Content-Type: multipart/mixed; boundary="-------------0611132308609"

This is a multi-part message in MIME format.

---------------0611132308609
Content-Type: text/plain; name="06-329.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="06-329.keywords"

Pauli-Fierz, infrared problem, non-relativistic QED, translation invariant, absence of ground states, electron, ion, photons
---------------0611132308609
Content-Type: application/x-tex; name="movingions5.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="movingions5.tex"

\documentclass[12pt]{article}
\usepackage{amsthm,amsmath,amsfonts,amssymb}
%\usepackage{amsthm,amsmath,amsfonts,amssymb,cases,showkeys}
\usepackage{amsmath,amsfonts,amsthm}
\setlength{\textwidth}{16cm} \setlength{\textheight}{22cm}
\setlength{\topmargin}{0cm} \addtolength{\evensidemargin}{-1.5cm}
\addtolength{\oddsidemargin}{-1.5cm}
%\numberwithin{equation}{section}

%%%%SETS

\newcommand{\R}{{\mathord{\mathbb R}}}
\newcommand{\Z}{{\mathord{\mathbb Z}}}
\newcommand{\N}{{\mathord{\mathbb N}}}
\newcommand{\C}{{\mathord{\mathbb C}}}
\newcommand{\I}{{\mathord{\mathbb I}}}
\newcommand{\HS}{{\mathord{\mathbb H}}}
\newcommand{\E}{{\mathord{\mathbb E}}}
\newcommand{\D}{{\mathord{\mathbb D}}}

%%%GREEKS

\def\om {\omega}
\def\eps {\epsilon}
\def\lam {\lambda}
\def\g   {\gamma}
\def\a   {\alpha}
\def\b   {\beta}

%%%%MATHCAL

\newcommand{\HH}{\mathcal{H}}
\newcommand{\DD}{\mathcal{D}}
\newcommand{\FF}{\mathcal{F}}
\newcommand{\CC}{\mathcal{C}}
\newcommand{\VV}{\mathcal{V}}
\newcommand{\WW}{\mathcal{W}}
\newcommand{\hh}{\mathfrak{h}}

%%%%MATHROMAN


\def\i {{i}}
\def\e {{e}}
\def\d {{d}}
\newcommand{\ran}{{\rm Ran}}

%%%%SPECIAL

\newcommand{\w}[1]{\widehat{#1}}
\newcommand{\ov}[1]{\widetilde{#1}}
\newcommand{\overl}[1]{\overline{#1}}

\renewcommand{\baselinestretch}{1.3}

%%%ONE

\def\one{{\sf 1}\mkern-5.0mu{\rm I}}


%%%MATHENVIRONMENT

\newcommand{\ben}{\begin{displaymath}}
\newcommand{\een}{\end{displaymath}}
\newcommand{\beqn}{\begin{equation}}
\newcommand{\eeqn}{\end{equation}}
\newcommand{\beqna}{\begin{eqnarray*}}
\newcommand{\eeqna}{\end{eqnarray*}}

%%%MATHCOMMANDS

\def\inf{{\rm inf}\,}
\def\tr{\operatorname{tr}}
\def\Tr{\operatorname{Tr}}
\def\const{{\rm const}\,}
\def\dist{{\rm dist}\,}
\def\supp{\operatorname{supp}}
\def\interior {\mathrm{intr}}
\def\boundary  {\mathrm{bdry}}
\newcommand{\der}[1]{\frac{\partial}{\partial #1}}
\newcommand{\sfrac}[2]{\textrm{\footnotesize $\frac{#1}{#2}$}}
\newcommand{\ssfrac}[2]{\textrm{\scriptsize $\frac{#1}{#2}$}}
\newcommand{\restricted}{|\grave{}\,}

%%%MATHTHEOREM

\newtheorem{lemma}{Lemma}
\newtheorem{theorem}[lemma]{Theorem}
\newtheorem{remark}[lemma]{Remark}
\newtheorem{proposition}[lemma]{Proposition}
\newtheorem{corollary}[lemma]{Corollary}
\newtheorem{definition}[lemma]{Definition}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title{Absence of Ground States for a Class of
Translation Invariant Models of Non-relativistic QED}

\date{ }

\author{D. Hasler, I. Herbst}


\begin{document}




\maketitle


\begin{abstract}
We consider translation invariant models of non-relativistic QED
with net charge. Under certain natural assumptions we prove that
ground states do not exist.
\end{abstract}



\section{Introduction}




Over the years there has been much interest in trying to develop an
appropriate mathematical framework to describe the interaction of
charged particles with the quantized electromagnetic field, see for
example \cite{SPOHN04} and references given therein.  Of course
relativistic quantum electrodynamics (QED) is a very successful
theory but has not been shown to provide a Hilbert space framework
for describing the states of charged particles interacting with
photons.  In spite of this there are certainly prescriptions for
getting correct answers to the ``right'' questions \cite{IZ80}.

One of the first questions which arises is perhaps the most
elementary:  Are there ``dressed one-electron states'' of fixed
momentum which are eigenstates of the appropriate Hamiltonian. These
states should of course have an adhering photon cloud.   In
\cite{FK} Faddeev and  Kulish gave a suggestion what form such
states should take.  As far as we know however, this specific
suggestion has not been shown to be a rigorous mathematical outcome
of a basic theory. The Fadeev-Kulish states do not live in Fock
space because of the nature of the photon cloud.

In recent years the simpler mathematical model of non-relativistic
QED has been studied by many authors (after some pioneering early
work of Fr\"{o}hlich \cite{F73,F74}), see for example
\cite{SPOHN04}. This model suffers from various difficulties but it
is hoped that it may serve as a reasonably realistic model for low
energies, and a testing ground for understanding the infrared
problem.  One of the main difficulties is that this model is neither
Galilean nor Poincar\'{e} covariant.  The charged particles are
treated non-relativistically while the photons are relativistic.
There remains an ultra-violet cutoff in the photon field to produce
a well defined theory, but the theory is well defined without an
infrared cutoff.

In this work we define our Hamiltonians on the Hilbert space
consisting of the Fock space for photons tensored with the usual
Hilbert space for the non-relativistic charged particles.  It may
turn out that there are more suitable Hilbert spaces where the
theory makes good physical and mathematical sense and the ground
states, which we show do not exist, are resurrected.  In any case,
it seems to us that there is much work which needs to be done to
describe scattering in a Hilbert space framework.  Even in the case
considered where one-electron ground states do not exist for fixed
total momentum  an elegant theory may yet exist where a dressed
one-electron state is thought of as a limiting situation of a
non-normalizable state of fixed energy in the continuum just as a
state of fixed momentum is also a limiting concept.

In this paper  we consider translation invariant models of
non-relativistic QED having a total net charge. The generator of
translations defines the operator of total momentum. Translation
invariance implies that the Hamiltonian commutes with this operator.
We can thus  restrict the Hamiltonian to any subspace of fixed total
momentum $\xi$. This restricted Hamiltonian is denoted by $H(\xi)$.
For any momentum $\xi$, $H(\xi)$ is bounded from below. We denote
the infimum of its spectrum by $E(\xi)$. One can easily show the the
function $E(\cdot)$ is almost everywhere differentiable. In this
paper we show that for momenta $\xi$ at which $E(\cdot)$ has a
non-vanishing derivative, $H(\xi)$ does not admit a ground state. We
do not impose an infrared cutoff, which in fact is the reason for
the absence of ground states. The coupling constant is arbitrary,
but nonzero.

First we consider an electron (with spin $1/2$) coupled to the
quantized electromagnetic field. We show that  $H(\cdot)$ does not
admit a ground state at points where $E(\cdot)$ has a non-vanishing
derivative. This model has been previously investigated in
\cite{C01}, where it has been shown that $H(0)$  does have a ground
state provided the coupling is sufficiently small. However, the
result given there does not prove the absence of ground states for
nonzero $\xi$. Moreover, in
 \cite{C01,BCFS05} it has been shown that $E(\cdot)$
has a non-vanishing derivative for small coupling constant and small
but nonzero $\xi$. Combining this with our result we see that for
such small $\xi$ and small coupling, $H(\xi)$ does not have a ground
state. The model is introduced and the result is stated in Section
\ref{sec:eldef}. The proof of the result is presented in Section
\ref{sec:elpro}.

We then generalize the above result to a positive ion. More
specifically, we consider a spinless nucleus with nuclear charge
$Ze$ and $N$ electrons each with charge $-e$ where the interaction
between the particles includes the Coulomb potential. If $Z \neq N$,
we show that $H(\cdot)$ does not admit a ground state at points
where $E(\cdot)$ has a non-vanishing derivative. This model has been
recently investigated in \cite{LMS06,AGG06}, where it was shown that
under natural assumptions $H(\xi)$ does have a ground state provided
$N=Z$. In contrast to our result,  Coulomb systems without coupling
to the quantized electromagnetic field  do have positive ions,  with
fixed nonzero total momentum. In Section \ref{sec:eldef} we
introduce the model describing an ion  and state the result. Its
proof is presented in Section \ref{sec:elpro}.

In the next section  we introduce the Fock space of photons.

