf \, - \, \c \Af \, )^2
\, + \, H_f \,
\eeqn
%We introduce the direct integral notation
%\eqn
% \int^\oplus dp \, := \, \int dP_{\Ptot}(p) \,,
%\eeqn
%where $dP_{\Ptot}$ denotes the spectral measure associated to $\Ptot$.
We then have that
\eqn
\Hn \, = \, \int e^{i(p-\Pf)x}\Hn(p)e^{-i(p-\Pf)x}dP_{\Ptot}(p)
\eeqn
which is the spectral representation of $\Hn$
with respect to the spectral measure associated to $\Ptot$.}
%
%and observe that $\Hn$ is unitarily
%equivalent to the operator
%\eqn
% \lefteqn{e^{-i\Pf x}\Hn e^{i\Pf x}}
% \\
% &&
% \, = \,
% \frac12( \, i\nabla_x \otimes \1_f \, - \, \1_{el}\otimes\Pf \, - \, \g \Af(0) \, )^2
% \, + \, \1_{el} \otimes H_f \,.
% \nonumber
%\eeqn
%Notably, the operator $i\nabla_x\otimes\1_f$ commutes with all
%other operators on the rhs.
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Local Decay?}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of Theorem \ref{thm:isospF}} \label{sec:isosprel}
In this appendix we \textit{omit the subindex} $\rho$ at $\chi_{\rho}$ and $\bchi_{\rho}$, and \textit{replace the subindex} $\rho$ in other operators by the subindex $\chi$. Moreover, we replace $H-\lam$ by $H$. Though $\chi$ and $\bchi$ we deal with are projections, we often keep the powers $\chi^2$ and $\bchi^2$, which occur often below, having in mind showing possible generalization to $\chi$ and $\bchi$ which are 'almost (or smooth) projections' satisfying $\chi^2+\bchi^2=1$ (see Appendix \ref{sec:sfm}).
First we note that the relation between $\psi$ and $\varphi$ in Theorem \ref{thm:isospF} (ii) is $\vphi=\chi\psi,\ \psi=Q_{\chi}( H)\vphi,$
and between $H^{-1}$ and $F_{\chi}(H)^{-1}$ in (iv) is
%then so does the other and these inverses are related by
%\medskip $ $
\begin{equation} \label{resolvrel}
H^{-1} = Q_{\chi} ( H) \: F_{\chi}
H^{-1} \: Q_{\chi} ( H)^\# + \; \bchi \, H_{\bchi}^{-1} \bchi ,
\end{equation}
%\medskip
where $H_{\bchi}:=\bchi_{\rho}H\bchi_{\chi}$ and $Q_{\chi} ( H)$ and $Q_{\chi} ( H)^\#$ are the operators, given by
\medskip
$Q_{\chi} (H) := \chi \: - \: \bchi \, H_{\bchi}^{-1}
\bchi H \chi ,$
\medskip
$Q_{\chi} ^\#(H) := \chi \: - \: \chi H \bchi \,
H_{\bchi}^{-1} \bchi .$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{itemize} \item[(i)] ; % i.e. $H_{g} - \lambda$ is bounded invertible on
%$\cH$ if and only if $F_{\rho} (H_{g} - \lambda)$ is bounded
%invertible on $\Ran\, \chi$; \item[(ii)] %$\psi \in \cH \setminus \{0\}$ solves
%
\DETAILS{\item[(iii)] If $\vphi
\in \Ran\, \pi \setminus \{0\}$ solves $F_{\rho} (H - \lambda)
\, \vphi = 0$ then $\psi := Q_{\pi} (H - \lambda) \vphi \in \cH
\setminus \{0\}$ solves $H\psi = \lambda \psi$;}
%
%The multiplicity of the spectral value $\{0\}$ is conserved in the sense that
%If one of the inverses,
%\hspace{8mm}
%\\ \label{eq-II-7}
%and $$ F_{\rho} (H_{g}- \lambda)^{-1} = \pi \, (H-
%\lambda)^{-1} \, \pi \; + \; \bpi \, (H- \lambda)^{-1} \bpi .
%$$
% \hspace{8mm}
%\end{eqnarray}
%\end{itemize} %\noindent
\emph{Proof of Theorem \ref{thm:isospF}. }
Throughout the proof we use the notation $F := F_\chi(H)$,
$Q := Q_\chi(H)$, and $Q^\# := Q_\chi^\#(H)$. Note that (i) ($ 0 \in \rho(H ) \Leftrightarrow 0 \in \rho(F_{\chi}(H))$) follows from (iv) ($H^{-1}$ exists $ \Leftrightarrow$ $F_{\chi}(H)^{-1}$
exists) and (iv) follows from \eqref{resolvrel}, so we start with the latter.
\noindent
\textbf{Proof of \eqref{resolvrel}.}
The next two identities,
%
\begin{equation} \label{eq-II-11}
H \, Q \ = \ \chi \, F
\hspace{8mm} \mbox{and} \hspace{8mm}
Q^\# \, H \ = \ F \, \chi ,
\end{equation}
%
are of key importance in the proof. They both derive from a simple
computation, which we give only for the first equality in (\ref{eq-II-11}). We observe the relations
%
\begin{equation} \label{Hchi} %{eq-II-9}
H \, \chi \ = \ \chi \, H_\chi \; + \; \bchi^2 \, H \chi ,
\hspace{8mm} \mbox{and} \hspace{8mm}
H \, \bchi \ = \ \bchi \, H_\bchi \; + \; \chi^2 \, H \bchi ,
\end{equation}
%
which follow from $\chi^2 + \bchi^2 = \one$. Now, using the definition of the operator $Q$ and the relations \eqref{Hchi}, we obtain
%
\begin{eqnarray} \label{eq-II-12}
H \, Q
%& = & H \chi \, - \, H \bchi \, H_\bchi^{-1} \bchi H \chi \nonumber \\
& = &
\chi H_\chi \, + \, \bchi^2 H \chi \, - \,
\big( \bchi H_\bchi + \chi^2 H \bchi \big)
\, H_\bchi^{-1} \bchi H \chi.
%\nonumber \\ & = & \chi \, F .
\end{eqnarray}
Canceling the second term on the r.h.s. with the first term in the parentheses in the third term, we see that the r.h.s is equal $\chi \, F$, which gives the first equality in (\ref{eq-II-11}).
%
%It suffices to check that (\ref{eq-II-6}) holds if $F$ is bounded invertible on $\Ran\, \chi$ and that (\ref{eq-II-7}) holds if $H$ is bounded invertible on $\cH$. Thus, suppose first that the restriction of $F$ to $\Ran\, \chi$ has bounded invertible on $\Ran\, \chi$,
Now, suppose first that the operator $F$ has bounded invertible and define
%
\begin{equation} \label{eq-II-13}
R \ := \ Q \: F^{-1} \: Q^\#
\; + \; \bchi \, H_\bchi^{-1} \bchi .
\end{equation}
%
Using (\ref{eq-II-11}) and \eqref{Hchi}, we obtain
%
\begin{eqnarray} \label{eq-II-14}
H \, R
& = &
H \, Q \, F^{-1} \, Q^\#
\; + \; \big( \bchi H_\bchi + \chi^2 H \bchi \big)
\, H_\bchi^{-1} \bchi
\\ \nonumber
& = &
\chi \, Q^\# \; + \; \bchi^2
\; + \; \chi^2 H \bchi \, H_\bchi^{-1} \bchi
\\ \nonumber
& = &
\chi^2 \; + \; \bchi^2 \ = \ \one ,
\end{eqnarray}
%
and, similarly, $R H = \one$. Thus $R = H^{-1}$,
and \eqref{resolvrel} holds true.
Conversely, suppose that $H$ is bounded invertible. %, and define\begin{equation} \label{eq-II-15}\tR \ := \ \chi \, H^{-1} \, \chi. %\; + \; \bchi \, T^{-1} \bchi .\end{equation}
Then, using the definition of $F$ and the relation $\chi^2 + \bchi^2=\one$, we obtain
%
\begin{eqnarray} \label{eq-II-16}
%F \, \tR & = &
F \, \chi \, H^{-1} \, \chi %\; + \; F \, \bchi \, T^{-1} \bchi \\ \nonumber
& = &
\chi H \, \chi^2 \; H^{-1} \, \chi %+ \; H_\chi \, \bchi \, T^{-1} \bchi\;
- \; \chi H \bchi \, H_\bchi^{-1} \bchi H \chi^2 \, H^{-1} \chi
\\ \nonumber
& = &
\chi H \, \chi^2 \; H^{-1} \, \chi - \; \chi H \bchi \, H_\bchi^{-1} \bchi H \, H^{-1} \chi + \; \chi H \bchi \, H_\bchi^{-1} \bchi H \bchi^2 \, H^{-1} \chi
\\ \nonumber
& = &
\chi H \, \chi^2 \; H^{-1} \, \chi + \; \chi H \bchi^2 \, H^{-1} \chi=\chi^2 .
\end{eqnarray}
%
Similarly, one checks that $\chi \, H^{-1} \, \chi F = \one$. Thus $F$ is invertible on $\Ran\, \chi$ %(globally) invertible on $\cH$
with inverse
$F^{-1} = \chi \, H^{-1} \, \chi$. %Moreover, we observe that \begin{equation} \label{eq-II-16-1}F \ = \ \chi \, F \, \chi \; + \; \bchi \, T \, \bchi \end{equation}is block-diagonal w.~r.~t.\ $\cH = \Ran\, \chi \oplus (\Ran\, \chi)^\perp$. Hence, the global invertibility of $F$ on $\cH$ implies the invertibility of its restriction to $\Ran\, \chi$, proving \eqref{resolvrel}.
\noindent
\textbf{Proof of (ii) ($H\psi = \lambda \psi\ \Longleftrightarrow$
%$\vphi := \pi \psi \in \Ran\, \pi \setminus \{0\}$ solves
$F_{\rho} (H - \lambda) \, \vphi = 0$).} %We write $H=T+W$, where $T$ commutes with $\chi, \bchi$ and is invertible on $\Ran\, \bchi$.
If $\psi \in \cH \setminus \{0\}$ solves $ H \psi = 0$
then (\ref{eq-II-11}) implies that
%
\begin{equation} \label{eq-II-16-2}
F \chi \psi \ = \ Q^\# \, H \, \psi \ = \ 0 .
\end{equation}
%
Furthermore, by \eqref{Hchi}, %\begin{equation} \label{eq-II-17}
$0 \ = \ \bchi \, H \, \psi \ = \
H_\bchi \, \bchi \psi \: + \: \bchi H \chi^2 \psi ,$ %\end{equation}
%
and hence
%
\begin{equation} \label{eq-II-18}
Q \, \chi \psi
\ = \
\chi^2 \psi \, - \, \bchi H_\bchi^{-1} \bchi H \chi^2 \psi
\ = \
\chi^2 \psi \, + \, \bchi^2 \psi \ = \ \psi .
\end{equation}
%
Therefore, $\psi \neq 0$ implies $\chi \psi \neq 0$.
If $\vphi \in \Ran\, \chi \setminus \{0\}$ solves
$F \vphi = 0$ then the definition of $Q$ implies that
\begin{equation} \label{eq-II-22}
\chi Q\vphi =\chi\vphi= \vphi,
\end{equation}
%
%
\DETAILS{(\ref{eq-II-11}) implies that\begin{equation} \label{eq-II-19}
H \, Q \vphi \ = \ \chi \, F \vphi \ = \ 0 .
