Content-Type: multipart/mixed; boundary="-------------0111050943986"

This is a multi-part message in MIME format.

---------------0111050943986
Content-Type: text/plain; name="01-408.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="01-408.keywords"

scaling limits, Pauli-Fierz Hamiltonian, dipole approximation, weak coupling limit, effective potential, Symplectic structure, Bogoliubov transformation
---------------0111050943986
Content-Type: application/x-tex; name="1104.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="1104.tex"

\documentstyle[12pt,two side]{article}

\topmargin=0truein
\oddsidemargin=0truein
\evensidemargin=0truein
\textheight=21cm
\textwidth=15cm


\catcode`\@=11
\def\newsymbol#1#2#3#4#5{\let\next@\relax%
 \ifnum#2=\@ne\else%
 \ifnum#2=\tw@\let\next@\msyfam@\fi\fi%
 \mathchardef#1="#3\next@#4#5}
\def\mathhexbox@#1#2#3{\relax%
 \ifmmode\mathpalette{}{\m@th\mathchar"#1#2#3}
 \else\leavevmode\hbox{$\m@th\mathchar"#1#2#3$}\fi}
\def\hexnumber@#1{\ifcase#1 0\or 1\or 2\or 3\or 4\or 5\or 6\or 7\or 8%
\or 9\or A\or B\or C\or D\or E\or F\fi}
\font\tenmsy=msbm10
\font\sevenmsy=msbm7
\font\fivemsy=msbm5
\newfam\msyfam
\textfont\msyfam=\tenmsy
\scriptfont\msyfam=\sevenmsy
\scriptscriptfont\msyfam=\fivemsy
\edef\msyfam@{\hexnumber@\msyfam}
\def\Bbb#1{\fam\msyfam\relax#1}
\catcode`\@=\active
\load{\footnotesize}{\sf}



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

\newtheorem{assumption}[theorem]{Assumption}





\newcommand{\cmp}[5]{{#1}, {#2}, {\it Commun. Math. Phys.} {\bf #3} (#4), #5}
\newcommand{\jmp}[5]{{#1}, {#2}, {\it J. Math. Phys.} {\bf #3} (#4), #5}
\newcommand{\jfa}[5]{{#1}, {#2}, {\it J. Funct. Anal.} {\bf #3} (#4),
#5}
\newcommand{\rmp}[5]{{#1}, {#2}, {\it Rev. Math. Phys.} {\bf #3} (#4), #5}


\newcommand{\lk}{\left(}
\newcommand{\rk}{\right)}
\newcommand{\lkk}{\left\{}
\newcommand{\rkk}{\right\}}
\newcommand{\mat}[4]{\lk\!\!\begin{array}{cc}
#1&#2\\
#3&#4
\end{array}
\!\!\rk}
\renewcommand{\v}[2]{\lk\!\!\bea{c}
#1\\
#2
\ena\!\!\rk}
\newcommand{\ren}{H_\e^{\rm ren}(\k)}
\newcommand{\TT}{\widetilde{T}}
\newcommand{\Tmat}[4]{^T\!\!\lk\!\!\begin{array}{cc}
#1&#2\\
#3&#4
\end{array}
\!\!\rk}
\renewcommand{\v}[2]{\lk\!\!\bea{c}
#1\\
#2
\ena\!\!\rk}


\newcommand{\ov}[1]{\overline{#1}}
\renewcommand{\sc}{Schr\"odinger\ } 




\newcommand{\limn}{\lim_{n\rightarrow\infty}}
\newcommand{\limN}{\lim_{N\rightarrow\infty}}
\newcommand{\limk}{\lim_{\k\rightarrow\infty}}
\newcommand{\limt}{\lim_{t\rightarrow\infty}}
\newcommand{\limm}{\lim_{m\rightarrow\infty}}
\newcommand{\lime}{\lim_{\eta\downarrow 0}}

\newcommand{\proof}{{\noindent \it Proof:\ }}
\newcommand{\qed}{\hfill $\Box$\par\medskip}
\newcommand{\BR}{{{\Bbb R}^{d}}}


\newcommand{\cu}{{\cal U}}
\newcommand{\ii}{\infty}


\newcommand{\AAA}{\jjj \mhalf \lk\vp\otimes \I-\frac{\al}{c}\I\otimes   {A}_j\rk^2}
\newcommand{\AAAx}{\jjj \mhalf \lk\vp\otimes \I-\frac{\al}{c}   {A}_j(x_j)\rk^2}