\section{Fock Space of  Photons}

\label{sec:nonqed}

 The degrees of freedom of the photons are
described by a symmetric Fock space, introduced as follows. Let
$$
\hh := L^2(\Z_2  \times \R^3 ) \cong L^2(\R^3 ; \C^2 )  \;
$$
denote the Hilbert space  of a transversally polarized photon. The
variable $\underline{k} = (\lambda,k) \in  \Z_2  \times \R^3$ consists
of the wave vector $k$ or momentum of the particle and  $\lambda$
describing the polarization. The symmetric Fock space, $\FF$, over
$\hh$ is defined by
$$
\FF = \C \oplus \bigoplus_{n=1}^\infty S_n( \hh^{\otimes n} ) \; ,
$$
where $S_n$ denotes the orthogonal projection onto the subspace of
totally symmetric tensors. The vacuum  is the vector $\Omega :=
(1,0,0, ... ) \in \FF$. The  vector $\psi \in \FF$ can be identified
with sequences $(\psi_n)_{n=0}^\infty$ of $n$-photon wave functions,
$\psi_n(\underline{k}_1, ... , \underline{k}_n) \in L^2((
\Z_2 \times \R^3)^n)$, which for $n \geq 1$ are totally symmetric in their $n$
arguments. The Fock space inherits a scalar product from $\hh$,
explicitly
$$
( \psi , \varphi )_\FF =  \overline{\psi}_0 \varphi_0 +
\sum_{n=1}^\infty \int \overline{\psi}_n(\underline{k}_1, ... ,
\underline{k}_n) \varphi_n(\underline{k}_1, ... , \underline{k}_n )
d \underline{k}_1 ...  d \underline{k}_n \; ,
$$
where we used the abbreviation $\int d \underline{k} =
\sum_{\lambda=1,2} \int dk$. The number operator $N$ is defined by
$(N \psi)_n = n \psi_n$. It is self-adjoint on the domain $D(N) :=
\{ \psi \in \FF | N\psi \in \FF\}$. For each function $f \in \hh$
one associates an annihilation operator $a(f)$ as follows.  For a
vector $\psi \in \FF$ we define
$$
(a(f) \psi )_n(\underline{k}_1, ... , \underline{k}_n ) =
(n+1)^{1/2} \int  \overline{f}(\underline{k})
\psi_{n+1}(\underline{k}, \underline{k}_1, ... , \underline{k}_n ) d
\underline{k} \quad \forall \ n \geq 0 \;  .
$$
The domain of $a(f)$ is the set of all $\psi$ such that $a(f)\psi
\in \FF$. Note that $a(f) \Omega = 0$. The creation operator
$a^*(f)$ is defined to be the adjoint of $a(f)$. Note that
 $a(f)$ is anti-linear, and $a^*(f)$ is linear in $f$. They are well
 known to satisfy the canonical commutation relations
$$
[ a^*(f) , a^*(g) ] = 0 \quad , \quad [a(f) , a(g) ] = 0 \quad ,
\quad [a(f) , a^*(g) ] =  (f , g) \; .
$$
where $f , g \in L^2( \Z_2 \times R^3 )$ and  $(f , g)$ denotes the
inner product of $L^2( \Z_2 \times R^3 )$. Since $a(f)$ is
anti-linear, and $a^*(f)$ is linear in $f$, we will write
$$
a(f) = \int     \overline{f}(\underline{k})   a_{\underline{k}}
d\underline{k} \quad , \quad a^*(f) = \int      f(\underline{k})
a^*_{\underline{k}} d\underline{k} \; ,
$$
where the right hand side is merely a different notation for the
expression on the left. For a function $f \in L^2(\R^3)$, we will
write $a_\lambda(f) := a(f_\lambda)$ and $a^*_\lambda(f) :=
a^*(f_\lambda)$, where $f_\lambda$ is the function defined by
$f_\lambda(\mu,k) := f(k) \delta_{\lambda,\mu}$. The field energy
operator denoted by $H_f$ is given by
$$
(H_f \psi)_n(\underline{k}_1, ... \underline{k}_n) = \left(
\sum_{i=1}^n |k_i| \right) \psi_n(\underline{k}_1, ...
\underline{k}_n) \; .
$$
It is self-adjoint on its natural domain
$D(H_f) := \{ \psi \in \FF | H_f \psi \in \FF \}$.
The operator of momentum $P_f$ is given by
$$
(P_f \psi)_n(\underline{k}_1, ... \underline{k}_n) = \left(
\sum_{i=1}^n k_i \right) \psi_n(\underline{k}_1, ...
\underline{k}_n) \; .
$$
Its components $(P_f)_j$ are each self-adjoint on the domain
$D((P_f)_j) := \{ \psi \in \FF | (P_f)_j \psi \in \FF\}$.

\section{The Electron: Model and Statement of Result}

\label{sec:eldef}

 At first  we consider a single free electron
interacting with the quantized electromagnetic field. The Hilbert
space describing the system composed of an electron and the
quantized field is
$$
\HH = L^2( \R^3 ; \C^2)  \otimes \FF \; .
$$
The Hamiltonian is
$$
H =   \left\{  \sigma \cdot ( p  + e A (x) ) \right\} ^2 + H_f \; ,
$$
where
\begin{equation} \label{eq:44}
A(x) = \sum_{\lambda=1,2} \int \frac{\rho(k)}{\sqrt{2|k|}} \left(
a_{\lambda,k} e^{ik \cdot x} \varepsilon_{\lambda,k}  +
a_{\lambda,k}^* e^{-ik \cdot x} \varepsilon_{\lambda,k}  \right) dk
\; ,
\end{equation}
where the $\varepsilon_{\lambda,k} \in \R^3$ are vectors, depending
measurably on $\widehat{k} = k /|k|$,  such that $(k/|k|,
\varepsilon_{1,k}, \varepsilon_{2,k})$ forms an orthonormal basis;
and  $\sigma = (\sigma_1, \sigma_2 , \sigma_3)$, where $\sigma_i$
denotes the $i$-th Pauli matrix:
$$
\sigma_1 = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0   \end{array}
\right) \ \ , \ \ \sigma_2 = \left(
\begin{array}{cc}  0 & - i \\ i & 0  \end{array} \right) \ \ , \ \ \sigma_3 =
\left( \begin{array}{cc} 1 & 0 \\ 0 & -1    \end{array} \right) \; .
$$
By $x$ we denote the position of the electron and its canonically conjugate
momentum by $p = - i \nabla_x$.
We have introduced the function
$$
\rho(k) = \frac{1}{(2 \pi)^{3/2}} \chi_\Lambda(|k|) \; ,
$$
where $\chi_{\Lambda}$ is the characteristic function of the set
$[0,\Lambda]$. Since we are interested in the infrared problem we
fix the ultraviolet cutoff $0 < \Lambda < \infty$. The Pauli
matrices satisfy the commutation relations $[\sigma_1 ,\sigma_2] = 2
i \sigma_3$ and cyclic permutations thereof. Using these commutation
relations, we can write the Hamiltonian as
$$
H = ( p + e A(x) )^2 +  e \sigma \cdot B(x) +  H_f \; ,
$$
where
$$
B(x) = (\nabla \wedge A)(x) =   \sum_{\lambda=1,2} \int \frac{
\rho(k) ( i k \wedge \varepsilon_{\lambda,k}) }{\sqrt{2|k|}} \left(
   a_{\lambda,k} e^{ik \cdot x}
   -
a_{\lambda,k}^* e^{-ik \cdot x} \right) dk \; .
$$
The
Hamiltonian is translation invariant and commutes with the generator
of translations, i.e., the operator of total momentum
$$
P_{\rm tot} = p + P_f  \; .
$$
Let $F$ be the Fourier transform in the electron variable $x$, i.e.,
on $L^2(\R^3)$,
$$
( F \psi)(\xi ) = \frac{1}{(2\pi)^{3/2}} \int_{\R^3} e^{-i \xi \cdot
x } \psi(x) dx \; .
$$
Set
$$
W = \exp(ix \cdot P_f) \; .
$$
Note $W P_{\rm tot} W^* = p $ so that in the new representation  $p$
is total momentum. We compute
$$
W H W^* =  \left\{ \sigma \cdot ( p - P_f  +  e A ) \right\}^2 + H_f
\; ,
$$
where $A := A(0)$. Then the composition $U=FW$ yields the fiber
decomposition of the Hamiltonian and the Hilbert space
$$
U H U^* = \int_{\R^3}^{\oplus} H(\xi) d\xi \quad , \quad U : \HH \to L^2(\R^3; \C^2 ) \otimes \FF
\cong \int_{\R^3}^\oplus   \C^2  \otimes \FF  d\xi
$$
with
$$ H(\xi) = \left\{ \sigma \cdot ( \xi - P_f  +  e A )
\right\}^2 + H_f
 \;
$$
 an operator on $\C^2 \otimes \FF$. Note that $H(\xi)$ can also be
written as
$$
H(\xi) = (\xi - P_f + e A)^2 +  e \sigma \cdot B + H_f \; ,
$$
where $B := B(0)$. % We define an explicit self-adjoint realization
%of $H(\xi)$ in terms of the Friedrichs extension, which is
%summarized in the following Lemma.
The explicit self-adjoint realization of $H(\xi)$ is given by the
following Lemma.
\begin{lemma} \label{lem:op1} The operator $H(\xi)$ is self-adjoint on $D( P_f^2 + H_f ) =
\{ \psi \in \FF | (P_f^2 + H_f ) \psi \in \FF \}$ and essentially
self-adjoint on any core of $P_f^2 + H_f$.
\end{lemma}
%\begin{lemma}  \label{lem:form1} The quadratic form
%$$q_\xi(\varphi, \psi ) = \sum_j ( v(\xi )_j \varphi, v(\xi)_j \psi )
%+ ( H_f^{1/2}   \varphi, H_f^{1/2} \psi )  + ( \varphi, e \sigma
%\cdot B \psi ) $$
%for all $\varphi, \psi$ in the form domain
% $Q(H(\xi)) := Q(H_f) \cap Q(P_f^2)$,
%is closed and bounded from below. There exists a unique self-adjoint operator
%$H(\xi)$ with domain $D(H(\xi)) \subset Q(H(\xi))$, such that
%$$(\varphi, H(\xi) \psi) = q_\xi(\varphi,\psi) \quad
%\forall  \ \varphi, \psi \in D(H(\xi)) \; .
%$$
%\end{lemma}
%\begin{lemma} \label{lem:op1} (i) $D(H(\xi)) = D( P_f^2 + H_f ) =
%\{ \psi \in \FF | (P_f^2 + H_f ) \psi \in \FF \}$ .
%(ii) Any core of $D(P_f^2 + H_f)$ is a core for $H(\xi)$.
%\end{lemma}
For a proof of Lemma \ref{lem:op1} see \cite{H00}. It is elementary
to show that $H(\xi)$ is bounded from below.
%By Lemma \eqref{lem:form1} $H(\xi)$ is bounded from
%below.
We write
$$
E(\xi) := \inf \sigma(H(\xi)) \; .
$$
\begin{proposition} \label{thm:lip1} The function $E(\cdot)$
is Lipschitz continuous and hence a.e. differentiable.
\end{proposition}
By spherical symmetry $E(\cdot) $ is invariant under rotations. We
want to point out that for small $e$ and $\xi$ in a neighborhood of
0, it has been shown that $E(\cdot)$ is twice continuously
differentiable with positive Hessian \cite{C01,BCFS05}. Using
functional integration it has been shown that  for any value of $e$,
$E(0) \leq E(\xi)$, \cite{G72,SPOHN04}. In \cite{F73,SPOHN04} it is
shown that for large $\xi$, $E(\xi) = |\xi| + O(1)$.  It seems
probable that for all $e$ and $\xi \neq 0$, $E(\cdot)$ is
differentiable with non-vanishing derivative.