\end{equation}
Since $T$ is invertible on $\Ran\, \bchi$, the trivial
identity $F = T + \chi W Q$ implies that
%
\begin{equation} \label{eq-II-20}
\bchi \ = \ T^{-1} \, \bchi \, T
\ = \ T^{-1} \, \bchi \, (F \, - \, \chi W Q) ,
\end{equation}
%
which, together with $\chi = \chi Q$, gives
%
\begin{equation} \label{eq-II-21}
\one \ = \ \bchi \, + \, \chi \ = \
T^{-1} \bchi F \; + \;
\big(\chi \, - \, T^{-1} \bchi \chi W \big) \, Q .
\end{equation}
%
Applying (\ref{eq-II-21}) to $\vphi$, we obtain that
\begin{equation} \label{eq-II-22}
\vphi \ = \ \big( \chi \, - \, T^{-1} \bchi \chi W \big) \,
Q \vphi ,
\end{equation}}
%
%
which implies that $Q \vphi \neq 0$ provided $\vphi \neq 0$.
\noindent
\textbf{Proof of (iii) ($\dim \cern (H - \lambda) = \dim \cern F_{\rho} (H -
\lambda)$).}
By (i), $\dim \cern H =0$ is equivalent to $\dim \cern F =0$,
assuming that $H\in D(F)$. We may
therefore assume that $\cern H \neq 0$ and $\cern F \neq 0$ are
both nontrivial.
Eq.~(\ref{eq-II-18}) shows that $\chi: \cern H \to \cern F$ is injective,
hence $\dim \cern H \leq \dim \cern F$, and
Eq.~(\ref{eq-II-22}) shows that $Q: \cern F \to \cern H$ is injective,
hence $\dim \cern H \geq \dim \cern F$.
This establishes (iv) and moreover that
$\chi: \cern H \to \cern F$ and $Q: \cern F \to \cern H$ are
actually bijections.
\QED
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Smooth Feshbach-Schur Map}\label{sec:sfm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Definition and isospectrality}\label{sec:sfm-def-iso}
We define the smooth Feshbach-Schur map and formulate its important isospectral property
%\textbf{Smooth Feshbach-Schur map.}
Let $\chi$, $\bchi$ be a partition of unity on a separable Hilbert
space $\cH$, i.e. $\chi$ and $\bchi$ are positive operators on
$\cH$ whose norms are bounded by one, $0 \leq \chi, \bchi \leq
\mathbf{1}$, and $\chi^{2}+ \bchi^{2} = \mathbf{1}$. We assume that
$\chi$ and $\bchi$ are nonzero. Let $\tau$ be a (linear) projection
acting on closed operators on $\cH$ with the property that operators
in its image commute with $\chi$ an
%d $\bchi$. We also assume that
$\tau(\textbf{1}) =\textbf{1}$.
%which leaves the domains of definition invariant, .
Let $\overline{\tau}:= \mathbf{1} - \tau$ and define
%
\begin{equation} \label{Htauchi} %{II-1}
%\label{eq-II-1-1}
%W_\chi \ \; := & \chi \, W \, \chi
%\comma \hspace{15mm}
%W_\bchi \ \; := & \bchi \, W \, \bchi \comma
%H_{\tau,\chi} \ \; := \tau(H) \: + \: \chi\overline{\tau}(H)\chi
%\comma \hspace{15mm}
H_{\tau,\chi^{\#}} \ \; := \tau(H) \: + \: \chi^{\#}
\overline{\tau}(H)\chi^{\#} ,
\end{equation}
%
where $\chi^{\#}$ stands for either $\chi$ or $\bchi$.
Given $\chi$ and $\tau$ as above, we denote by $D_{\tau,\chi}$ the
space of closed operators, $H$, on $\cH$ which belong to the domain
of $\tau$ and satisfy the following three conditions:
(i) $\tau$ and $\chi$ (and therefore also $\btau$ and $\bchi$) leave
the domain $D(H)$ of $H$ invariant:
\begin{equation}\label{domtauchi} %{II-2}
D(\tau(H))=D(H)\ \mbox{and}\ \chi D(H)\subset D(H),
\end{equation}
(ii)
\begin{equation}\label{HtaubchiInvert} %{II-3}
H_{\tau,\bchi}\ \mbox{is (bounded) invertible on}\
\Ran \, \bchi,
\end{equation}
(iii)
\begin{equation}\label{btauHbound} %{II-3}
\overline{\tau}(H) \chi\ \mbox{and}\ \chi
\overline{\tau}(H)\ \mbox{extend to bounded operators on}\ \cH.
\end{equation}
(For more general conditions see \cite{bcfs1, %{BachChenFroehlichSigal2003,
GriesemerHasler1}.)
% $|H_{\tau,\bchi}|^{-1/2} U^{-1}$ $\bchi
%\overline{\tau}(H) \chi$ and $\chi \overline{\tau}(H) \bchi
%|H_{\tau,\bchi}|^{-1/2}$ extend to bounded operators on $\cH$ and
%$\Ran \, \bchi$, respectively,
%
%\begin{equation} \label{eq-II-1-1.1}
%\big\| H_{\tau,\bchi}^{-1} \big\|_{\cB[\Ran \, \bchi]} \comma
%\hspace{2mm} \big\| |H_{\tau,\bchi}|^{-1/2} \, U^{-1} \, \bchi
%\overline{\tau}(H) \chi \big\|_{\cB[\cH]} \comma
%\end{equation}
%$$\big\| \chi
%\overline{\tau}(H) \bchi \, |H_{\tau,\bchi}|^{-1/2}
%\big\|_{\cB[\Ran \bchi, \cH]} \ < \ \infty \comma$$
%
%where $H_{\tau,\bchi} = U |H_{\tau,\bchi}|$ is the polar
%decomposition on $\Ran \bchi$.
The \textit{smooth Feshbach-Schur map (SFM)} maps operators %on $\cH$ belonging to
from $D_{\tau,\chi}$ into operators on $\cH$ by %$H \ \mapsto \ F_{\tau,\chi} (H)$, where
%
\begin{equation} \label{sfm}
F_{\tau,\chi} (H) \ := \ H_0 \, + \, \chi W\chi \, -
\, \chi W \bchi H_{\tau,\bchi}^{-1} \bchi W \chi ,
\end{equation}
%
where $H_0 := \tau(H)$ and $W := \overline{\tau}(H)$. Note that $H_0$
and $W$ are closed operators on $\cH$ with coinciding domains, $
D(H_0)= D(W)=D(H)$, and $H = H_0 + W$. We remark that the domains of
$\chi W\chi$, $\bchi W\bchi$, $H_{\tau,\chi}$, and $H_{\tau,\bchi}$
all contain $D(H)$.
Define operators $Q_{\tau,\chi} (H) := \chi \: - \: \bchi \, H_{\tau,\bchi}^{-1}
\bchi W \chi$ and $Q_{\tau,\chi} ^\#(H) := \chi \: - \: \chi W \bchi \,
H_{\tau,\bchi}^{-1} \bchi$. The following result (\cite{bcfs1}) generalizes Theorem \ref{thm:isospF} above; its proof is similar to the one of that theorem:
%
\begin{theorem}[Isospectrality of SFM] \label{sfmisosp}
{\em Let $0 \leq \chi \leq \one$ and $H\in D_{\tau,\chi}$ be an operator on a separable Hilbert space $\cH$. Then we
have the following results:
%
\begin{itemize}
\item[(i)]
$H$ is bounded invertible on $\cH$ if and only if
$F_{\tau,\chi} (H)$ is bounded invertible on $\Ran\, \chi$.
In this case
\begin{eqnarray} \label{Hinvrepr} %{eq-II-6}
H^{-1} & = & Q_{\tau,\chi} (H) \: F_{\tau,\chi} (H)^{-1} \: Q_{\tau,\chi} (H)^\#
\; + \; \bchi \, H_\bchi^{-1} \bchi , \hspace{8mm}
\\ \label{FHinvrepr} %{eq-II-7}
F_{\tau,\chi} (H)^{-1} & = &
\chi \, H^{-1} \, \chi \; + \; \bchi \, \tau(H)^{-1} \bchi . \hspace{8mm}
\end{eqnarray}
%
\item[(ii)]
If $\psi \in \cH \setminus \{0\}$ solves $H \psi = 0$
then $\vphi := \chi \psi \in \Ran\, \chi \setminus \{0\}$
solves $F_{\tau,\chi} (H) \, \vphi = 0$.
\item[(iii)]
If $\vphi \in \Ran\, \chi \setminus \{0\}$ solves
$F_{\tau,\chi} (H) \, \vphi = 0$
then $\psi := Q_{\tau,\chi} (H) \vphi \in \cH \setminus \{0\}$ solves $H \psi = 0$.
\item[(iv)]
The multiplicity of the spectral value $\{0\}$ is conserved in the sense that
$\dim \Null H $ $= \dim \Null F_{\tau,\chi} (H)$.
%
\DETAILS{\item[(v)]
Assume, in addition, that $H = H^*$ and $\tau(H)=\tau(H)^*$ are self-adjoint,
and introduce the bounded operators
%
\begin{eqnarray} \label{eq-II-8.1a}
M & := & H_{\tau,\chi}^{-1} \, \bchi \, (H-\tau (H)) \, \chi
\hspace{8mm} \mbox{and}
\\ \label{eq-II-8.2}
N & := &
\big( \one \, + \, M^* M \big)^{-1/2} .
\end{eqnarray}
%
Then, for any $\psi \in \cH$,
%
\begin{eqnarray} \label{eq-II-8.3}
\lefteqn{
\lim_{\eps \searrow 0} \; \rIm
\big\la \psi , \: (H - i \eps)^{-1} \, \psi \big\ra
\ = \
} \\ \nonumber & &
\lim_{\eps \searrow 0} \; \rIm
\Big\la N \, Q_{\tau,\chi} (H)^* \, \psi , \:
\big( N \, F_{\tau,\chi} (H) \, N \, - \, i \eps \big)^{-1}
\: N \, Q_{\tau,\chi} (H)^* \, \psi \Big\ra
\end{eqnarray}
and
\begin{eqnarray} \label{eq-II-8.4}
\lefteqn{
\lim_{\eps \searrow 0} \; \rIm
\big\la \psi , \:
\big( N \, F_{\tau,\chi} (H) \, N \, - \, i \eps \big)^{-1}
\, \psi \big\ra
\ = \
} \\ \nonumber & &
\lim_{\eps \searrow 0} \; \rIm
\big\la \chi \, N^{-1} \, \psi , \:
( H \, - \, i \eps \big)^{-1} \, \chi \, N^{-1}
\, \psi \Big\ra .
\end{eqnarray}}
%
\end{itemize}}
\end{theorem}
%
We also mention the following useful property of $ F_{\tau, \chi}$:
\begin{equation} \label{Hsa}
H\ \mbox{is self-adjoint}\ \quad \Rightarrow \quad F_{\tau, \chi}(H)\ \mbox{is self-adjoint}.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{ Transfer of local decay}\label{sec:locdectransf}
%\textbf{Transfer of local decay.}
%One can transfer through the renormalization group also other properties of Hamiltonians, e.g.
We have shown above that the smooth Feshbach-Schur map is isospectral. In fact, under certain additional conditions it preserves (or transfers) much stronger spectral property -
the limiting absorption principle (LAP) (\cite{fgs3}), which is defined as follows. %, cf. Statement (v) of Theorem \ref{sfmisosp}). which in particular implies the local decay property %We formulate an important property of the smooth Feshbach-Schur map which
Let $\Delta\subset \R$ be an interval, $\nu>0$ and $B$, a self-adjoint operator.