\newcommand{\k}{\kappa}
\renewcommand{\P}{P_\epsilon}
\newcommand{\hk}{\he(\kappa)}
\newcommand{\he}{H_\epsilon}
\newcommand{\hz}{H_\epsilon^0}
\newcommand{\heq}{\hz(q)}
\newcommand{\zcr}{z\in{\Bbb C}\setminus\RR} 
\newcommand{\heqq}{\he(q,\delta,a,L)}


\newcommand{\pp}{\vec{p}}
\newcommand{\ff}{\vec{f}}
\newcommand{\vv}{{v}_\mu}

\newcommand{\vvvv}{{\bf V}}


\newcommand{\vvv}[1]{{v}_{\mu_{#1}}}
\newcommand{\curl}{{\rm curl}\ }
\newcommand{\supp}{{\rm supp}\ }
\newcommand{\sumn}{\sum_{n=1}^\infty}
\newcommand{\sumN}{\sum_{N=1}^\infty}
\newcommand{\sumi}{\sum_{i=1}^\infty}
\newcommand{\mf}[1]{\frac{1}{m_{j_{#1}}}}
\newcommand{\af}[1]{\alpha_{j_{#1}}}
\newcommand{\so}{\sqrt{\omega}}
\newcommand{\sa}{(s^2+A_0)\f}
\newcommand{\am}[1]{\frac{\alpha_{#1}}{m_#1}}
\newcommand{\atm}[1]{\frac{\alpha_{j_{#1}}^2}{{m_{j_{#1}}}}}
\newcommand{\atmm}[1]{\frac{\alpha_{#1}^2}{m_#1}}

\newcommand{\ww}{(-1)^{n-1}}
\newcommand{\www}{(-1)^n}
\newcommand{\rI}{{\rm (I)}}
\newcommand{\rII}{{\rm (II)}}
\newcommand{\ie}{{\rm I}_\lambda^\alpha(\e)}
\newcommand{\iie}{{\rm II}_\lambda^\alpha(\e)}
\newcommand{\oo}{\|\la/\so\|^2}
\newcommand{\ooj}{\|\laj/\so\|^2}


\newcommand{\rIl}{${\rm I_\lambda}$}
\newcommand{\rIIl}{${\rm II_\lambda}$}

\newcommand{\cons}{{\rm CONS}}



\newcommand{\qq}{\vec{q}}
\newcommand{\dd}{K}
\newcommand{\ddd}{{\cal D}}
\newcommand{\domain}{D(\jjjj\otimes\I)\cap D(\I\otimes \hf)}
\newcommand{\od}{\omega_\delta}

\newcommand{\th}{\theta(\xi,L)}

\newcommand{\tr}{{\rm tr}}

\newcommand{\vp}{{p_j}}
\newcommand{\jjj}{\sum_{j=1}^N}
\newcommand{\jjjjj}{\sum_{j=1}^Np_j^2}
\newcommand{\jjjjq}{\sum_{j=1}^N\frac{q_j^2}{2\mj}}
\newcommand{\jjjj}{\sum_{j=1}^N\frac{p_j^2}{2\mj}}
\newcommand{\xxx}{\sum_{j=1}^N{p_j^2}/({2\mj})}
\newcommand{\xx}{\sum_{j=1}^N{\e \al^2A_j^2}/({2\mj})}
\newcommand{\vr}{\vec{R}}
\newcommand{\vq}{\vec{Q}}
\newcommand{\fq}{H_{\rm quad}}
\newcommand{\vl}{L(q)}
\newcommand{\tvp}{{\varphi}_{ij}}
\newcommand{\tvvp}{\widetilde{\varphi}_{ij}}
\newcommand{\ttvp}{{\widetilde{\varphi}}_{ij}}
\newcommand{\tttvp}{{\widetilde{\varphi}}_{ij}}

\newcommand{\tlm}{{\Delta}}
\newcommand{\ttlm}{\widetilde{\Delta}}
\newcommand{\lm}{\Delta}


\newcommand{\vA}{{A_j}}
\newcommand{\aij}{\alpha_i\al}
\newcommand{\diag}{D_{\rm diag}}
\newcommand{\eff}{H_{\rm eff}}
\newcommand{\efff}{F_{\rm eff}}
\newcommand{\ft}{F_{T\f}}