\begin{theorem} \label{thm:main1} Let $e \neq 0$. If $E(\cdot)$ is differentiable at
 $\xi$ and has a
nonzero derivative, then $H(\xi)$ does not have a ground state
\end{theorem}


We want to relate this to a result obtained in \cite{C01}. Where $A
= A(0)$ in \eqref{eq:44} is replaced by an infrared regularized
$A_\sigma(0) = \sum_\lambda \int_{\sigma \leq |k|} \rho(k)
(2|k|)^{-1/2} ( a_{\lambda,k}^* \varepsilon_{\lambda,k} +
a_{\lambda,k} \varepsilon_{\lambda,k}) dk $. It is shown that if $e$
and $\xi$ are sufficiently small then for any $\sigma >0$, there
exists a normalized ground state $\psi_\sigma$. For $\xi=0$,
$\psi_\sigma$ converges weakly as  $\sigma \to 0$ to a nonzero
vector. If $\xi \neq 0$ it was shown that $(\Omega, \psi_\sigma )
\to 0$ as $\sigma \to 0$. This indicates that if there is a ground
state for $\xi \neq 0$ the overlap with the vacuum should be zero,
but does not prove this.


\section{The Electron: Proof of Results} \label{sec:proofelec}



\label{sec:elpro}

First we give a proof of the Lipschitz property of the ground state
energy. We note that this follows immediately from the inequality
$E(k+q) \leq E(k) + |q|$, a proof of which is  given for example in
\cite{F73,SPOHN04}. However, we will give an independent proof of
the  Lipschitz property, which easily generalizes to the case of an
ion. We write
$$
v(\xi) = ( \xi - P_f + e A)  \; .
$$

\noindent {\it Proof of Proposition  \ref{thm:lip1}.}
%First
%observe that for any $a>0$ there exists a  finite constant $C_0$,
%such that for all $\varphi \in D(H_f^{1/2})$,
%\begin{eqnarray} \label{eq:lip01}
%|( \varphi , e \sigma \cdot B \varphi ) | \leq a (\varphi , H_f
%\varphi ) + C_0 \| \varphi \|^2  \; .
%\end{eqnarray}
Choose $\varphi_n \in \ran \chi_{[E(\xi_1), E(\xi_1) +
1/n)}(H(\xi_1))$ with $\| \varphi_n \| =1$. Then
\begin{eqnarray}
E(\xi_2) \leq ( \varphi_n , H(\xi_2) \varphi_n ) &&= ( \varphi_n , (
H(\xi_1) + ( \xi_2 - \xi_1 ) \cdot v(\xi_1) ) \varphi_n ) + ( \xi_2
- \xi_1 )^2  \nonumber \\
&& \leq E(\xi_1) + \frac{1}{n} + (\xi_2 - \xi_1 ) \cdot ( \varphi_n,
v(\xi_1) \varphi_n )  + ( \xi_2 - \xi_1 )^2  \; . \label{eq:lip01}
\end{eqnarray}
 We estimate for some finite constants $C_0$ and $C_1$,
\begin{eqnarray}
| ( \varphi_n, v(\xi_1) \varphi_n ) |^2 && \leq (\varphi_n ,
v(\xi_1)^2 \varphi_n ) \nonumber \\
&& \leq ( \varphi_n , ( v(\xi_1)^2 + e \sigma \cdot B + H_f + C_0 )
\varphi_n ) \nonumber \\
&& \leq  E(\xi_1) + \frac{1}{n} + C_0  \nonumber \\
&& \leq  (\Omega , H(\xi_1) \Omega ) + C_0 + \frac{1}{n} \nonumber \\
&& \leq C_1 ( 1 + | \xi_1|^2)  \label{eq:lip3} \; ,
\end{eqnarray}
 where  in the second inequality we used that $\sigma \cdot B$ is
infinitesimally form-bounded with respect to $H_f$. By
\eqref{eq:lip3} and \eqref{eq:lip01},
$$E(\xi_2) - E(\xi_1) \leq \frac{1}{n} + | \xi_2 - \xi_1 |
C_1^{1/2} ( |\xi_1| + 1 )  + ( \xi_2 - \xi_1 )^2  \; .
$$
Since this holds for all $n \geq 1$ and all $\xi_1$ and $\xi_2$ we
obtain for some finite constant $C_2$,
$$
| E(\xi_2) - E(\xi_1) | \leq C_2 | \xi_2 - \xi_1 | ( |\xi_1| +
|\xi_2| + 1 )  \; .
$$
This proves the Lipschitz continuity. By Rademacher's Theorem we
know that Lipschitz continuity implies a.e. differentiability. \qed

\vspace{0.5cm}

\noindent Before we present the proof of Theorem \ref{thm:main1}, we
need a few Lemmas. For $E(\cdot)$ differentiable at $\xi$, we fix
$\xi$ and consider the following subset of the unit sphere,
$$
S_\epsilon := \{ \omega \in S^2 |  \  \omega \cdot \nabla E(\xi)
\leq 1 -\epsilon \} \; .
$$
We denote normalized vectors by  $\widehat{k} = k /|k |$.

\begin{lemma} \label{lem:o(k)} Assume that $E(\cdot)$ is
differentiable at $\xi$.
%Given $\epsilon > 0$, then
For $\widehat{k}   \in S_\epsilon $ , we have
 $$
H(\xi - k) + |k| - E(\xi) \geq  \epsilon |k| + o(|k|)  \; .
$$
\end{lemma}