We say that a $C^1$ family of self-adjoint operators, s.t. $H(\lambda) \in D_{\tau,\chi}$ has the ($\Delta, \nu, B, \theta$) \textit{limiting absorption principle} (LAP) property iff
\begin{eqnarray} %{eqn:7}
\lim_{\eps\rightarrow 0+}\la B\ra^{-\theta}\big(H(\lambda)-i\eps\big)^{-1}\la B\ra^{-\theta}\ \mbox{exists and}\ \in C^\nu(\Delta).\label{BHinvB'}
\end{eqnarray}
%We say that a self-adjoint operator $H$ has the ($\Delta, \nu, B, \theta$) \textit{local decay} (LD) property iff for any function $f(\lambda)$ with $\supp f \subseteq\Delta$, we have that \begin{equation} \|\la B\ra^{-\theta}e^{-i H t}f(H)\la B\ra^{-\theta}\|\le C t^{-\nu}. \end{equation}
Usually LAP holds for $\nu< \theta-\frac{1}{2}$.
One can show that the LAP implies the local decay property (see, e.g. \cite{RS}, vol III; recall that the definition of the local decay property is given in Section \ref{sec:relat-res}).
\begin{theorem} \label{thm:LAPtransfer}
{\em Let $\Delta\subset \R$ and $,\ \forall \lambda\in\Delta,\ H(\lambda)$ be a $C^1$ family of self-adjoint operators, s.t. $H(\lambda) \in D_{\tau,\chi}$. %$W(\lambda) := \overline{\tau}(H(\lambda))\in C^1$
Assume that there is a self-adjoint operator $B$ s.t. % the operators $H_{\tau,\bchi}(\lambda)$ are
%invertible, the operators $\chi\overline{\tau}(H)$ and
%$\overline{\tau}(H)\chi$ are bounded and
%\begin{itemize}\item
\begin{equation} \label{commbnds} %{eqn:5}
\adj_B^j(A)\ \mbox{is bounded and differentiable in}\ \lam,\ \forall j\le 2,
\end{equation}
where $A$ is one of the operators
$\chi,\ \overline{\chi},\ \chi \bar\tau(H(\lambda)),\ \bar\tau(H(\lambda))\chi$, $\partial_\lambda^k (\bchi H_{\tau,\bchi}(\lambda)^{-1} \bchi),$ $k=0, 1$.
%%\partial_\lambda^k\tau(H(\lambda)),\partial_\lambda^k\overline{\tau}( H(\lambda)),\
%\item For some $0\le \nu\le 1$ and $0 < \theta \le 1$, the operator norm limit \begin{equation}\label{BFB'}\lim_{\eps\rightarrow 0+}\la B\ra^{-\theta}[F_{\tau,\chi}(H(\lambda))-i\eps]^{-1}\la B\ra^{-\theta}=:B_\theta[F_{\tau,\chi}(H(\lambda))-i0]^{-1}\end{equation} exists and satisfies \begin{eqnarray}\label{Finv-conv} \la B\ra^{-\theta}\left[F_{\tau, \chi}(H(\lambda))-i0\right]^{-1}\la B\ra^{-\theta}\in C^\nu(\Delta).\end{eqnarray}\end{itemize}
Then, for any $0\le \nu\le 1$ and $0 < \theta \le 1$ and in the operator norm, we have
%\begin{itemize}\item $\la B\ra^{-\theta}[F_{\tau,\chi}(H(\lambda)-i\eps)]^{-1}\la B\ra^{-\theta}$ converges in norm, as $\eps\rightarrow 0+$; %, so if its limit is denoted as $B_\theta[F_{\tau,\chi}(H(\lambda))-i0]^{-1}$, we have
\item %With the notation \begin{equation}\label{BFB}\la B\ra^{-\theta}[F_{\tau,\chi}(H(\lambda))-i0]^{-1}\la B\ra^{-\theta}:=\lim_{\eps\rightarrow 0+}\la B\ra^{-\theta}[F_{\tau,\chi}(H(\lambda)-i\eps)]^{-1}\la B\ra^{-\theta},\end{equation}
%we have, for any $0\le \nu\le 1$ and $0 < \theta \le 1$,
\begin{eqnarray} %{eqn:7}
&&\lim_{\eps\rightarrow 0+}\lefteqn{\la B\ra^{-\theta}\left(F_{\tau, \chi}(H(\lambda))-i\eps\right)^{-1}\la B\ra^{-\theta}\ \mbox{exists and}\ \in C^\nu(\Delta)} \label{BFinvB} %\nonumber
\\ &&\Rightarrow
\lim_{\eps\rightarrow 0+}\la B\ra^{-\theta}\big(H(\lambda)-i\eps\big)^{-1}\la B\ra^{-\theta}\ \mbox{exists and}\ \in C^\nu(\Delta).\label{BHinvB}
\end{eqnarray}}
%\end{itemize}
\end{theorem}
This allows one to reduce the proof of the LAP for the original operator, $H-\lam$, to the proof of this property for a much simpler one, $\cR_\rho^n(H-\lam)$.
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{ Transfer of analyticity}\label{sec:anal-transf}
\begin{theorem} %\label{LAPtransfer}
{\em Let $\Lam$ be an open set in $\C$ and
$H(\lambda)$, $\lambda\in\Lam$, a family of operators with a fixed domain, which belong to the domain of $F_{\tau\chi}$. Assume $H(\lambda)$ and $\tau(H(\lambda))$, with the same domain, are analytic in the sense of Kato (see e.g. \cite{RS}, vol IV). %\textbf{(needs technical conditions, something like there ia an operator $T\ge \one$, s.t. $T\bchi_\rho H(\lambda)_{\tau,\bchi_\rho}^{-1}\bchi_\rho T$ and $ T^{-1} H(\lambda)_{\tau,\bchi_\rho}T^{-1}$ are bounded and $ T^{-1} H(\lambda)_{\tau,\bchi_\rho}T^{-1}$ is analytic)},
Then we have that
\begin{itemize}
\item $F_{\tau, \chi}(H(\lambda))$ is an analytic in $\lambda\in\Lam$ family of operators.
\end{itemize}}
\end{theorem}
\begin{proof} Since that $H(\lambda)$, $\ H_0(\lambda):=\tau(H(\lambda))$ and $W(\lambda) \chi\ \mbox{and}\ \chi
W(\lambda)$, where $W(\lambda):=\overline{\tau}(H(\lambda))$, are analytic in
$\lambda \in \Lam,$ we see from the
definition of the smooth Feshbach-Schur map $F_{\tau\chi}$ in \eqref{sfm}
that $F_{\tau\chi}(H(\lambda))$ is analytic in $\lambda \in
\Lam$, provided $\bchi_\rho H(\lambda)_{\tau,\bchi_\rho}^{-1}$ $ \bchi_\rho$
%W^{(j-1)}(\lambda) \chi_\rho
is analytic in $\Lam$. The analyticity of the latter family
follows by the Neumann series argument. %expanding it in the Neumann series as $\bchi_\rho H(\lambda)_{\tau,\bchi_\rho}^{-1}\bchi_\rho=\sum_{n=0}^\infty\bchi_\rho \big[\bchi_\rho H_0(\lambda)^{-1}\bchi_\rho W(\lambda)\big]^n\bchi_\rho $. %and Proposition~\ref{analytic00}.
\end{proof}
%One can also prove analyticity in the operator $H$ and its various parts.
One can generalize the above result to $\Lam$'s which are open sets in a complex Banach space. Recall that a complex vector-function $f$ in an open set $\Lam$ in a
complex Banach space $\cW$ is said to be \textit{analytic} iff it is
locally bounded and G\^{a}teaux-differentiable. One can show that
$f$ is analytic iff $\forall \xi \in \cW,\ f(H+ \tau \xi)$ is
analytic in the complex variable $\tau$ for $|\tau|$ sufficiently
small (see \cite{Berger, HillePhillips}). Furthermore if $f$ is
analytic in $\Lam$ and $g$ is an analytic vector-function from an
open set $\Omega$ in $\mathbb{C}$ into $\Lam$, then the composite
function $f\circ g$ is analytic on $\Omega$.
%This definition implies that if $f$ in
%analytic on a set $\cD$ and $H: \Omega \rightarrow \cD$ is analytic
%on an open set $\Omega \subset \mathbb{C}$, then $f(H(z))$ is
%analytic on $\Omega$. The last property is the one we need here.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
%\section{Anisotropic Banach Spaces of Hamiltonians} \label{sec:anisotr-B-sp}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Mass Renormalization} \label{sec:mass-renorm}
%\newpage
%\section{Mass Renormalization } %Particles and Electromagnetic Field}
%\label{sec:mass}
As the free electron is surrounded by virtual 'soft' photons its effective (inertial0 mass is greater than the value ('bare' mass) entering its Hamiltonian. One calls this electron \textit{mass renormalization}. We begin with analyzing the definition of (inertial) mass in Classical Mechanics. Consider a classical particle with the Hamiltonian $h(x, k):=K(k)+V(x)$, where $K(k)$ is some function describing the kinetic energy of the particle. To find the particle mass in this case we have to determine the relation between the force and acceleration at very low velocities. The Hamilton equations give $\dot x=\p_k K$ and $\dot k=F$, where $F=-\p_x V$ is the force acting on the particle. Assuming that $K$ has a minimum at $k=0$ and expanding $\p_k K (k)$ around $0$, differentiating the resulting relation $\dot x= K''(0)k$, where $K''(0)$ is the hessian of $K$ at $k=0$, w.r. to time and using the second Hamilton equation, we obtain $\ddot x= K''(0)F(x)$. This suggests to define the mass of the particle as $m=K''(0)^{-1}$, i.e. as the inverse of the Hessian of the energy, in the absence of external forces, as a function of of momentum at $0$. ($K(k)$ is called the dispersion relation.) We adopt this as a general definition: \textit{the (effective) mass of a particle interacting with fields is the inverse of the Hessian of the energy of the total system as a function of of the total momentum at $0$.}
Now, we consider a single non-relativistic electron coupled to quantized electromagnetic field. Recall that the charge of electron is denoted by $-e$ and its \textit{bare} mass in our units is $m$. The corresponding Hamiltonian is
\begin{equation}\label{H-single}
\Hn \, := \, \frac{1}{2m}( i\nabla_x \otimes 1_f \, - \, \c \Af(x) \, )^2
\, + \, \1_{el} \otimes H_f ,%\frac12
\end{equation}
acting on the space $L^2({\mathbb R}^{3})\otimes \cF\equiv \cH_\part \otimes \cH_f$. It is the generator for the dynamics of a single non-relativistic electron, and of
the electromagnetic radiation field, which interact via minimal coupling. Here recall $\Af(x)$ and $H_f$ are the quantized electromagnetic vector potential with ultraviolet cutoff and the field Hamiltonian and are defined in \eqref{A} and \eqref{Hf}
%, where $\kappa_\sigma$ is an approximate
%characteristic function for the interval $[\sigma,1]$, which we can, for instance, to assume
%to have the form
%$\kappa_\sigma(y)=\frac y\sigma$ for $y\in[0,\sigma]$,
%$\kappa_\sigma(y)=0$ for $y>2$, and smooth and monotone on $(1,2)$.