\begin{proof} Using that $E(\xi - k)$ is a lower bound for
$H(\xi - k)$ and the differentiability of $E(\cdot)$ at $\xi$,
 we have
\begin{eqnarray*}
H(\xi - k) + |k| - E(\xi) \geq E(\xi - k) - E(\xi) + |k| = - k \cdot
\nabla E(\xi) + |k| + o(|k|)  \geq \epsilon |k| + o(|k|) \; .
\end{eqnarray*}
\end{proof}
Let $P_0 = P_0(\xi)$ denote the orthogonal projection onto the
kernel of $H(\xi) - E(\xi)$.
%Let $\chi_\Lambda(k)$ be the the
%function which is zero if $|k| > \Lambda$ and one everywhere else.
For $\varphi \in \FF$, we set
\begin{eqnarray} \label{eq:annihilation}
(a_{\underline{k}} \varphi )_n(\underline{k}_1, ... ,
\underline{k}_n) = (n+1)^{1/2} \varphi_{n+1}(\underline{k},
\underline{k}_1, ... , \underline{k}_n ) \; .
\end{eqnarray}
For a.e. $k$ and all $n$, the expression in \eqref{eq:annihilation}
is in $L^2(\R^{3n})$. Its relation to $a(f)$ is outlined in the
following Lemma
\begin{lemma} \label{lem:new22}
Let $\Omega  \subset \R^3$ and $\varphi \in \FF$. If
$a_{\underline{k}} \varphi \in L^2(\Omega; \FF)$, then for all $f
\in L^2(\Omega; \C)$ and $\eta \in \FF$
$$
( \eta , a(f) \varphi ) = \int \overline{f}(\underline{k}) ( \eta ,
a_{\underline{k}} \varphi ) d\underline{k} \; .
$$
\end{lemma}
\begin{proof}
%(a)
We have
\begin{eqnarray*}
(\eta , a(f) \varphi ) && =  \sum_{n=0}^\infty \int
\overline{\eta}_n( \underline{k_1},  ... , \underline{k}_{n})
(n+1)^{1/2} \overline{f}(\underline{k}) \varphi_{n+1}(
\underline{k}, \underline{k}_1, ... , \underline{k}_{n+1})   d
\underline{k} d\underline{k}_1 ...
d\underline{k}_n  \\
&& = \int  \overline{f}(\underline{k} ) \left( \sum_{n=0}^\infty
\int \overline{\eta}_n( \underline{k_1}, ... , \underline{k}_{n})
(n+1)^{1/2} \varphi_{n+1}( \underline{k}, \underline{k}_1, ... ,
\underline{k}_{n+1})
d\underline{k}_1 ...  d\underline{k}_n \right) d\underline{k} \\
&& =  \int \overline{f}(\underline{k}) ( \eta , a_{\underline{k}}
\varphi ) d\underline{k} \; ,
\end{eqnarray*}
where the interchange of the order of integration and summation is
justified since
\begin{eqnarray*}
 \lefteqn{ \sum_{n=0}^\infty \int \left| \overline{\eta}_n(
\underline{k_1},  ... , \underline{k}_{n}) (n+1)^{1/2}
\overline{f}(\underline{k}) \varphi_{n+1}( \underline{k},
\underline{k}_1, ... , \underline{k}_{n+1})  \right|  d
\underline{k}
d\underline{k}_1 ... d\underline{k}_n  } \\
&&  \leq  \| f \| \| \eta \| \left( \int_\Omega \| a_{\underline{k}}
\varphi \|^2 d \underline{k} \right)^{1/2} < \infty \; .
\end{eqnarray*}
\end{proof}
\begin{lemma} \label{lem:new23} For each $\varphi \in D(H_f^{1/2})$, the function
$k  \mapsto a_{\underline{k}} \varphi$ is in  $L^2_{\rm loc}(\R^3
\setminus \{ 0 \} ;  \FF )$.
\end{lemma}
\begin{proof}
Since $\varphi \in D(H_f^{1/2})$, we conclude that
\begin{eqnarray*}
\lefteqn{\sum_{n=0}^\infty \int |k| | ( a_{\underline{k}}
\varphi)(\underline{k}_2, ..., \underline{k}_{n+1})
|^2 d\underline{k}_2 ...  d\underline{k}_{n+1} d\underline{k}} \\
&& = \sum_{n=0}^\infty \int \sum_{j=1}^{n+1} |k_j| |
\varphi_{n+1}(\underline{k}_1, \underline{k}_2, ... ,
\underline{k}_{n+1}) |^2  d\underline{k_1} d\underline{k}_2 ...
d\underline{k}_{n+1}  \\
&& = \| H_f \varphi \|^2 < \infty \; .
\end{eqnarray*}
This implies that $\| a_{\underline{k}} \varphi \|^2$ is integrable
over any compact subset of $\R^3 \setminus \{ 0 \}$.
\end{proof}
\begin{lemma} \label{lem:basicom1} Suppose $E(\cdot)$ is
differentiable at $\xi$ and and that $\psi$ is a  ground state of
$H(\xi)$.  Let $\epsilon > 0$. Then there exists a $\delta>0$ such
that for all $\eta \in \FF$,
\begin{equation} \label{eq:last888}
\left( \eta , a_{\underline{k}} \psi \right) =
 \frac{e \rho(k)}{\sqrt{2|k|} }
\left( H(\xi , k )^{-1} \eta, \left( - 2  \epsilon_{\underline{k}}
\cdot v(\xi) + i (  k \wedge  \epsilon_{\underline{k}})  \cdot
\sigma \right) \psi \right) \; ,
\end{equation}
for a.e. $k$, with $0< |k| < \delta $ and $\widehat{k} \in
S_\epsilon$, where $H(\xi,k) := H(\xi - k) + |k| - E(\xi)$.
\end{lemma}

\begin{proof} Let $f \in C_0^\infty(\R^3 \setminus \{ 0 \} )$ .
Let $\varphi \in \ran (P_{[0,n]}(N))$ be a state having less or
equal to $n$ photons and assume each $\varphi_n$ has compact
support. Then for real $f$,
\begin{eqnarray*}
\left( ( a_\lambda^*(f) H(\xi,k) - (H(\xi) - E(\xi)) a_\lambda^*(f)
) \varphi , \psi \right) = \left( ( A^*(f) + R_0^*(f)    + R_1^*(f)
) \varphi , \psi \right) \; ,
\end{eqnarray*}
with
\begin{eqnarray*}
R_0(f) &&:= \int  f(y) 2 ( y - k) \cdot  v(\xi)  a_{\lambda , y} \,
dy
 +
\int   f(y) ( k^2 - y^2) a_{\lambda,y} \, dy
 + \int  f(y) ( |k| - | y| ) a_{\lambda,y} \, dy \\
R_1(f) &&:=   \int  f(y)   \frac{ e \rho(y)}{\sqrt{2|y|}}
( k \cdot \varepsilon_{\lambda,y} )  \, dy \\
A(f) &&:= \int   f(y) \frac{e \rho(y)}{\sqrt{2|y|}} \left( - 2
\varepsilon_{\lambda,y} \cdot  v(\xi) + i (  y \wedge
\varepsilon_{\lambda,y}) \cdot  \sigma \right) \, dy  \; .
\end{eqnarray*}
Since $\psi \in D(H_f + P_f^2) \subset D(a_\lambda(f))$,
\begin{equation} \label{eq:proofgg}
( H(\xi,k) \varphi , a_\lambda(f) \psi ) = (\varphi , (A(f) + R_0(f)
+R_1(f)  ) \psi )
\end{equation}
Note that this holds for all $\varphi$ in an operator core for
$H(\xi,k)$.  For any $\epsilon
> 0$, there exists by Lemma \ref{lem:o(k)} a $\delta>0$ such that
for all $k$ with $0<|k|<\delta$ and $\widehat{k} \in S_\epsilon$,
$H(\xi,k)$ has a bounded inverse. This and equation
\eqref{eq:proofgg} imply that in fact for all such $k$ and all $\eta
\in \FF$,
\begin{eqnarray} \label{eq:proofgg2}
 ( \eta , a_\lambda(f) \psi ) =
(H(\xi,k)^{-1} \eta , (A(f) + R_0(f) + R_1(f)  ) \psi ) \; .
\end{eqnarray}
Now fix $\eta \in \FF$. For $k$, with $0<|k|<\delta$ and
$\widehat{k} \in S_\epsilon$, we choose a $\delta$--sequence
centered at $k$. Explicitly, we choose a nonnegative function $g \in
C_0^\infty(\R^3)$ with $\int g(y) dy =1$ and support in the unit
ball. We set $f_{k,m}(y) := m^3 g(m(y-k))$. By Lemmas
\ref{lem:new22} and \ref{lem:new23} it follows  that the left hand
side of \eqref{eq:proofgg2} yields, $\lim_{m\to \infty} (\eta ,
a_\lambda(f_{m,k}) \psi ) = (\eta, a_{\lambda,k} \psi)$ a.e. $k$.
The term $(H(\xi,k)^{-1}\eta, A(f_{m,k}) \psi)$ converges to the
right hand side of \eqref{eq:last888}.
 Below we  will show that the terms $(H(\xi,k)^{-1} \eta,
R_0(f_{m,k}) \psi)$ and $(H(\xi,k)^{-1} \eta, R_1(f_{m,k}) \psi)$
vanish as $m$ tends to infinity for a.e. $k$.
 The expression containing $R_1$  vanishes since
$k \cdot \epsilon_{\lambda,k} = 0$. To show that the expression
involving $R_0$ vanishes we will only consider one term. The other
terms will follow similarly. We set $\phi_l := v_l(\xi)
H(k,\xi)^{-1} \eta $ and estimate
\begin{eqnarray*}
\lefteqn{ R_{0,1}(f_{m,k}) := |( H(k,\xi)^{-1} \eta , \int f_{k,m}(
y) 2 ( y - k)
\cdot  v(\xi) a_{\lambda,y} \psi ) dy | }  \\
&&\leq \sum_{l=1}^3  \left|(  v_l(\xi) H(k,\xi)^{-1} \eta ,
\int  f_{k,m}(y) 2 ( y - k)_l  a_{\lambda,y} \psi ) dy \right|  \\
&& \leq \sum_{l=1}^3 \sum_{n=0}^\infty \int  \left|
\overline{\phi}_l(\underline{k}_1, ... , \underline{k}_n) \int
f_{k,m} (y) 2 ( y - k)_l (n+1)^{1/2} \psi_{n+1}(\lambda, y,
\underline{k}_1, ... , \underline{k}_n )  dy \right|
d\underline{k}_1 ...
d\underline{k}_n\\
&& \leq   \| \phi \| \int   f_{k,m} (y) 2 | y - k| h_\lambda(y) dy
\; ,
\end{eqnarray*} where
$$
h_\lambda(y) = \left( \sum_{n=0}^\infty  \int  (n+1) \left|
\psi_{n+1}(\lambda, y, \underline{k}_1, ... , \underline{k}_n )
\right|^2    d\underline{k}_1 ... d\underline{k}_n  \right)^{1/2} \;
$$
and $$ \| \phi \|^2  = \sum_{l=1}^3 \| \phi_l \|^2 \; .
$$
 Since $\psi$ is in $D(H_f^{1/2})$,
$$
\int \left| |y|^{1/2} h_\lambda(y) \right|^2 dy \leq
\sum_{n=0}^\infty   \sum_{\mu=1,2}  \int (n+1) |y| \left|
\psi_{n+1}(\mu, y, \underline{k}_1, ... , \underline{k}_n )
\right|^2 dy d\underline{k}_1 ... d\underline{k}_n  = ( \psi , H_f
\psi ) \; .
$$
Thus $h_\lambda \in L^1_{\rm loc}(\R^3 \setminus \{ 0 \} )$.
Therefore a.e. point is a Lebesgue point of $h_\lambda$. At such
points $k$,
$$ \int f_{k,m}(y) h_\lambda(y) dy \to h_\lambda(k) \; ,$$
by Lebesgue's differentiation theorem, see for example \cite{SW71}
Theorem 1.25. Thus $R_{0,1}(f_{k,m})$ tends to zero  as
 $m \to \infty$, a.e. $k$.
\end{proof}


\begin{lemma} \label{lem:nablae1} If $E(\cdot )$ is differentiable at $\xi$, then
$$
P_0 2 v(\xi) P_0 = \nabla E(\xi) P_0 \; .
$$
\end{lemma}


\begin{proof}
Suppose $\psi \in \ran P_0$, with $\|\psi \|=1$, then
\begin{eqnarray*}
E(\xi + k) - E(\xi) \leq ( \psi, (H(\xi + k) - H(\xi) ) \psi ) =  2
k \cdot ( \psi , v(\xi) \psi ) + |k|^2 \; .
\end{eqnarray*}
This implies
$$
k  \cdot \nabla E(\xi) \leq  2 k  \cdot (\psi , v(\xi) \psi ) +
o(|k|) \quad , \quad |k| \to 0 \; .
$$
Since $k$ can have any direction we conclude that
$$
\nabla E(\xi) = 2 ( \psi, v(\xi) \psi ) \; .
$$
Since this holds for any $\psi \in \ran P_0$ the claim follows by
polarization.
\end{proof}


We set
$$
Q(k) = |k| ( H(\xi - k) + |k| - E(\xi))^{-1} \; ,
$$
whenever this exists. And for $|k| > 0$, we set
$$
Q_0(k) = |k| ( H(\xi) + |k| - E(\xi) )^{-1} \; .
$$
By the spectral theorem
$$
P_0 = P_0(\xi) = s-\lim_{|k| \to 0} Q_0(k) \; .
$$

\begin{lemma} \label{lem:weak} Let $E(\cdot)$ be differentiable at $\xi$. Given
$\epsilon > 0$, then
$$
w- \lim_{\widehat{k} \in S_\epsilon, |k| \to 0} \left( Q(k) - ( 1 -
\widehat{k} \cdot \nabla E(\xi) )^{-1} P_0 \right) = 0 \; .
$$
\end{lemma}


\begin{proof}
Fix $\xi$

\vspace{0.5cm}

\noindent \underline{Step 1:} $v(\xi) Q_0(k)$ is uniformly bounded
for small $|k|$.

\vspace{0.5cm} Since $B$ is $H_f^{1/2}$ operator bounded, we see
that there exists a finite constant $C_0$ such that
\begin{eqnarray}
\label{ineq:55}
 v(\xi)^2 \leq   H(\xi) + C_0 \leq  ( H(\xi) +  |k| -
E(\xi) ) + ( E(\xi) + C_0 ) \; .
\end{eqnarray}
On the other hand
\begin{eqnarray*}
 \sum_{l=1}^3 ( v(\xi) Q_0(k))_l^* ( v(\xi) Q_0(k) )_l
= \frac{|k|}{H(\xi) - E(\xi) + |k| } v(\xi)^2 \frac{|k|}{H(\xi) -
E(\xi) + |k| } \; .
\end{eqnarray*}
By inequality \eqref{ineq:55} we see that the right hand side is
uniformly bounded for small $|k|$. This shows Step 1.



\vspace{0.5cm}

\noindent \underline{Step 2:} We have $s-\lim_{|k| \to 0} v(\xi)
Q_0(k) = v(\xi) P_0 $.

\vspace{0.5cm} \noindent By the resolvent identity
\begin{eqnarray}
\lefteqn{ v(\xi) \frac{|k|}{H(\xi) - E(\xi) + |k|} } \nonumber \\
&& = v(\xi)  \frac{|k|}{H(\xi) - E(\xi) + |k|  + 1} - v(\xi)
\frac{1}{H(\xi) - E(\xi) + |k| + 1 } \frac{|k|}{H(\xi) - E(\xi) +
|k|} \; .  \label{step2:conv1}
\end{eqnarray}
Again using the resolvent identity and an argument similar to the
one in Step 1,
\begin{eqnarray*}
\lefteqn{ \left\| v(\xi) \frac{1}{H(\xi) - E(\xi) + |k| + 1} -
v(\xi) \frac{1}{H(\xi) - E(\xi) + 1 } \right\| } \\
&& =\left\|  v(\xi) \frac{1}{H(\xi) - E(\xi)  + 1}  |k |
\frac{1}{H(\xi) - E(\xi) + |k| + 1 }   \right\| \\
&&  \stackrel{ |k| \to 0 }{\longrightarrow} 0 \; .
\end{eqnarray*}
This implies that the first term on the right hand side in
\eqref{step2:conv1} converges in norm to zero and the second term
converges strongly to $v(\xi)P_0$.


 \vspace{0.5cm}



\noindent \underline{Step 3:}  Uniformly for $\widehat{k} \in
S_\epsilon$,
$$P_0 Q(k) P_0 -  \left( P_0 2 \widehat{k} \cdot v(\xi) P_0 \right)
(P_0 Q(k) P_0 ) \stackrel{w}{\longrightarrow} P_0 \;  \ \  {\rm and}
\ \ \ Q(k) - P_0Q(k) P_0 \stackrel{w}{\longrightarrow} 0 \; .
$$


\vspace{0.5cm} \noindent Using the second resolvent identity twice
we obtain for small $|k|$ and $\widehat{k} \in S_\epsilon$,
\begin{eqnarray}
Q(k) &=& Q_0(k)  + Q_0(k) ( 2  \widehat{k}\cdot  v(\xi) - |k| ) Q(k)
\label{solventone1} \\
&=& Q_0(k) + Q_0(k)( 2 \widehat{k} \cdot v(\xi) - |k| ) Q_0(k)  \nonumber \\
&& + Q_0(k) ( 2 \widehat{k}\cdot v(\xi) - |k| ) Q(k )  ( 2
\widehat{k} \cdot  v(\xi) - |k| ) Q_0(k)   \label{resolventtwo1}
\end{eqnarray}
Now using \eqref{resolventtwo1} and the results of Step 1 and Step
2, we find
$$
Q(k) ( 1 - P_0 ) \stackrel{w}{\longrightarrow} 0 \quad , \quad ( 1 -
P_0 ) Q(k)
 \stackrel{w}{\longrightarrow} 0  \; ,
$$
where the limit is uniform for $\widehat{k} \in  S_\epsilon$. It
follows that
$$
Q(k) - P_0Q(k) P_0 \stackrel{w}{\longrightarrow} 0 \; ,
$$
uniformly for $\widehat{k} \in S_\epsilon$. Now this and
\eqref{solventone1} show Step~3.

 \vspace{0.5cm}

The claim of the Lemma is now an immediate consequence of Lemma
\ref{lem:nablae1}  and Step~3.
\end{proof}

Now we are ready to prove Theorem \ref{thm:main1}.

\vspace{0.5cm}

\noindent {\it Proof of Theorem \ref{thm:main1}.} Suppose $H(\xi)$
has a ground state $\psi$ with $\| \psi \| = 1$. We want to show
this leads to a contradiction. We  choose an $\eta \in
D((N+1)^{1/2})$ such that $( \eta ,\psi) \neq 0$. Choose $\epsilon$
with $0< \epsilon < 1$ and $\delta >0$ sufficiently small. Then  by
Lemma \ref{lem:basicom1}  for a.e. $k$ with  $\widehat{k} \in
S_\epsilon$ and $|k| < \delta$,
\begin{eqnarray*}
 (\eta , a_{\lambda,k}  \psi ) && =  \left( \eta , H(\xi,k)^{-1}
({2|k|})^{-1/2} \rho(k) \left( - 2 \varepsilon_{\lambda,k} \cdot
v(\xi) +
 i ( \varepsilon_{\lambda, k} \wedge k ) \cdot \sigma \right) \psi
\right) \\
&& = \frac{ e \rho(k) } {\sqrt{2}|k|^{3/2}}  \left( \eta , Q(k)
\left( -  2 \varepsilon_{\lambda,k} \cdot v(\xi) + i (
\varepsilon_{\lambda, k} \wedge k ) \cdot \sigma \right) \psi
\right) \; .
\end{eqnarray*}
Now uniformly for $\widehat{k} \in S_\epsilon$,
\begin{eqnarray*}
\left( \eta , Q(k) \left( - 2  \varepsilon_{\lambda,k} \cdot v(\xi)
+ i ( \varepsilon_{\lambda, k} \wedge k ) \cdot \sigma \right) \psi
\right)   \stackrel{|k| \to 0}{\longrightarrow} && -
 (1 - \widehat{k} \cdot \nabla E )^{-1}  \varepsilon_{\lambda,k} \cdot (P_0 \eta,
2 v(\xi) \psi ) \\
&&  = -  ( \varepsilon_{\lambda,k} \cdot \nabla E ) (1 - \widehat{k}
\cdot  \nabla E )^{-1}
 (\eta, \psi ) \; ,
\end{eqnarray*}
where in the last step we used  Lemma \ref{lem:nablae1}.  Thus we
find for a.e. $k$ with  $|k|<\delta$ and $\widehat{k} \in
S_\epsilon$,
\begin{eqnarray*}
| ( \eta , a_{\lambda , k} \psi ) | \geq \frac{1}{2} \frac{ |e
\rho(k)|}{\sqrt{2}|k|^{3/2}} ( 1 - \widehat{k} \cdot \nabla E )^{-1}
\left| ( \eta , \psi )  ( \varepsilon_{\lambda,k} \cdot \nabla E
)\right| \; .
\end{eqnarray*}
 We introduce the set
$$
K := \{ \omega \in S^2 | - \sfrac{1}{2} | \nabla E | \leq \omega
\cdot \nabla E \leq 0 \} \subset S_\epsilon \; .
$$
Then there exists a positive constant $c_0 > 0$ such that for all
$\widehat{k} \in K$,
$$
\sum_{\lambda=1,2} | ( \varepsilon_{\lambda, k} \cdot \nabla E ) |^2
\geq c_0
> 0  \; .
$$
Therefore, there exists a $c_1 > 0$ such that for a.e. small $k$
with $\widehat{k} \in K$, we have
$$
\frac{| e \rho(k) c_1 |^2}{|k|^{3}} \leq \sum_{\lambda=1,2}| ( \eta
,a_{\lambda,k} \psi ) |^2 \leq \| ( 1 + N)^{1/2} \eta \|^2 \left(
\sum_{\lambda=1,2} \sum_{n=0}^\infty  \int | \psi_{n+1}(\lambda, k,
\underline{k}_1,  ... , \underline{k}_n )|^2 d\underline{k}_1 ...
d\underline{k}_n \right)  \; .
$$
Integrating over the set of all $k$ with $\widehat{k} \in K$ and
$|k| \leq \delta$,  we see this is inconsistent with $\psi$ being in
$\FF$. Thus $H(\xi)$ does not have a ground state.
 \qed