\bigskip
%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Translation invariance}\label{subsec:transl} A crucial role in what follows will be played by the fact that
The system considered is \textit{translationally invariant} \index{translation ! invariance} in the sense that $\Hn$ commutes with the translations, $T_y$, %\eqref{Ty},
%
%
\DETAILS{In the Fock space representation this gives: $T_{y}: \oplus_n\Psi_n(x, k_1, \dots, k_n)$\\ $\ra \oplus_n e^{ iy\cdot (k_1+\dots k_n)} \Psi_n(x+y, k_1, \dots, k_n)$ and therefore can be rewritten as $T_{y}: \Psi(x)\ra e^{ iy\cdot P_\re} \Psi(x+y)$, where $ P_\re$ is the momentum
operator associated to the quantized radiation field,
\[\Pf=\sum_\lambda\int dk \, k \, a_\lambda^*(k)a_\lambda(k).\] %$P_\re := \int_{\mathbb{R}^3} k a^*(k) a(k) \d k.$
As it is straightforward to show $T_y$ are unitary operators and that they satisfy the relations
$ T_{x+y} = T_xT_y,$
and therefore $y\ra T_{y}$ is a unitary Abelian representation of $\R^3$. We also note that $T_y$ commutes with $H$,
\begin{equation*} T_y \Hn = \Hn T_y. \end{equation*}
Indeed, using \eqref{Ty} and the definitions of the operators $A(x)$ and $E(x)$, we arrive at $T_y A(x) = (t_{y}A)(x) T_y$ and $T_y E(x) = (t_{y}E)(x) T_y$, which, due to the definition of $\Hn$, gives $T_y \Hn = \Hn T_y$.
Finally, we observe that the group $T_y$ is generated by}
%
%
\begin{equation*} T_y \Hn = \Hn T_y, \end{equation*}
which in the present case take the form
\begin{equation}\label{Ty'}
T_{y}: \Psi(\ux)\to e^{ iy\cdot P_\re} \Psi(\ux+\uy), %\Psi(x, A)\to \Psi(x+y, t_{y}A), %\Psi(x)\ra e^{ iy\cdot P_\re} \Psi(x+y); \oplus_n\Psi_n(x, k_1, \dots, k_n)\ra \oplus_n e^{ iy\cdot (k_1+\dots k_n)} \Psi_n(x+y, k_1, \dots, k_n).
\end{equation}
%where $(t_{y}A)(x, A)=A(x-y)$.
This as before leads to $\Hn$ commuting with the total momentum operator,
\eqn
\Ptot \, := \, P_\el\otimes\1_f \, + \, \1_{el}\otimes \Pf,
\eeqn
of the electron and the photon field: $[H, \Ptot]=0$. Here
$P_\el:=-i\nabla_x$ and $ \Pf=\sum_\lambda\int dk \, k \, a_\lambda^*(k)a_\lambda(k)$ are electron and field momenta. %: $T_y= e^{ iy\cdot \Ptot}$. %. ($\Ptot$ is the selfadjoint operator on $\cH$.) %associated to the total momentum of the system containing the electron and the photon field
%\begin{equation} P_\re := \int_{\mathbb{R}^3} k a^*(k) a(k) \d k.\end{equation}Hence
Again as in Appendix \ref{sec:transl-Ham}, %Section \ref{subsec:fiber},
this leads to
%%%%%%%%%%%%%%%%
%
%
\DETAILS{\bigskip
%%%%%%%%%%%%%%%%%%%%%%
\subsection{Fiber decomposition with respect to total momentum}\label{subsec:fiber}
%\subsection{Direct Integral Representation}
%For each $P \in \R^3$, We define the Hilbert space
Let $\cH$ be the direct integral
%\begin{equation*}
$\cH = \int_{\R^3}^\oplus \cH_P dP, $ %\end{equation*}
with the fibers $\cH_P := \cF$ (this means that $\cH = L^2(\R^3, dP; \cF)$) and %\begin{enumerate}[(a)] \item \end{enumerate}
define $U : \cH_\el \otimes \cH_{\re} \to \cH$ on smooth functions with compact domain by the formula
\begin{equation}\label{U}
(U \Psi)(P) = \int_{\mathbb{R}^3}e^{i(P-P_\re)\cdot y}\Psi(y) \d y. %\int_{y \in \R^3} \chi_k^{-1}(y) T_y \Psi(x) dy,
\end{equation}
%where $\chi_k$, $k \in \R^3$, are the characters of given explicitly by %\begin{equation*}
% $\chi_k(y) = e^{ik\cdot y}.$ %\end{equation*}
Then $U$ extends uniquely to a unitary operator (see below). Its converse is written, for $\Phi (P)\ \in \mathcal{F}$, as
\begin{equation}\label{Uinv} %{Psidecomp}
(U^{-1}\Phi)(x)=\int_{\mathbb{R}^3} e^{-\i x\cdot(P-P_\re)}\Phi (P) \d P.
\end{equation}
The functions $\Phi (P)\ =\int_{\mathbb{R}^3} dye^{i(P-P_\re)\cdot y}\Psi(y)$ are called fibers of $\Psi$. We denote $\widehat\Psi(P) \, = \, (U \Psi)(P)$ and $ \check\Phi (x)%^\vee(x)
\, = \, (U^{-1} \Phi)(x)$
%
%
\DETAILS{ For $k \in \R^3$, let $H_{ k}$ be the operator $H$ acting on $\cH_k$ with domain consisting of those $v \in \cH_k \cap H^2$ such that $\Psi$
satisfies the boundary conditions $T_y \Psi(x) = \chi_k(t) \Psi(x)$. Then
\begin{equation}
\label{K-decomposition}
UH U^{-1} = \int_{\R^3}^\oplus H_{ k} dk.
\end{equation}
and
\begin{equation}
\label{KKkspecrelat}
\sigma(H) = \bigcup_{k\in\R^3} \sigma(H_{ k}).
\end{equation}}
%
%
%
%
\DETAILS{\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Spectral decomposition with respect to total momentum}
We consider the direct integral decomposition
\eqn
\cH \, = \, \int^\oplus dp \, \cH_p \,,
\eeqn
with respect to $\Ptot$, where each fiber $\cH_p\cong {\rm Ker}(\Ptot-p)$ is isomorphic to $\Fo$.
It corresponds to the natural isomorphism
\eqn
\cH \, \cong \, L^2(\R^3,\Fo) \,
\eeqn
which we will use in the sequel.
%obtained from
%\eqn
% (x,\Psi) \, \mapsto \, \iota_{x}\Psi \,\in \Fo
%\eeqn
%where $\iota_x$ is the evaluation map, for $x\in\R^3$.
For $\psi\in\cH$, we define the Fourier transform with respect to the electron variable,
\eqn
\widehat\psi(p) \, = \, \int dx \, e^{-i(p-\Pf)x} \psi(x)
\eeqn
with inverse transform
\eqn
\check\phi%^\vee(x)
\, = \, \int dp \, e^{i(p-\Pf)x}\phi(p) \,.
\eeqn}
%
%
\begin{lemma}
The operations \eqref{U} %{eq-FT-def-1}
and \eqref{Uinv} %{eq-FTinv-def-1}
define unitary maps $L^2({\mathbb R}^{3})\otimes \cF\rightarrow\cH$ and $\cH\rightarrow L^2({\mathbb R}^{3})\otimes \cF$, and are mutual inverses.
\end{lemma}
\prf
By density, we may assume that $\Psi$ is a generalized Schwartz class function,
\eqn
{\mathcal S}_{\Fo} \, :=\,
\big\{ \, \Psi\in L^2({\mathbb R}^{3})\otimes \cF \, \big| \, \sup_x\|x^\alpha\partial_x^\beta\Psi\|_{ \Fo} \, < \, \infty
\; \; \; , \; \; \; \forall \alpha,\beta\in\N_0^3 \, \big\} \,.
\eeqn
Then, it follows from standard arguments in Fourier analysis that
\eqn\label{eq-FT-def-1}
(\widehat\psi)^\vee(x) &=&
\int \d P \, e^{-i(P-\Pf)x} \int dx' \, e^{i(p-\Pf)x'} \, \Psi(x')
\nonumber\\
&=&
\int dx' \, \int \d P \, e^{-ip(x-x')}e^{i\Pf(x-x')} \, \Psi(x')
\nonumber\\
&=&
\int dx' \, \delta(x-x') \, e^{i\Pf(x-x')} \, \Psi(x')
\nonumber\\
&=&\Psi(x) \,.
\eeqn
On the other hand, for $\psi\in {\mathcal S}_{\Fo} $,
\eqn\label{eq-FTinv-def-1}
(\check\Phi)^{\widehat{\;}}(P)
&=&
\int dx \, e^{i(P-\Pf)x} \int dq \, e^{i(q-\Pf)x} \, \Phi(q)
\nonumber\\
&=&
\int dq \, \int dx \, e^{i(p-q)x} \, \Phi(q)
\nonumber\\
&=&
\int dq \, \delta(P-q) \, \Phi(q)
\nonumber\\
&=&\Phi(P) \,.
\eeqn
From the density of ${\mathcal S}_{\Fo}$ in $L^2({\mathbb R}^{3})\otimes \cF$, we infer that \eqref{eq-FT-def-1}
and \eqref{eq-FTinv-def-1} define bounded maps %$\cH\rightarrow\cH$, and
which are mutual inverses. Unitarity can be checked easily.
\endprf
%
\DETAILS{We observe that
\eqn
(\Ptot\psi)^{\widehat{\;}}(p) & = & \int dx \, e^{i(p-\Pf)x} (-i\nabla_x+\Pf)\psi(x)
\nonumber\\
&=& \int dx \, \big( \, (-i\nabla_x+\Pf)e^{i(p-\Pf)x}\,\big) \, \psi(x)
\nonumber\\
&=&p \, \widehat\psi(p) \,.
\eeqn
Hence, $\widehat\psi(p)\in\cH_p\cong {\rm Ker}(\Ptot-p)$, for any $p\in\R^3$.}
%In particular, \eqn \psi(x) \, = \, \int dP_{\Ptot}(p) \, e^{i(p-\Pf)x}\widehat\psi(p)\eeqn where $dP_{\Ptot}$ denotes the spectral measure associated to $\Ptot$.
%
It follows that $\Hn$ admits}
%
%
the fiber decomposition
\begin{equation}
U\Hn U^{-1} = \int_{\mathbb{R}^3}^{\oplus} \Hn(P) \d P,
\end{equation}
where the fiber operators $\Hn(P)$, $P \in \mathbb{R}^3$, are self-adjoint operators on $\mathcal{F}$. Using $a(k)e^{-\i x\cdot P_\re}=e^{-\i x\cdot (P_\re+k)}a(k)$ and $a^*(k)e^{-\i x\cdot P_\re}=e^{-\i x\cdot (P_\re-k)}a^*(k)$, we find $\nabla_x e^{\i x\cdot(P-P_\re)}\Af(x) e^{\i x\cdot(P-P_\re)} =0$ and therefore
\begin{equation}
\Af(x) e^{\i x\cdot(P-P_\re)} = e^{\i x\cdot(P-P_\re)}\Af(0).
\end{equation}
Using this and \eqref{Uinv}, we compute $\Hn (U^{-1}\Phi)(x)=\int_{\mathbb{R}^3} e^{\i x\cdot(P-P_\re)}\Hn(P)\Phi (P) \d P$, where $\Hn(P)$ are Hamiltonians on the fibers $\cH_P := \cF$ given explicitly by
\begin{equation}
\Hn(P) = \frac{1}{2m}\big ( P - P_\re -e\Af)^2 + H_\re
\end{equation}
where $\Af:=\Af(0)$. Explicitly, $\Af$ is given by
\begin{equation}\label{Achi}
\Af \, = \,
\sum_{\lambda}\int \d k \, \frac{\chi(|k|)}{|k|^{1/2}} \,
\e_\lambda(k) \,\{ \, a_\lambda(k) \, + \, a^*_\lambda(k) \, \}. %\frac{ \kappa(k) }{ |k|^{\frac{1}{2}}}
\end{equation}
%We define $E(P) := \inf \sigma( H(P) )$. If $g=0$, and if $|P|$ is less than the bare electron mass (equal to 1 in the units used in this paper), then $E(P) = P^2 /2$ is an \emph{eigenvalue} of $H(P)$. If $|P| > 1$, then $E(P) = |P| - 1/2$, and $E(P)$ is \emph{not} an eigenvalue of $H(P)$.