\section{Positive Ion: Model and Statement of Results}

\label{sec:iondef}

We consider an atom consisting of a spinless nucleus of mass $m_0$
and charge $Ze$ and $N$ spin $1/2$ electrons having charge $-e$ and
mass $1$. The energy of this system is described by  the operator
$$
H = \frac{1}{2m_0} \left( p_0 - Z e A(x_0) \right)^2 + \sum_{j=1}^N
\frac{1}{2}\{ \sigma_j \cdot (p_j + e A(x_j) ) \}^2 + H_f +  V(x_0,
... , x_N) \; ,
$$
acting on the Hilbert space
$$
\HH = L^2(\R^3 ) \otimes  \left( \bigwedge^N_{j=1} L^2(\R^3; \C^2 )
\right)  \otimes \FF \; ,
$$
where $p_0= - i \nabla_{0}$ acts on the first factor  and $p_j = - i
\nabla_j$ and $\sigma_j$, the three-vector of Pauli matrices, act on
the $j$-th factor of the antisymmetric tensor product. We take the
spin of the nucleus to be zero only to simplify notation. We will
make the following assumptions about the potential $V$:
\begin{eqnarray*}
V(x_0, ... , x_N) =
%&=& \sum_{1 \leq i < j \leq N} v(x_i - x_j)  + \sum_{j=1}^N w(x_0 - x_j) \\
%&=:&
\sum_{0 \leq i < j \leq N} V_{ij}(x_i - x_j ) \; .
\end{eqnarray*}
Each $V_{ij}$ is infinitesimally bounded with respect to the
Laplacian in three dimensions,
which we denote by $-\Delta$, i.e.,
there exists for any $a > 0$ a  finite constant $b$ such that for
 all $f$ in the domain
of $-\Delta$,
$$
\| V_{ij} f \| \leq a  \|-\Delta f \| + b \| f \| \ \; .
%, \quad f \in D(-\Delta) := \{ f \in L^2(\R^3) | - \Delta f \in L^2(\R^3) \} \; .
$$
The Hamiltonian is translation
invariant and therefore commutes with the generator of translations,
i.e., the operator of total momentum
$$
P_{\rm tot} = \sum_{j=0}^N  p_j + P_f \; .
$$
Let $F$ be the Fourier transform
in the variable $x_0$, i.e., on $L^2(\R^3)$,
$$
( F \psi)(\xi ) = \frac{1}{(2\pi)^{3/2}} \int_{\R^3} e^{-i \xi \cdot
x_0 } \psi(x_0) dx_0 \; .
$$
%Let $\HH_{\rm trans} = \left( \bigwedge^N L^2(\R^3) \right) \otimes
%\FF $, then $\HH = L^2(\R^3) \otimes \HH_{\rm trans}$.
Let
$$
W = \exp(ix_0 \cdot (P_f  + \sum_{j=1}^N p_j ) )  \; .
$$
Note that $W P_{\rm tot} W^* = p_0$ so that in a new representation,
$p_0$ is the total momentum. Then the composition $U=FW$ yields the
decomposition of the Hamiltonian
$$
U H U^* = \int_{\R^3}^{\oplus} H(\xi) d\xi \; ,
$$
with
$$
H(\xi) = \frac{1}{2m_0}( \xi - \sum_{j=1}^N p_j - P_f  - Z e A(0)
)^2 +  \frac{1}{2} \sum_{j=1}^N \left\{ \sigma_j \cdot ( p_j + e
A(x_j) ) \right\}^2 + H_f + \widetilde{V} \;
$$
acting on $\left( \bigwedge^N L^2(\R^3; \C^2) \right) \otimes \FF $
and where we have set $\widetilde{V} = V\vert_{x_0=0}$. Let us cite
the following Theorem \cite{LMS06,H02}.

\begin{theorem} \label{thm:domain2} The operator $H(\xi)$ is self-adjoint on
$$ \bigcap_{j=1}^N D( p_j^2 ) \cap D(P_f^2 + H_f )$$
 and essentially
self-adjoint on any core of $\sum_{j=1}^N p_j^2  + P_f^2 + H_f$.
\end{theorem}

It is easy to show that for every $\xi$ the operator $H(\xi)$ is
bounded below. Let $E(\xi) = \inf \sigma( H(\xi))$ be the infimum of
the spectrum. We prove the following results.




\begin{proposition} \label{thm:lip2} The function $E(\cdot)$ is
Lipschitz continuous and
hence almost everywhere differentiable.
\end{proposition}


\begin{theorem} \label{thm:main22} Suppose $N \neq Z$ and $e\neq 0$.
If $E( \cdot )$ is differentiable at $\xi$ with non-vanishing
derivative then $H(\xi)$ does not have a ground state.
\end{theorem}


\section{Positive Ion: Proof of Result}

\label{sec:ionpro}

First we show the following lemma.

\begin{lemma}   \label{lem:formbound} $|\widetilde{V}|$ is
relatively form bounded with respect to  $H(\xi)$.
\end{lemma}

\begin{proof} By Theorem \ref{thm:domain2}, we know that
$H(\xi)$ is self-adjoint on the domain of $P_f^2 + \sum_{j=1}^Np_j^2
+ H_f$. Therefore there exist finite constants $c_0$ and $c_1$ such
that
$$
P_f^2 + \sum_{j=1}^Np_j^2 + H_f \leq c_0 H(\xi) + c_1  \; .
$$
By assumption $\widetilde{V}$ is infinitesimally small with respect
to $\sum_{j=1}^N p_j^2$. Therefore, $|\widetilde{V}|$ is
infinitesimally form bounded with respect to  $\sum_{j=1}^N p_j^2$.
Hence  for any $a>0$ there exists a finite $b$ such that
\begin{eqnarray*}
|\widetilde{V}| \leq  a  \sum_{j=1}^N p_j^2 +  b \leq a \left( P_f^2
+ \sum_{j=1}^N p_j^2 + H_f \right) + b  \leq a c_1 H(\xi) + a c_2 +
b \; .
\end{eqnarray*}
\end{proof}
The proof of Proposition \ref{thm:lip2} is omitted since it is a
minor modification  of the proof of Proposition \ref{thm:lip1}, but
note that Lemma   \ref{lem:formbound} is needed to obtain the
corresponding bound \eqref{eq:lip3}. We will prove Theorem
\ref{thm:main22} using a sequence of Lemmas. For notational
convenience we set
$$
v(\xi) =  \xi - \sum_{j=1}^N p_j - P_f - Z e A(0) \; .
$$
Recall the definitions $S_\epsilon := \{ \omega \in S^2 | \ \omega
\cdot \nabla E(\xi) \leq 1 - \epsilon \}$ and  $\widehat{k} :=
k/|k|$, which are the same as in Section \ref{sec:proofelec}.
\begin{lemma}\label{lem:E(k)2} Assume that $E(\cdot)$ is differentiable in $\xi$. Given $
\epsilon > 0$, then for $\widehat{k}  \in S_\epsilon$,  we have
 $$
H(\xi - k) + |k| - E(\xi) \geq  \epsilon |k| + o(|k|)  \; .
$$
\end{lemma}
The proof of  Lemma \ref{lem:E(k)2} is the same as the proof of
Lemma \ref{lem:o(k)}.