%
%
\DETAILS{For an observable $A$ on $ \mathcal{H}_{\mathrm{el}} \otimes \mathcal{H}_\re$, we consider the operator $UA U^{-1}$, acting on the space $\cH = \int_{\R^3}^\oplus \cH_P dP$. It can be written as
$$(UA U^{-1}\Phi)(P)=\int_{\mathbb{R}^3} A_{PQ}\Phi(Q)dQ,$$
for some operator-function $A_{PQ}$ on $\cF$. If $A\in \cA$, then $A_{PQ}\in {\frak W}(L^{2}_0)$.
We write $UA U^{-1}=\int A_{PQ} dPdQ.$ Note that the evolution, $\al^t(A)$, of $A$ is then given by
\begin{equation}
\al^t(A)= \int_{\mathbb{R}^3\times \R^3}dPdQ\alpha_{PQ}^t (A_{PQ}),
\end{equation}
where $\alpha_{PQ}^t (A_{PQ}):=e^{itH_P}A_{PQ} e^{-itH_Q}$. If an observable $A$ is translation invariant, then we can write it as a fiber integral, $A = \int_{\mathbb{R}^3}^{\oplus} A_P \d P.$ If $A$ is translation invariant, then so is the observable $\al^t(A)$ and
\begin{equation}
\al^t(A)=\int_{\mathbb{R}^3} \al^t_{P}(A_{P})dP,\ \mbox{where}\ \al^t_{P}(A_{P}):=e^{itH_P}A_{P} e^{-itH_P}.
\end{equation}
We write $\al^t=\int \al^t_{P}dP$. An example of a translation invariant observable is the particle momentum $P_\el=-i\nabla=i[H, x]$. In this case we have $P_\el = \int_{\mathbb{R}^3}^{\oplus} (P-P_\re) \d P = \int_{\mathbb{R}^3}^{\oplus} P \d P-P_\re.$}
%
%
%
%
\DETAILS{\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Next, we consider the Hamiltonian \eqref{eq-Hn-def-1}.
Clearly, $[\Hn,\Ptot]=0$, by translation invariance.
Then,
\eqn
(\Hn \psi)^{\widehat{\;}}(p) \, = \, \Hn(p) \widehat\psi(p)
\eeqn
where $\Hn(p) = \Hn\big|_{\cH_p}$ is the fiber Hamiltonian associated to total momentum $p$. Let $\Af:=\Af(0) $. We compute
\eqn
\Hn(p) =\frac12( \, p \, - \, \Pf \, - \, \c \Af \, )^2
\, + \, H_f \,
\eeqn
%We introduce the direct integral notation
%\eqn
% \int^\oplus dp \, := \, \int dP_{\Ptot}(p) \,,
%\eeqn
%where $dP_{\Ptot}$ denotes the spectral measure associated to $\Ptot$.
We then have that
\eqn
\Hn \, = \, \int e^{i(p-\Pf)x}\Hn(p)e^{-i(p-\Pf)x}dP_{\Ptot}(p)
\eeqn
which is the spectral representation of $\Hn$
with respect to the spectral measure associated to $\Ptot$.}
%
%and observe that $\Hn$ is unitarily
%equivalent to the operator
%\eqn
% \lefteqn{e^{-i\Pf x}\Hn e^{i\Pf x}}
% \\
% &&
% \, = \,
% \frac12( \, i\nabla_x \otimes \1_f \, - \, \1_{el}\otimes\Pf \, - \, \g \Af(0) \, )^2
% \, + \, \1_{el} \otimes H_f \,.
% \nonumber
%\eeqn
%Notably, the operator $i\nabla_x\otimes\1_f$ commutes with all
%other operators on the rhs.
%\bigskip
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Renormalized electron mass}
%\textbf{Renormalized electron mass.}
Consider the infimum $E(P):=\inf\sigma (\Hn(P))$ of the spectrum of the fiber Hamiltonian $\Hn(P)$. Note that for $e=0$, $E(P)|_{e=0} \; =: \; E_0(P)$ is the ground state energy of $H_0(P):=\Hn(P)|_{e=0} = \frac{1}{2m}\big ( P - P_\re \big)^2 + H_\re$ with the ground state $\vac$ and is $E_0(P) \; = \frac{|P|^2}{2 m}$. The renormalized electron mass %found in the literature (see, e.g., \cite{cotann,sak})
is defined as
\eqnn
E(P) \; = \, \frac{|P|^2}{2 m_{ren}} \, + \, O(|P|^3)
\eeqnn
where the left hand side is computed perturbatively up the second order in the coupling constant (charge).
Provided that $E(P)$ is spherically symmetric and $C^2$ at $P=0$,
and therefore, in particular, $\partial_{|P|} E(0) = 0$,
we define the renormalized electron mass at zero total momentum %(again, for the IR regularized model)
as
\eqnn
m_{ren} \; := \; \frac{1}{ \partial_{|P|}^2 E(0) } \;.
\eeqnn
The kinematic meaning of this expression is as follows.
The ground state energy $E(P)$ can be considered as an effective Hamiltonian
of the electron in the ground state.
(The propagator $\exp(-itE(P))$ determines the propagation properties of a
wave packet formed of dressed one-particle states with a wave function supported near $p = 0$
-- which exist as long as there is an infrared regularization.)
The first Hamilton equation gives the expression for the electron velocity as
\eqnn
v \; = \; \partial_P E(P) \;.
\eeqnn
Expanding the right hand side in $P$ we find $v = {\rm Hess} \, E(0) P + O(P^2)$,
where
\eqn\label{eq:Hess-E-def-1}
\big({\rm Hess}\,E(P)\big)_{ij} \; = \;
\Big(\delta_{ij}-\frac{P_i P_j}{|P|^2}\Big)
\frac{\partial_{|P|}E(P)}{|P|}
\, + \, \frac{P_i P_j}{|P|^2}
\partial_{|P|}^2E(P) \;
\eeqn
is the Hessian of $E(P)$ at $P\in\R^3$ (given that $E(P)$ is spherically symmetric,
and $C^2$ in $|P|$ near $P=0$).
It follows from (\ref{eq:Hess-E-def-1}) and the fact $\partial_{|P|} E(0) = 0$ that
${\rm Hess} \, E(0) = \partial_{|P|}^2 E(0) \, \1$, so that
\eqnn
v \; = \; \partial_{|P|}^2 E(0,\sigma) \, P \, + \, O(P^2) \;.
\eeqnn
This suggests taking $(\partial_{|P|}^2 E(0) )^{-1}$ as the renormalized
electron mass at $P=0$.
%
\DETAILS{For sufficiently small momenta $p$ we define the renormalized electron mass as
%\eqn\label{mren-def-1} %{massincr-1}
$m_{ren}(p) \; := \; \frac{1}{\partial_{|p|}^2 E(p)} \;. $ %\eeqn
We remark that a different notion of the renormalized electron mass
in non-relativistic QED can be introduced through the binding of an electron to a nucleus, see \cite{hasei,lilo1}.}
%
%The most common definition of the renormalized electron mass found in the literature
%(see e.g. \cite{sp1}) is $m_{ren} := (\partial_{|p|}^2 E(0,\sigma) )^{-1}$
%(we write it out for the IR regularized model).
%This expression can be understood as follows. The ground state energy
%$E(p,\sigma)$ can be considered as an effective Hamiltonian of the electron
%in the ground state. The Hamilton equation gives the expression
%for the electron velocity as $v = \partial_p E(p,\sigma)$.
%Expanding the right hand side in $p$ we find $v = {\rm Hess} \, E(0,\sigma) p + O(|p|^2)$,
%where ${\rm Hess} \, E(p,\sigma)$ is the Hessian of $E(p,\sigma)$. Since $E(p,\sigma)$
%is spherically symmetric in $p$, a simple calculation gives that
%${\rm Hess} \, E(0,\sigma) p = \partial_{|p|}^2 E(0,\sigma) p$, which suggests
%taking $(\partial_{|p|}^2 E(0,\sigma) )^{-1}$ as the renormalized electron mass near
%$p=0$.
%More generally, for sufficiently small momenta $p\in \R^3$,
%we define the renormalized electron mass as follows.
%The propagation properties of a wave packet formed of dressed one-particle states (which exist as long
%as $\sigma$ is positive) with a wavefunction supported near $p = 0$ are determined by the propagator
%$\exp(-itE(p,\sigma)$. The eigenvalues of
%${\rm Hess}\,E(p,\sigma)$ is of central importance
%for the construction of the scattering theory of infraparticles
%because it determines the dispersive behavior of the free infraparticle
%propagation.
%We compute the Hessian of $E(p,\sigma)$ at $p\in\R^3$,
%%As a linear map, it
%has a simple radial eigenvalue $\partial_{|p|}^2E(p,\sigma)$,
%and a tangential eigenvalue $\frac{\partial_{|p|}E(p,\sigma)}{|p|}$
%of multiplicity two.
%If $p=0$ it simplifies to a multiple of the identity,
%$\partial_{|p|}^2E(0,\sigma){\bf 1}_3$, as noted above.
%We shall refer to the inverse of the radial eigenvalue
%\subsection{Discussion}
% has the following crucial properties, as,cfp2
The following result is proven in \cite{bcfs2,ch,cfp1}: %that
\begin{theorem}
{\em For any $P$, s.t. $|P|<\frac13$, the infimum of the spectrum $\Eg(P)=\inf{\rm spec}(\Hn(P))$ is twice differentiable and satisfies $ 1 \; \leq \; m_{ren} \; \leq \; 1 \, + \, c \, g^2 \;$ for some $c>0$.}
%\eqn \label{m-ren-pert} m_{ren} \; \approx \; m\big( 1 \, + \, 8\pi \, \c^2\frac{\kappa}{m}\big) \;. \eeqn
\end{theorem}
%\begin{thm}
%has the following properties:
%\begin{enumerate}
%\item For any $\sigma>0$, $\Eg(p)$ is a simple eigenvalue.\\
%\item For any $p\in\cS$, $|\nabla_p \Eg(p)|< 1+C\alpha$.\\ %, uniformly in $\sigma\geq0$.
%\item
%For any $p\in\cS$, the infimum of the spectrum $\Eg(p)=\inf{\rm spec}(\Hn(p))$ is twice differentiable. %and satisfies $|\partial_{|p|}^2 \Eg(p)|< C$. %, uniformly in $\sigma\geq0$.
%\end{enumerate}
%\end{thm}
\bigskip
%
\DETAILS{We obtain this result by a formal perturbation theory in $e$. To this end we compute $\Eg(P)$ (by perturbation theory in $e$). We write $\Hn(P)$ as $\Hn(P)=H_0(P)+W$ where $H_0(P)= \big ( P - P_\re )^2 + H_\re$. Let $P$ be the orthogonal projection onto the
eigenspace $\Null(H_0 - \lam_0)$ spanned by the vacuum $\vac$, let
$\oP := \bfone - P$ and let $H_{0\oP}(P) := \oP H_0(P) \oP \restriction_{\Ran \oP}$ . Then a general result of perturbation theory (see Appendix \ref{subsec:pert}, \eqref{pert-exp})
gives
\begin{equation}\label{eq:pertexp}
\Eg(P) = \E0(P) + \sum_{n=0}^\infty(-1)^n \lan \vac, W' (\bar{R}_0 W')^n \vac \ran,
%+ e^2 \lan \vac,W \bar{R}_0 W \vac \ran + O(|e|^3)
\end{equation}
where $\E0(P)=|P|^2/2$, $W':=\oP(W+ \Eg(P) - \E0(P))\oP$ and $\bar{R}_0 = \oP ( H_{0 \oP}(P) - \E0(P) )^{-1} \oP$.