\begin{lemma} \label{lem:new222} Let $\HH_0$ be any Hilbert space.
Let $\Omega  \subset \R^3$ and $\varphi \in  \HH_0 \otimes \FF$. If
$a_{\underline{k}} \varphi \in L^2(\Omega; \HH_0 \otimes \FF)$. Then
for all $f \in L^2(\Omega; \C)$ and $\eta \in \HH_0 \otimes \FF$
$$
( \eta , a(f) \varphi ) = \int \overline{f}(\underline{k}) ( \eta ,
a_{\underline{k}} \varphi ) d\underline{k} \; .
$$
\end{lemma}
The proof of this Lemma is analogous to the proof of  Lemma
\ref{lem:new22}. We merely have to replace the inner product of
$\mathbb{C}$ by the inner product of $\HH_0$. Likewise, one
generalizes the proof of Lemma \ref{lem:new23} to prove  the next
lemma. Anticipating our application we set henceforth $\HH_0:=\left(
\bigwedge^N L^2(\R^3; \C^2) \right) $.
\begin{lemma} \label{lem:new233} Let $\varphi \in D(H_f^{1/2})$.
The function $k  \mapsto a_{\underline{k}} \varphi$ is in  $L^2_{\rm
loc}(\R^3 \setminus \{ 0 \} ; \HH_0 \otimes \FF )$
\end{lemma}

\begin{lemma}
\label{lem:basicom} Suppose $E(\cdot)$ is differentiable at $\xi$
and  that $\psi$ is a  ground state of $H(\xi)$. Let $\epsilon > 0$.
Then there exists a $\delta > 0$ such that for all $\eta \in \HH_0
\otimes \FF$,
\begin{eqnarray}
\lefteqn{ \left(  \eta  , a_{\lambda,k} \psi   \right) }
\label{eq:lem:basicom}
\\ &&=      \frac{e \rho(k)}{\sqrt{2|k|}}
 \left( H(\xi,k)^{-1} \eta  ,  \left(  \frac{Z}{m_0} v(\xi)
-  \sum_{j=1}^N  e^{-ik \cdot x_j} (  \frac{1}{2} i  k \wedge
\sigma_j + p_j + e A(x_j) ) \right) \cdot \varepsilon_{\lambda,k}
\psi \right) \; , \nonumber
\end{eqnarray}
for a.e. $k$, with  $0 < |k| < \delta$ and $\widehat{k} \in
S_\epsilon$, where $H(\xi,k) := H(\xi - k) + |k| - E(\xi)$.
\end{lemma}
\begin{proof} Let $f \in C_0^\infty(\R^3 \setminus \{ 0 \} )$.
Let $\varphi \in \ran (P_{[0,n]}(N))$ be a state having less or
equal to $n$ photons and assume $\varphi_n$ is smooth and has
compact support. Then a straightforward calculation using the
canonical commutation relations, yields for $f$ real,
\begin{eqnarray*}
\left( ( a^*_\lambda(f) H(\xi,k) - (H(\xi) - E(\xi) a^*_\lambda(f) )
\varphi , \psi \right) = \left( ( A^*(f) + R_0^*(f) + R^*_1(f) )
\varphi , \psi \right) \; ,
\end{eqnarray*}
with
\begin{eqnarray*}
R_0(f) &&:= \int ( |k| - |y|)  f(y) a_{\lambda,y} \, dy
 + m_0^{-1}  \int f(y) (y-k) \cdot v(\xi) a_{\lambda,y} \, dy \\
 &&
+  (2 m_0)^{-1} \int  f(y) ( k^2 - y^2 )   a_{\lambda,y}  \, dy \\
R_1(f) &&:=  -    \frac{Z}{2m_0} e  \int  \frac{
\rho(y)}{\sqrt{2|y|}} f(y)  k  \cdot \varepsilon_{\lambda,y} \, dy
\\
A(f) &&:=   -  \sum_{j=1}^N  e \int  \frac{ \rho(y)}{\sqrt{2|y|}}
e^{-i y \cdot x_j } f(y)  \varepsilon_{\lambda,y}  \cdot (p_j + e
A(x_j)) \, dy \\
&&
 +   \frac{Z}{m_0} e \int  \frac{ \rho(y)}{\sqrt{2|y|}}   f(y)
  \varepsilon_{\lambda,y} \cdot v(\xi) \, dy \\
 &&  + \frac{1}{2} \sum_{j=1}^N e \int   \frac{ \rho(y)}{\sqrt{2|y|}}
 e^{- i y \cdot x_j} f(y)  ( i k \wedge \varepsilon_{\lambda,y} )
 \cdot  \sigma_j  \, dy
 \; .
\end{eqnarray*}
Since $\psi \in \bigcap_{j=1}^N D(p_j^2) \cap D(P_f^2 + H_f)
\subset D(a_\lambda(f))$,
$$( H(\xi,k) \varphi , a_\lambda(f) \psi )
= (\varphi , (A(f) + R_0(f) + R_1(f) ) \psi ) \;.$$
%Rewriting this, we have
%$$
%(  H(\xi,k) \varphi  , a(f) \psi ) = (H(\xi,k)^{-1}  ( H(\xi,k) \varphi ) , (A(f) + R_0(f) +R_1(f)  ) \psi ) \; .
%$$
Note that this holds for all $\varphi$ in an operator core for
$H(\xi,k)$. For $\epsilon > 0$, there exists by Lemma
\ref{lem:E(k)2} a $\delta
> 0$ such that for all $k$ with $0<|k|<\delta$ and $\widehat{k} \in
S_\epsilon$, $H(\xi,k)$ has a bounded inverse. Thus we conclude by
density that for all such $k$ and all $\eta \in \HH_0 \otimes \FF$,
\begin{equation} \label{eq:lhs}
( \eta , a_\lambda(f) \psi ) = (H(\xi,k)^{-1}  \eta  , (A(f) +
R_0(f) + R_1(f) ) \psi ) \; .
\end{equation}
Now fix $\eta \in \HH_0 \otimes \FF$. For  $k$, with $0<|k|<\delta$
and $\widehat{k} \in S_\epsilon$, we choose a $\delta$-sequence,
$f_{m,k}$, centered at $k$ as in the proof of Lemma
\ref{lem:basicom1}. We insert $f_{m,k}$ for $f$ in equation
\eqref{eq:lhs}. As $m \to \infty$, it follows by Lemmas
\ref{lem:new222} and \ref{lem:new233} that the left hand side of
\eqref{eq:lhs} converges to the left hand side of
\eqref{eq:lem:basicom} for a.e. $k$. In the same limit the term
involving $A$ converges to the right hand side of
\eqref{eq:lem:basicom}. As demonstrated in the proof of Lemma
\ref{lem:basicom1} the terms involving $R_0$ and $R_1$ vanish as $m$
tends to infinity for a.e. $k$. This implies the assertion of the
Lemma.
\end{proof}
The next lemma would follow easily from the formal commutation
relation
$$
[H(\xi), i x_j] = -  \frac{1}{m_0} v(\xi) +    p_j + e A(x_j) \; .
$$


\begin{lemma} \label{lem:virial} Let $P_0$ be the projection onto the kernel
of $H(\xi) - E(\xi)$. Then for all integer $j$ with $1 \leq j \leq
n$,
$$
P_0   \frac{1}{m_0} v(\xi) P_0 = P_0   ( p_j + e A(x_j) )P_0 \; .
$$
\end{lemma}



\begin{proof} Fix a $j \in \{ 1,2, ... , n\}$.
Let $\chi \in C^\infty(\R_+ ;[0,1])$ with $\chi
\restricted{[0,1]}=1$ and $\chi \restricted {[2,\infty)} = 0$. We
set $\chi_n(x_j) = \chi(|x_j|/n)$. Let $\psi \in \ran P_0$, then for
all $n$
\begin{eqnarray*}
0 && = \langle \psi , H(\xi) i \chi_n (x_j) x_j \psi \rangle -
\langle \psi , i \chi_n(x_j) x_j H(\xi) \psi \rangle \\
 && = \langle
\chi_n(x_j) \psi , (- \frac{1}{m_0} v(\xi) + p_j + e A(x_j) ) \psi
\rangle \\
&&
 +  {\rm Re} \langle \psi , \frac{1}{n} (\nabla \chi)(|x_j|/n) x_j \cdot (-
\frac{1}{ m_0} v(\xi) + p_j + e A(x_j) ) \psi \rangle \\
&& - i \left( \psi , \left( \frac{1}{2 m_0} + \frac{1}{2} \right)
\frac{1}{n} (\nabla
\chi)(|x_j|/n) \psi \right) \\
  &&\stackrel{n\to \infty}{\longrightarrow} \langle \psi ,  (-
\frac{1}{m_0} v(\xi) +    p_j + e A(x_j)  ) \psi \rangle \; .
\end{eqnarray*}
The limit as $n$ tends to infinity follows from dominated
convergence.
 By polarization this yields the
claim.
\end{proof}
The proof of the next lemma is the same as the proof of Lemma
\ref{lem:nablae1}.
\begin{lemma} \label{lem:nablae} Let $P_0$ be the projection onto the
the kernel of $H(\xi) - E(\xi)$. If $E(\cdot )$ is differentiable at
$\xi$, then
$$
P_0 \frac{1}{m_0} v(\xi) P_0 = \nabla E(\xi) P_0 \; .
$$
\end{lemma}







We set
$$
Q(k) = |k| ( H(\xi - k) + |k| - E(\xi))^{-1} \; ,
$$
whenever this exists. And for $|k| > 0$, we set
$$
Q_0(k) = |k| ( H(\xi) + |k| - E(\xi) )^{-1} \; .
$$
Let $P_0$ be the orthogonal projection onto the kernel of $H(\xi) -
E(\xi)$. By the spectral Theorem
$$
P_0 = P_0(\xi) = s-\lim_{|k| \to 0} Q_0(k) \; .
$$


\begin{lemma} \label{lem:weak2} Let $E(\cdot)$ be differentiable at $\xi$. Given
$\epsilon > 0$. Then for $\widehat{k} = k /|k|$,
$$
w- \lim_{\widehat{k} \in S_\epsilon, |k| \to 0} \left( Q(k) - ( 1 -
\omega \nabla E(\xi) )^{-1} P_0 \right) = 0 \; .
$$
\end{lemma}
The proof follows the steps of Lemma \ref{lem:weak}, where  Step 1
uses Lemma \ref{lem:formbound}. We now present the proof of Theorem
\ref{thm:main22}.