%Our expression for the renormalized mass, (\ref{unifmassbd}) - (\ref{unif-mass-const-1}), is confirmed by formal,
Using that $W=\, -e\frac{1}{m}( \, P \, - \, \Pf ) \, \cdot \Af +\frac{1}{2m}e^2\Af^2$ and therefore in particular $\lan \vac, W \vac \ran=0$. %The perturbative calculations, with %Indeed, for a sharp ultraviolet cutoff at $\Lambda$,
%and setting the bare mass equal to 1,
Now, we compute the second term on the r.h.s: %the ground state energy has the form
\eqn \label{E-ren-pert-lead-1}
&&\lan W \vac, \bar{R}_0 W \vac \ran \;= \; \frac{e^2}{m^2}\lan P \, \cdot \Af \vac, \bar{R}_0 P \, \cdot \Af \vac \ran \;\nonumber\\
& = & \; \frac{e^2}{m^2}P_iP_j\sum_{\lambda}\int dk \, \frac{\kappa_\sigma(|k|)}{|k|^{1/2}} \,
\e^i_\lambda(k) \,\sum_{\lambda'} \int dk' \, \frac{\kappa_\sigma(|k'|)}{|k'|^{1/2}} \,
\e^j_{\lambda'}(k') \, \;\nonumber\\
& \times & \lan \, \vac, a_\lambda(k) ( H_{0}(P) - \E0(P) )^{-1} a^*_{\lambda'}(k') \vac \ran.
\eeqn
To compute the inner product on the r.h.s., we move the operators $a(k)$ to the extreme right and
the operators $a^*(k)$ to the extreme left. In doing this
we use the following rules:
\begin{enumerate}
\item
$a(k)$ is pulled through $a^* (k)$ according to the
relation
\[
a(k) a^* (k') = a^* (k') a(k) + \delta (k-k')
\]
\item
$a(k)$ and $a^*(k)$ are pulled through $R_{0,\oP}$ according
to the relations
\begin{eqnarray*}
a(k) ( H_{0} - \E0(P) )^{-1} &=& ( H_{0}(P) +\omega(k)- \E0(P) )^{-1} a(k),\\
( H_{0} - \E0(P) )^{-1}a^*(k) &=& a^*(k)( H_{0}(P) +\omega(k)- \E0(P) )^{-1}
\end{eqnarray*}
(see the equations \eqref{eq:pull} and of \eqref{eq:pull2} of Appendix \ref{sec:pullthrough}).
\end{enumerate}
Using that $a(k)\vac=0$ and $H_{0}(P)\vac=\frac{|P|^2}{2}\vac$, we obtain, to leading order in $\c$,
\eqn \label{E-ren-pert}
E(P) \; = %\approx \; \frac{2 \, \pi \, \c^2 \, \kappa^2}{3} \, + \,
\frac{|P|^2}{2m}\Big(1 - 8\pi \, \c^2\frac{\kappa}{m} \Big)
%\Big(1 - \frac{16\pi \c^2}{3} \, \log\Big(1 \, + \, \frac\kappa2\Big)\Big)
\, + \, O(\c^4)
\eeqn
so that we infer \eqref{m-ren-pert}.
%which agrees with (\ref{unifmassbd}) and (\ref{unif-mass-const-1})
%(It is a common convention to include a factor $\frac{1}{\sqrt2}$ in the definition (\ref{Aks-def}); then, $\c$ would correspond to $\frac{\tilde \c}{\sqrt2}$, and $\frac{16\pi \c^2}{3}$ to $\frac{8\pi \widetilde \c^2}{3}$.) A presentation of the leading order calculations producing these results can for instance be found in \cite{sp1}. An important result of this paper is that the right hand side of (\ref{m-ren-pert-lead-1}) is the correct value of the renormalized mass, up to an error $o(\c^2)$.
Note that (\ref{E-ren-pert}) and (\ref{m-ren-pert}) depend
on the ultraviolet cutoff $\Lambda$.
\[\mbox{Representation of \eqref{E-ren-pert-lead-1} and more generally terms in \eqref{eq:pertexp} in terms of Feynman diagrams.}\]}
%
A presentation of the leading order calculations %producing these results
can be found in \cite{HiroshimaSpohn2}.
%Throughout this paper, we choose $\Lambda$ to be $O(1)$. For some preliminary results about the dependence of $E_0$ and $m_{ren}(0)$ when $\Lambda\rightarrow\infty$, see \cite{hasei,hispo,lilo,lilo1}, and \cite{sp1} for a recent survey.
\begin{remark}
{\em The estimate $ 1 \; \leq \; m_{ren} \; \leq \; 1 \, + \, c \, g^2 \;$ %Expression \eqref{m-ren-pert}
reflects the fact that the mass of the electron is increased
by interactions with the photon field.}
\end{remark}
%
\DETAILS{\begin{remark}
The existence of the ground state at $p=0$
(see also \cite{mol}), and
renormalization of the electron mass is an important ingredient for the
phenomenon of {\em enhanced binding}, \cite{cvv} and \cite{hispo1,hvv}. A
Schr\"odinger operator with a non-confining potential
can exhibit a bound state when the interaction of the
electron with the quantized electromagnetic field is included.
Binding to a shallow potential can be energetically more favorable for the electron
than forming an infraparticle through binding of a cloud of soft photons.
\end{remark}}
%
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{One-particle States}\label{sec:One-particle states} %\textbf{One-particle states.}
First we note that for $e=0$, the one-particle states of the Hamiltonian $H_0:=\Hn|_{e=0} = |P_\el|^2 + H_\re$ are the generalized eigenfunctions
\begin{equation}\label{eq:2.4}
e^{-iP\cdot x} \otimes \, \Omega,
\end{equation}
corresponding to the spectral points $E (P) = \frac{|P|^2}{2m}$. This corresponds to the true ground state $\vac$ of the fiber Hamiltonians $H_0(P):=\Hn(P)|_{e=0} = \frac{1}{2m} ( P - P_\re )^2 + H_\re$. The generalization of such a state for the interacting system would be the ground state of $\Hn(P)$, if it existed. However, we have
\begin{thm}
$\Hn(P)$ has a ground state if and only if $P=0$.
%has the following properties:
%\begin{enumerate}
\item %For any $\sigma>0$, $\Eg(p)$ is a simple eigenvalue.\\
%\item For any $p\in\cS$, $|\nabla_p \Eg(p)|< 1+C\alpha$.\\ %, uniformly in $\sigma\geq0$.
%\item
%For any $p\in\cS$, the infimum of the spectrum $\Eg(p)=\inf{\rm spec}(\Hn(p))$ is twice differentiable. %and satisfies $|\partial_{|p|}^2 \Eg(p)|< C$. %, uniformly in $\sigma\geq0$.
%\end{enumerate}
\end{thm}
To define one particle states for the interacting model, we first introduce IR regularization,
$\ks\in C_0^\infty([0,\kappa];\R_+)$ is assumed to be a smooth cutoff function obeying %$0 \; < \sup_{x\in\R_+}|x^{-\sigma}\ks(x)|< \; 10 \;$ and
%\eqn $0 \; < \; \|\ks\|_\sigma $ %+ \|x^\sigma\partial_x\big(x^{-\sigma}\ks\big)\|_\sigma \; < \; 10 \label{kappaassump} \eeqn
%\eqn
$\lim_{x\rightarrow0}\frac{\ks(x)}{x^\sigma} \; = \; 1\;.$ %\label{kappasig-norm-def-1}\eeqn
%where \eqn $\|f\|_\sigma \; := \; \sup_{x\in\R_+}|x^{-\sigma}f(x)|\;.$ %\eeqn
The corresponding Hamiltonian is
\eqn\label{eq-Hn-def-1}
\Hns \, := \, \frac{1}{2m}\big( \, -i\nabla_x \otimes 1_f \, + \, \c \Afs(x) \, \big)^2
\, + \, \1_{el} \otimes H_f
\eeqn
with the quantized electromagnetic vector potential subjected, besides the ultra-violet
cutoff, also infrared regularization,
\eqn
\Afs(x) \, = \,
\sum_{\lambda}\int dk \, \frac{\kappa_\sigma(|k|)}{|k|^{1/2}} \,
\{ \, \e_\lambda(k) \, e^{-ikx} \otimes a_\lambda(k) \, + \, h.c. \, \}.
\eeqn
Let $\cS:=\{P\in \R^3\ |\ |P|<1/3\}$.
\begin{thm}
For $P\in\cS$ and for any $\sigma>0$, the infimum of the spectrum $\Egs(P)=\inf{\rm spec}(\Hns(P))$ is a simple eigenvalue.\\
\end{thm}
\begin{remark}
The upper bound on $|P|$ of $\frac13$ is not optimal, but we note that,
For $E(P)$ to be an eigenvalue, $|P|$ cannot exceed a critical value $P_c< 1$ (corresponding
to the speed of light). As $|P|\rightarrow P_c$,
it is expected that the eigenvalue at $E(P)$ dissolves in
the continuous spectrum, while a resonance appears.
This is a manifestation of a phenomenon analogous to
Cherenkov radiation.
\end{remark}
Let $\Psig(P)\in\Fo$ denote the associated normalized fiber ground state, $\|\Psig(P)\|_\Fo=1$,
for $P\in\cS$,
\eqn
\Hns(P)\Psig(P) \, = \, \Egs(P) \, \Psig(P) \,.
\eeqn
The vector
$\Psi(P,\sigma)$ is an {\em infraparticle state}, describing a compound
particle comprising the electron together with a cloud of low-energy (soft)
photons whose expected number diverges as $\sigma\rightarrow0$, unless $p=0$. %\subsection{Bogoliubov transformation}
For $P\in \cS$, we introduce the Weyl operators
\eqn
%\lefteqn{
W_{\nablE(P)}(x) % } \ &:=& \exp\Big[\alpha^{\frac12}\sum_\lambda\int dk \, \kappa_\sigma(|k|) \, \frac{\nablE(p)\cdot\e_\lambda(k)a_\lambda(k)e^{-ikx}-h.c.}{|k|^{1/2}(|k|-\nablE(p)\cdot k)}\Big] \,, %\nonumber
:= e^{D(x)-D^*(x)}
\eeqn
where
\eqn
D(x) \, := \, \sum_\lambda \int dk \, G_\lambda(k,p) \, e^{-ikx} \, a_\lambda(k) \,,
\eeqn
with
\eqn
G_\lambda(k,p) \, := \,\alpha^{\frac12} \, \kappa_\sigma(|k|) \,
\frac{\nablE(p)\cdot\e_\lambda(k) }{|k|^{1/2}(|k|-\nablE(p)\cdot k)} \,.
\eeqn
(Here and in the sequel, we will use the abbreviated notation %\eqn
$\nablE(p) \, \equiv \, \nabla_p \Eg(p) \,.$)
We observe that they commute with the total momentum operator,
\eqn
[\Ptot \, , \, W_{\nablE(p)}(x)] \, = \, 0 \,.
\eeqn
To see this, we note that $[\Ptot,D(x)]=0=[\Ptot,D^*(x)]$.
Indeed, we have that
\eqn
\lefteqn{
\Ptot \, \sum_\lambda \int dk \, G_\lambda(k,p) \, e^{-ikx} \, a_\lambda(k) \, \psi(x)
}
\nonumber\\
&=&
\sum_\lambda \int dk \, G_\lambda(k,p) \, (-i\nabla_x+\Pf) \, e^{-ikx} \, a_\lambda(k) \, \psi(x)
\nonumber\\
&=&
\sum_\lambda \int dk \, G_\lambda(k,p) \, e^{-ikx} \, a_\lambda(k) \, (-i\nabla_x+k+\Pf-k) \, \psi(x)
\nonumber\\
&=&
\sum_\lambda \int dk \, G_\lambda(k,p) \, e^{-ikx} \, a_\lambda(k) \, \Ptot \, \psi(x) \,.
\eeqn
Accordingly, we infer that $W_{\nablE(p)}(x) =\exp[D(x)-D^*(x)]$ commutes with $\Ptot$.
Furthermore, we observe that
\eqn
W_{\nablE(p)}(x) \, e^{i(p-\Pf)x} \, = \, e^{i(p-\Pf)x} \, W_{\nablE(p)}
\eeqn
holds. Here and in what follows, we will use the abbreviated notation
\eqn
W_{\nablE(p)} \, \equiv \, W_{\nablE(p)}(x=0) \,.
\eeqn
We define the maps
\eqn
(\Wmap\phi)(x) & := & \int dp \, W_{\nablE(p)}(x) \, e^{i(p-\Pf)x} \, \widehat\phi(p)
\nonumber\\
&=& \int dp \, e^{i(p-\Pf)x} \, W_{\nablE(p)} \, \widehat\phi(p) \,.
\eeqn
Likewise,
\eqn
(\Wmap^*\phi)(x)
& := & \int dp \, W_{\nablE(p)}^*(x) \, e^{i(p-\Pf)x} \, \widehat\phi(p) \,.
\eeqn
The associated Bogoliubov-transformed Hamiltonian is given by
\eqn
\Kns \, := \, (\Wmap\Hns\Wmap^{*}) \,.
\eeqn
We also introduce the Bogoliubov-transformed fiber Hamiltonians
\eqn
\Kns(p) \, := \, W_{\nablE(p)} \, \Hns(p) \, W_{\nablE(p)}^* \,.
\eeqn
Then,we observe that
\eqn
\Kns & = & (\Wmap \Hns \Wmap^{*})(x)
\nonumber\\
&=& \int W_{\nablE(p)}(x) \, e^{i(p-\Pf)x}\Hn(p)e^{-i(p-\Pf)x} \, dP_{\Ptot}(p) \, W_{\nablE(p)}^*(x)
\nonumber\\
&=& \int W_{\nablE(p)}(x) \, e^{i(p-\Pf)x}\Hn(p)e^{-i(p-\Pf)x} \, W_{\nablE(p)}^*(x) \, dP_{\Ptot}(p)
\nonumber\\
& = & \int e^{i(p-\Pf)x}\Kn(p)e^{-i(p-\Pf)x}dP_{\Ptot}(p).
\eeqn
In particular, we have that
\eqn
\Wmap (\Hns\psi) \, = \, \Kns(\Wmap\psi) \,,
\eeqn
as can be readily verified.
%We define the fiber Hamiltonians
%\eqn
% \Kn(p) & := & (e^{i\Pf x}\Hn e^{-i\Pf x})|_{\cH_p}
% \nonumber\\
% & = &
% \frac12( \, p \, - \, \Pf \, - \, \g \Af \, )^2
% \, + \, \1_{el} \otimes H_f
%\eeqn
%where
%\eqn
% \Af \, \equiv \, \Af(0) \, = \,
% \sum_{\lambda}\int dk \, \frac{\kappa_\sigma(|k|)}{|k|^{1/2}} \,
% \{ \, \e_\lambda(k) \, a_\lambda(k) \, + \, h.c. \, \} \,.
%\eeqn
Defining
\eqn
\Phsig(p) \, := \, W_{\nablE(p)} \, \Psig(p) \,,
\eeqn
we obtain
\eqn
\Kns(p) \, \Phsig(p) \, = \, \Egs(p) \, \Phsig(p) \,.
\eeqn
The following result is proven in \cite{cfp1}. %,cfp2},.
\begin{thm}
For any $P\in \cS$,
the ground state eigenvector $\Phsig(P)$ of $\Kns(P)$ converges strongly in $\Fo$:
$\Phi(P):=\lim_{\sigma\rightarrow0}\Phsig(P)$ exists in $\Fo$.
\end{thm} %}
%
%
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Pull-through formulae}\label{sec:pullthrough}
In this appendix we prove the very useful ``pull-through'' formulae
(see \cite{bfs1})
\begin{equation}\label{eq:pull}
a(k) f(H_f) = f(H_f + \omega(k)) a(k)
\end{equation}
and
\begin{equation}\label{eq:pull2}
f(H_f) a^*(k) = a^*(k)f(H_f+\omega(k)),
\end{equation}
valid for any piecewise continuous, bounded
function, $f$, on $\BR$.
%\begin{hw} {\em
First, using the commutation relations for $a(k),\ a^*(k)$,
one proves relations~(\ref{eq:pull})-~(\ref{eq:pull2})
for $f(H) = (H_f - z)^{-1}$,
$z \in \BC / \bar{\BR}^+$.
%} \end{hw}
Then using the Stone-Weierstrass theorem,
one can extend~(\ref{eq:pull})-~(\ref{eq:pull2})
from functions of the form
$f(\lambda) = (\lambda - z)^{-1}$,
$z \in \BC \backslash \bar{\BR}^+$,
to the class of functions mentioned above.
%
\DETAILS{Alternatively, (\ref{eq:pull}) -~(\ref{eq:pull2})
follow from the relation
\[
f(H_f) \prod_{j=1}^N a^*(k_j) \Omega
= f(\sum_{i=1}^N \omega(k_i)) \prod_{j=1}^N a^*(k_j) \Omega.
\]
\begin{hw}
{\em Prove this last relation, and
derive~(\ref{eq:pull})-~(\ref{eq:pull2})
from it.}
\end{hw}}
%
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Background on the Fock space***}\label{sec-SA}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%Let $ \fh$ be either $ L^2 (\RR^3, \mathbb{C}, d^3 k)$ or $ L^2
%(\RR^3, \mathbb{C}^2, d^3 k)$. In the first case we consider $ \fh$
%
%as the Hilbert space of one-particle states of a scalar Boson or a
%phonon, and in the second case, of a photon. The variable
%%$k\in\RR^3$ is the wave vector or momentum of the particle. (Recall
%that throughout this paper, the velocity of light, $c$, and Planck's
%constant, $\hbar$, are set equal to 1.)
%The Bosonic Fock space, $\cF$, over $L^2 (\R^3, \mathbb{C},d^3 k)$ (or $ L^2 (\mathbb{R}^3, \mathbb{C}^2, d^3 k)$) is defined by %\begin{equation} \label{eq-I.10}\cF \ := \ \bigoplus_{n=0}^{\infty} \cS_n \, L^2 (\mathbb{R}^3, \mathbb{C},d^3 k)^{\otimes n} ,\end{equation}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Supplement: Creation and Annihilation Operators}\label{sec:crannihoprs}
Let $ \fh$ be either $ L^2 (\RR^3, \mathbb{C}, d^3 k)$ or $ L^2
(\RR^3, \mathbb{C}^2, d^3 k)$. In the first case we consider $ \fh$
%
to be the Hilbert space of one-particle states of a scalar boson or
phonon, and in the second case, of a photon. The variable
$k\in\RR^3$ is the wave vector or momentum of the particle. (Recall
that throughout these lectures, the propagation speed $c$, of photon or
photons and Planck's constant, $\hbar$, are set equal to 1.) The
Bosonic Fock space, $\cF$, over $\fh$ is defined by
%
\begin{equation} \label{fock}
\cF \ := \ \bigoplus_{n=0}^{\infty} \cS_n \, \fh^{\otimes n} , %\comma
\end{equation}
%
where $\cS_n$ is the orthogonal projection onto the subspace of
totally symmetric $n$-particle wave functions contained in the
$n$-fold tensor product $\fh^{\otimes n}$ of $\fh$; and $\cS_0
\fh^{\otimes 0} := \CC $. The vector $\Om:= (1, 0, ... )$ %1\bigoplus_{n=1}^{\infty}0$
is called the \emph{vacuum vector} in
$\cF$. Vectors $\Psi\in \cF$ can be identified with sequences
$(\psi_n)^{\infty}_{n=0}$ of $n$-particle wave functions, which are
totally symmetric in their $n$ arguments, and $\psi_0\in\CC$. In the
first case these functions are of the form, $\psi_n(k_1, \ldots,
k_n)$, while in the second case, of the form $\psi_n(k_1, \lambda_1,
\ldots, k_n, \lambda_n)$, where $\lambda_j \in \{-1, 1\}$ are the
polarization variables.
%
\DETAILS{The Bosonic Fock space, $\cF$, over $L^2 (\R^3, \mathbb{C},
d^3 k)$ (or $ L^2 (\mathbb{R}^3, \mathbb{C}^2, d^3 k)$) is defined by
%
\begin{equation} \label{eq-I.10}
\cF \ := \ \bigoplus_{n=0}^{\infty} \cS_n \, L^2 (\mathbb{R}^3, \C^2,
d^3 k)^{\otimes n} ,
\end{equation}
where $\cS_n$ is the orthogonal projection onto the subspace of
totally symmetric $n$-particle wave functions contained in the
$n$-fold tensor product $L^2 (\RR^3, \mathbb{C}, d^3 k)^{\otimes n}$
of $L^2 (\RR^3, \mathbb{C}, d^3 k)$; and $\cS_0 L^2 (\RR^3,
\mathbb{C}, d^3 k)^{\otimes 0} := \BC $. The vector $\Omega:=1
\bigoplus_{n=1}^{\infty}0$ is called the \emph{vacuum vector} in
$\cF$. Vectors $\Psi\in \cF$ can be identified with sequences
$(\psi_n)^{\infty}_{n=0}$ of $n$-particle wave functions,
$\psi_n(k_1, \ldots, k_n)$, which are
totally symmetric in their $n$ arguments, and $\psi_0 \in \BC$.
and $a^*(k)$ and $a(k)$ denote the creation and annihilation
operators on $\cF$. The families $a^*(k)$ and $a(k)$ are
operator-valued generalized, transverse vector fields: (below $a_{\lambda}^{\#}= a_{\lambda}$ or $a_{\lambda}^*$.)
$$a^\#(k):= \sum_{\lambda \in \{-1, 1\}}
e_{\lambda}(k) a^\#_{\lambda}(k),$$ where $e_{\lambda}(k)$ are
polarization vectors, i.e. orthonormal vectors in $\mathbb{R}^3$
satisfying $k \cdot e_{\lambda}(k) =0$, and $a^\#_{\lambda}(k)$ are
scalar creation and annihilation operators, satisfying the \emph{canonical
commutation relations}:
%
\begin{equation} \label{ccr}
\big[ a_{\lambda}^{\#}(k) \, , \, a_{\lambda'}^{\#}(k') \big] \ = \ 0 , \hspace{8mm}
\big[ a_{\lambda}(k) \, , \, a_{\lambda'}^*(k') \big] \ = \ \del_{\lam\lam'}\delta^3 (k-k').
\end{equation}}
%
%
%In the first case while in the second case, of the form $\psi_n(k_1, \lambda_1, \ldots, k_n,
%\lambda_n)$, where $\lambda_j \in \{-1, 1\}$ are the polarization
%variables.