 \vspace{0.5cm}

\noindent {\it Proof of Theorem \ref{thm:main22}.} Suppose $H(\xi)$
has a ground state $\psi$ with $\| \psi \| = 1$. We want to show
that this leads to a contradiction. Choose $\epsilon$ with $0<
\epsilon < 1$, and choose $\eta \in D((N+1)^{1/2})$ with
$(\eta,\psi) \neq 0$. By Lemma \ref{lem:basicom} there exits a
$\delta
>0$ such that for a.e. $k$, with $0<|k|<\delta$ and $\widehat{k} \in
S_\epsilon$,
\begin{eqnarray*}
( \eta , a_{\lambda,k} \psi ) &&= \frac{e
\rho(k)}{\sqrt{2}|k|^{3/2}} \left[ \left( \eta , Q(k) \frac{1}{2}
\sum_{j=1}^N e^{- i k \cdot x_j }( i k \wedge
\varepsilon_{\lambda,k} ) \cdot \sigma_j \psi
\right) \right. \\
&& + \left. \varepsilon_{\lambda , k } \cdot \left\{ \left( \eta ,
Q(k)\frac{Z}{m_0} v(\xi) \psi \right)  +    \left( \eta , Q(k)
\sum_{j=1}^N ( - e^{-ik \cdot x_j})(p_j + e A(x_j) ) \psi \right)
\right\} \right] \; .
\end{eqnarray*}
Since $Q(k)$ is uniformly bounded on $S_\epsilon$  for small $|k|$,
$$
\left( \eta , Q(k)   \sum_{j=1}^N   e^{-ik \cdot x_j} i (k \wedge
\varepsilon_{\lambda,k} ) \cdot \sigma_j \psi \right) \stackrel{|k|
\to 0}{\longrightarrow} 0 \; \; ,
$$
uniformly for  $k \in S_\epsilon$.
 Using Lemma \ref{lem:weak}, we find uniformly for $\widehat{k} \in
 S_\epsilon$
as $|k| \to 0$,
\begin{eqnarray*}
 \left( \eta ,
Q(k)\frac{Z}{m_0} v(\xi) \psi \right) {\longrightarrow}&& ( 1 -
\widehat{k} \cdot \nabla
E )^{-1} \left( P_0 \eta , \frac{Z}{m_0} v(\xi) \psi \right) \; \\
 && =  Z ( \nabla E ) (1 - \widehat{k} \cdot \nabla
E)^{-1} (\eta, \psi ) \; ,
\end{eqnarray*}
where we used Lemma \ref{lem:nablae}. Again by Lemma \ref{lem:weak}
and using that $e^{-ik \cdot x_j}$ converges in the strong operator
topology to $1$, we find uniformly for $k \in S_\epsilon$ as $|k|
\to 0$,
\begin{eqnarray*}
 \left( \eta , Q(k)
\sum_{j=1}^N ( - e^{-ik \cdot x_j})(p_j + e A(x_j) ) \psi \right)
 {\longrightarrow}   && ( 1 -  \widehat{k} \cdot \nabla E )^{-1}
\left( P_0
\eta , - \sum_{j=1}^N (p_j + e A(x_j))  \psi \right) \\
&&= ( 1 - \widehat{k} \cdot \nabla E )^{-1} \left( P_0 \eta ,
\frac{-N}{m_0} v(\xi)  \psi \right) \\
&& =  - \frac{N}{m_0}  (\nabla E) (  1 - \widehat{k} \cdot  \nabla E
)^{-1} (\eta ,\psi)
\end{eqnarray*}
where in the second line we used Lemma \ref{lem:virial} and in the
last again Lemma \ref{lem:nablae}. Collecting the above estimates we
conclude that for small $|k|$ uniformly for  $\widehat{k} \in
S_\epsilon$,
$$
| ( \eta , a_{\lambda , k} \psi ) | \geq \frac{1}{2} \frac{e
\rho(k)}{\sqrt{2}|k|^{3/2}} ( 1 - \widehat{k} \cdot \nabla E )^{-1}
\left| \varepsilon_{\lambda,k}  \cdot  \nabla E \right| | Z - N | |
(\eta , \psi ) | \; .
$$
 We introduce the set
$$
K := \{ \omega \in S^2 | - \sfrac{1}{2} | \nabla E | \leq \omega
\cdot \nabla E \leq 0 \} \subset S_\epsilon \; .
$$
Then, since by assumption $\nabla E \neq 0$, there exists a positive
constant $c_0 > 0$ such that for all   $\widehat{k} \in K$,
$$
\sum_{\lambda=1,2} | \varepsilon_{\lambda, k} \cdot  \nabla E  |^2
\geq c_0
> 0  \; .
$$
By this and $N \neq Z$, there exists a $c_1 > 0$ such that for all
small $k$ with $\widehat{k} \in K$, we find
\begin{eqnarray*}
 \frac{|\rho(k) c_1 |^2}{|k|^{3}}  && \leq
\sum_{\lambda=1,2}| ( \eta ,a_{k,\lambda} \psi ) |^2 \\
&& \leq \| ( 1 + N)^{1/2} \eta \|^2 \left( \sum_{\lambda=1,2}
\sum_{n=0}^\infty \int \| \psi_{n+1}(\lambda,k, \underline{k}_1,
... , \underline{k}_n  ) \|^2 d\underline{k}_1 ... d\underline{k}_n
\right) \; ,
\end{eqnarray*}
where in the last inequality we used
 Cauchy-Schwarz.
 This is inconsistent with $\psi$ being in $\FF \otimes \HH_0$. Thus
$H(\xi)$ does not have a ground state. \qed

\section*{Acknowledgements}

D.H. wants to thank Marcel Griesemer, Volker Bach, and Michael Loss
for  helpful discussions. I.H. would like to acknowledge an
interesting conversation with Benoit Gr\'{e}bert.



\begin{thebibliography}{30}



\bibitem{SPOHN04} H.~Spohn, {\em Dynamics of charged particles and their radiation field}. Cambridge
University Press, Cambridge, (2004).

\bibitem{IZ80} C.~Itzykson, J.-B.~Zuber , {\em  Quantum Field Theory}.
McGraw-Hill, (1980).



\bibitem{FK} L.D.~Fadeev, P.~Kulish, {\em }, Theor. and Math. Phys,
vol. 4, (1970), p. 247.

\bibitem{F73}   J. Fr\"{o}hlich, {\em
 On the infrared problem in a model of scalar electrons and massless, scalar bosons}.
 Ann. Inst. H. Poincaré Sect. A (N.S.) 19 (1973), 1--103.

\bibitem{F74} J. Fr\"{o}hlich, {\em Existence of Dressed One Electron
States in a Class of Persistent Models}. Fortschritte der Physik
{\bf 22}, (1974), 159--198.

%\bibitem{GJ} J. Glimm, A. Jaffe, {\em Quantum physics. A functional
%integral point of view}. Second edition. Springer-Verlag, New York,
%(1987).

\bibitem{LMS06} M. Loss, T. Miyao, H. Spohn, {\rm Lowest Energy
States in Nonrelativistic QED: Atoms and Ions in Motion}. preprint
{\tt math-ph/0605005}

\bibitem{AGG05} L. Amour, B. Grebert, J. Guillot, {\em The dressed
nonrelativistic electron in a magnetic field}. C. R. Math. Acad.
Sci. Paris 340 (2005) 421--426.

\bibitem{AGG06} L. Amour, B. Grebert, J. Guillot, {\em The Dressed Mobile
Atoms and Ions}. preprint {\tt math-ph/0507052}

\bibitem{BCFS05} V. Bach, T. Chen, J. Fr\"{o}hlich, I.M. Sigal, {\em
The renormalized electron mass in non-relativistic quantum
eletrodynamics}. preprint {\tt math-ph/0507043}

\bibitem{C01} T. Chen, {\em Operator-theoretic infrared
renormalization and construction of dressed one-particle states in
non-relativistic QED}. preprint {\tt mp\_arc 01-310}

%\bibitem{RSI}  M. Reed, B. Simon, {\em Methods of Modern
%Mathematical Physics: I Functional Aanalysis}, New York, Academic
%Press, (1972).

%\bibitem{RSII} M. Reed, B. Simon, {\em Methods of Modern
%Mathematical Phsysics: II Fourier Analysis, Self-Adjointness}, New
%York, Academic Press, (1975).

\bibitem{G72} L. Gross, J. Funct. Analysis {\bf 10}, 52, (1972).

\bibitem{H00} F. Hiroshima, {\em Essential self-adjointness of
translation-invariant quantum field models for arbitrary coupling
constants}, Commun. Math. Phys. 211, (2000), 585--613.

\bibitem{H02} F. Hiroshima, {\em Self-adjointness of the Pauli-Fierz
Hamiltonian for arbitrary values of the coupling constants}, Ann.
Henri Poincare 3, (2002), 171--201.

\bibitem{SW71} E. Stein, G. Weiss, {\em Introduction to Fourier
Analysis on Euclidean Spaces}, Princeton Press, (1971).

\end{thebibliography}



\end{document}
---------------0611132308609--