In what follows we present some key definitions in the first case,
limiting ourselves to remarks at the end of this appendix on how
these definitions have to be modified for the second case. The
scalar product of two vectors $\Psi$ and $\Phi$ is given by
%
\begin{equation} \label{F-scalprod}
\langle \Psi \, , \; \Phi \rangle \ := \ \sum_{n=0}^{\infty} \int
\prod^n_{j=1} d^3k_j \; \overline{\psi_n (k_1, \ldots, k_n)} \:
\varphi_n (k_1, \ldots, k_n) .
\end{equation}
%
Given a one particle dispersion relation $\omega(k)$, the energy of
a configuration of $n$ \emph{non-interacting} field particles with
wave vectors $k_1, \ldots,k_n$ is given by $\sum^{n}_{j=1}
\omega(k_j)$. We define the \emph{free-field Hamiltonian}, $H_f$,
giving the field dynamics, by
%that the vector $\hf \Psi$, with $\Psi=(\psi_n)_{n=0}^{\infty}$ ($\psi_n=0$,
%except for finitely many $n$, and $\psi_n(k_1,\ldots,k_n)$ of rapid decrease in
%$k_1,\ldots,k_n$), corresponds to the sequence
%$(\vphi_n)^{\infty}_{n=0}$ of $n$-particle wave functions given by
%
\begin{equation} \label{Hfn}
(H_f \Psi)_n(k_1,\ldots,k_n) \ = \ \Big( \sum_{j=1}^n \omega(k_j)
\Big) \: \psi_n (k_1, \ldots, k_n) ,
\end{equation}
for $n\ge1$ and $(H_f \Psi)_n =0$ for $n=0$. Here
$\Psi=(\psi_n)_{n=0}^{\infty}$ (to be sure that the r.h.s. makes
sense we can assume that $\psi_n=0$, except for finitely many $n$,
for which $\psi_n(k_1,\ldots,k_n)$ decrease rapidly at infinity).
Clearly that the operator $H_f$ has the single eigenvalue $0$ with
the eigenvector $\Omega$ and the rest of the spectrum absolutely
continuous.
With each function $\varphi \in L^2 (\RR^3, \mathbb{C}, d^3 k)$ one
associates an \emph{annihilation operator} $a(\varphi)$ defined as
follows. For $\Psi=(\psi_n)^{\infty}_{n=0}\in \cF$ with the property
that $\psi_n=0$, for all but finitely many $n$, the vector
$a(\varphi) \Psi$ is defined by
%
\begin{equation} \label{a}
(a(\varphi) \Psi)_n (k_1, \ldots, k_n) \ := \ \sqrt{n+1 \,} \, \int
d^3 k \; \overline{\varphi(k)} \: \psi_{n+1}(k, k_1, \ldots, k_n).
\end{equation}
%
These equations define a closable operator $a(\varphi)$ whose
closure is also denoted by $a(\varphi)$. Eqn \eqref{eq-I.12} implies
the relation
%
\begin{equation} \label{aOm}
a(\varphi) \Omega \ = \ 0 ,
\end{equation}
%
The creation operator $a^*(\varphi)$ is defined to be the adjoint of
$a(\varphi)$ with respect to the scalar product defined in
Eq.~\eqref{F-scalprod}. Since $a(\varphi)$ is anti-linear, and
$a^*(\vphi)$ is linear in $\varphi$, we write formally
%
\begin{equation} \label{ak}
a(\varphi) \ = \ \int d^3 k \; \overline{\varphi(k)} \, a(k) ,
\hspace{8mm} a^*(\varphi) \ = \ \int d^3 k \; \varphi(k) \, a^*(k),
\end{equation}
%
where $a(k)$ and $a^*(k)$ are unbounded, operator-valued
distributions. The latter are well-known to obey the \emph{canonical
commutation relations} (CCR):
%
\begin{equation} \label{CCR}
\big[ a^{\#}(k) \, , \, a^{\#}(k') \big] \ = \ 0 , \hspace{8mm}
\big[ a(k) \, , \, a^*(k') \big] \ = \ \delta^3 (k-k') ,
\end{equation}
%
where $a^{\#}= a$ or $a^*$.
Now, using this one can rewrite the quantum Hamiltonian $H_f$ in
terms of the creation and annihilation operators, $a$ and $a^*$, as
%
%
\begin{equation} \label{Hfa}
H_f \ = \ \int d^3 k \; a^*(k)\; \omega(k) \; a(k) ,
\end{equation}
%
acting on the Fock space $ \cF$.
More generally, for any operator, $t$, on the one-particle space $
L^2 (\mathbb{R}^3, \mathbb{C}, d^3 k)$ we define the operator $T$ on
the Fock space $\cF$ by the following formal expression $T: = \int
a^*(k) t a(k) dk$, where the operator $t$ acts on the $k-$variable
($T$ is the second quantization of $t$). The precise meaning of the
latter expression can obtained by using a basis $\{\phi_j\}$ in the
space $ L^2 (\mathbb{R}^3, \mathbb{C}, d^3 k)$ to rewrite it as $T:
= \sum_{j} \int a^*(\phi_j) a(t^* \phi_j) dk$.
To modify the above definitions to the case of photons, one replaces
the variable $k$ by the pair $(k, \lambda)$ and adds to the
integrals in $k$ sums over $\lambda$. In particular, the creation-
and annihilation operators have now two variables: $a_
\lambda^\#(k)\equiv a^\#(k, \lambda)$; they satisfy the commutation
relations
\begin{equation}
\big[ a_{\lambda}^{\#}(k) \, , \, a_{\lambda'}^{\#}(k') \big] \ = \
0 , \hspace{8mm} \big[ a_{\lambda}(k) \, , \,
a_{\lambda'}^*(k') \big] \ = \ \delta_{\lambda, \lambda'} \delta^3
(k-k') .
\end{equation}
One can introduce the operator-valued transverse vector fields by
$$a^\#(k):= \sum_{\lambda \in \{-1, 1\}} e_{\lambda}(k) a_{\lambda}^\#(k),$$
where $e_{\lambda}(k) \equiv e(k, \lambda)$ are polarization
vectors, i.e., orthonormal vectors in $\mathbb{R}^3$ satisfying $k
\cdot e_{\lambda}(k) =0$. Then, in order to reinterpret the
expressions in this paper for photons, one either adds the variable
$\lambda$, as was mentioned above, or replaces, in appropriate
places, the usual product of scalar functions or scalar functions
and scalar operators by the dot product of vector-functions or
vector-functions and operator-valued vector-functions.
%The right side of \eqref{Hf} can be understood as a weak integral.
%is subject to an ultraviolet cut-off, Of course, in the physical case
%keep the general $d$ here as we use this notation also for the
%In (\ref{eq-I.18}) the dimension $d$ stands for $3$, but we use
%(\ref{eq-I.18}) also to denote the free field Hamiltonian in the
%Nelson model where we consider an arbitrary dimension.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
%$$XXXXXXXXXXXXXXXXXXXXXXXXX$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
\DETAILS{
\section{Related Problems}\label{sec:relatprobl}
Similar techniques are or can be used to obtain
\begin{itemize}
\item The mass renormalization for electrons,
\item Local decay of scattering states,
\item Existence and stability of thermal states.
\end{itemize}
We will discuss this at the end of these lectures. }
%
\bigskip
\DETAILS{
where $X_{\theta} := U_{\theta}e^{-ig F} $ with $F$, the
self-adjoint operator \textbf{defined below}. The
transformation $H \rightarrow e^{-ig F} H_{g}^{SM} e^{ig
F}$ is a generalization of the well-known Pauli-Fierz
transformation. Note that the operator-family $X_{\theta}$ has the
following two properties needed in order to establish the desired
properties of the resonances:
(a) $U_{\theta}$ are unitary for $\theta \in \mathbb{R}$;
(b) $U_{\theta_1 +\theta_2}= U_{\theta_1}U_{\theta_2}$ where
$U_{\theta}$ are unitary for $\theta \in \mathbb{R}$. }
%the complex plane $\mathbb{C}$.
%
%Now Eqn \eqref{HPF} implies that \begin{equation} \label{}
%\langle\psi_F, \ (H^{SM}_{g } -z)^{-1}\phi_F\rangle = \langle \psi
%, ( H^{PF}_{g } -z)^{-1}\phi \rangle \ .\end{equation}
%Doing the complex deformation of both sides of this equation gives,
%as above,
%\begin{equation} \label{}
%\langle\psi_{F\bar\theta}, \ (H^{SM}_{g \theta}
%-z)^{-1}\phi_{F\theta}\rangle = \langle \psi_{\bar\theta} , (
%H^{PF}_{g \theta} -z)^{-1}\phi_{\theta} \rangle \ .
%\end{equation}
%The last equation implies that the Hamiltonians $H^{PF} _{g \theta}$
%and $H^{SM} _{g \theta}$ have the same resolvent set and the same
%eigenvalues (real and complex). This
%yields the statement of Theorem II.1. \QED
%
%Fixed points of $\sigma_{cont} (H^{\#}_{g \theta})$ under small
%variations of $\theta$ will be called the {\it thresholds} of
%$H^{\#}_{g \theta}$.
\bigskip
\DETAILS{
To prove the results mentioned above we apply the
spectral renormalization group (RG) method
(BachChenFroehlichSigal2003,
BachFroehlichSigal1998a,BachFroehlichSigal1998b, GriesemerHasler2,
FroehlichGriesemerSigal2008b) to the Hamiltonians $H_{g \theta=0}^{SM}= e^{-ig F}
H_{g}^{SM} e^{ig F}$ (the ground state case) and $H_{g \theta}^{SM},\ \rIm \theta >0,$
(the resonance case).
%We change this method in a key aspect by introducing
Note that the version of RG needed in this work uses new --
anisotropic -- Banach spaces of operators, on which the
renormalization group acts.
%These Banach spaces codify anisotropic infra-red behaviour of
%Hamiltonians under consideration.
Using the RG technique we describe the spectrum of the operator
$H^{SM} _{g \theta}$ in $\{z \in \mathbb{C}^- |\ \epsilon_{0} < \rRe
z < \nu$\} from which we derive Theorems \ref{thm-main} and
\ref{thm-main2}. }
%
%belongs to a union of the wedges $S_j,\ j \le j(\nu)$,
%all of which but one are in $\mathbb{C}^-$,
%with the eigenvalues of $H^{SM} _{g \theta}$ residing at their
%vertices $\epsilon_j,\ \rIm \epsilon_j \le 0$.
%The exceptional cuspidal domain has it cusp at the ground state
%energy, $\epsilon_0$, which is real, and the rest in $\mathbb{C}^-$.
%Due to \eqref{I.7} this will imply Theorem \ref{thm-main2}.
%
%Our analysis does not apply directly to the QED model,
%Eq.~(\ref{eq1}): for it to work the ultra-violet cut-off function
%$\chi$ in Eq.~(\ref{eq3}) would have to obey Eq.~(\ref{I.7}) with
%$\mu >0$. So we transform the QED Hamiltonian appropriately, apply our approach
%
%Finally, we derive the desired properties of the original
%Hamiltonian from the obtained properties of the transformed
%Hamiltonian.
%However, our approach is well applicable to the Hamiltonian obtained
%from Eq.~(\ref{eq1}) by applying the
%transform (see \cite{BachFroehlichSigal1999}),
%%%%%%%%%%%35
%Note that in this approach a specific form of the interaction and
%the coupling functions becomes irrelevant.
%
%In particular, as was mentioned above, our approach can handle the
%perturbations quadratic in creation and annihilation operators, $a$
%and $a^*$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\bibliography{/home/vbach/BIB/volle}
%\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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