\newcommand{\effk}{H_{\rm eff}(\k)}

\newcommand{\veff}{V_{\rm eff}}
\newcommand{\Vp}{\theta(p)}
\newcommand{\mass}{m^{\rm eff}_j}
\newcommand{\masss}{m^{{\rm eff}}}
\newcommand{\mj}{m_j}
\newcommand{\mij}{m_{ij}}
\newcommand{\ms}{m^\ast}


\newcommand{\bfs}{Bach-Fr\"ohlich-Sigal}


\newcommand{\BC}{{{\Bbb C}}}


\newcommand{\pf}{H_{\rm PF}}
\newcommand{\LR}{L^2(\BRN)}
\newcommand{\LL}{{L^2(\BR)}}

\newcommand{\er}{E_{\rm ren}}

\renewcommand{\L}{L_l}
\newcommand{\R}{R_l}
\newcommand{\hm}{{\hhh_m}}
\newcommand{\rrm}{{\RR^m}}
\renewcommand{\sup}{Suppose \rIl or \rIIl}
\newcommand{\RR}{{\Bbb R}}
\newcommand{\BRN}{{\RR^{dN}}}
\newcommand{\BRM}{\RR^{dM}}
\newcommand{\W}{{\Bbb W}}

\newcommand{\fff}{{{\cal F}}}
\newcommand{\wick}[1]{:\!\!{#1}\!\!:}
\newcommand{\f}{^{-1}}



\newcommand{\half}{{\frac{1}{2}}}
\newcommand{\sh}{{\frac{1}{\sqrt{2}}}}
\newcommand{\mhalf}{\frac{1}{2m_j}}
\newcommand{\mmhalf}{\frac{1}{2m}}

\newcommand{\mr}{E_a^{\rm ren}(p)}
\newcommand{\hr}{\he^{\rm ren}(\k)}

\newcommand{\dm}{\delta m}
\newcommand{\dmj}{\delta \mj}

\newcommand{\hf}{H_{\rm f}}
\newcommand{\hi}{H_{\rm I}}
\newcommand{\ele}{H_{\rm el}}
\newcommand{\nf}{N_{\rm f}}


\newcommand{\add}[1]{a^{\dagger #1}}
\newcommand{\Bdd}{B^\dagger}
\newcommand{\bdd}{b^\dagger}
\newcommand{\bss}{b^\sharp}
\newcommand{\Bss}{B^\sharp}

\newcommand{\ass}[1]{a^{\sharp #1}}
\newcommand{\aaa}[1]{a^{#1}}
\renewcommand{\aa}{\{\alpha_1,\cdots,\alpha_N\}}

\newcommand{\ccc}{{C_0^\infty(\BRN)}}
\newcommand{\D}{\Delta}
\newcommand{\han}{{1\!/\!2}}


\newcommand{\I}{1} 
\newcommand{\hhh}{{\cal H}}
\newcommand{\kkk}{{\cal K}}


\newcommand{\mmm}{\sum_{\mu=1}^d}
\newcommand{\nnn}{\sum_{l=1}^m}
\newcommand{\rrr}{\sum_{r=1}^{d-1}}
\newcommand{\iii}{\sum_{i=1}^N}
\newcommand{\ab}[1]{\langle{#1}\rangle}

\newcommand{\al}{\alpha_j}
\newcommand{\all}{\alpha_1,\cdots,\alpha_N}
\newcommand{\at}{\alpha_j^2}
\newcommand{\kt}{\k^2}


\newcommand{\emr}{e_\mu^r}
\newcommand{\ens}{e_\nu^s}
\newcommand{\wprs}{W_+^{rs}}
\newcommand{\wpmrs}{W_\pm^{rs}}
\newcommand{\wmrs}{W_-^{rs}}

\newcommand{\dmn}{d_{\mu\nu}}
\newcommand{\tmn}{T_{\mu\nu}}
\newcommand{\wrs}{W^{rs}}

\newcommand{\la}{\hat{\lambda}}
\newcommand{\laj}{\hat{\lambda}_j}
\newcommand{\lai}{\hat{\lambda}_i}
\newcommand{\rj}{\rho_j}
\newcommand{\e}{\epsilon}

\newcommand{\eI}{E_{\rm I}}
\newcommand{\eII}{E_{\rm II}}
\newcommand{\gI}{g_{\rm I}}
\newcommand{\gII}{g_{\rm II}}
\newcommand{\gi}{g_{\rm I}}
\newcommand{\gii}{g_{\rm II}}


\newcommand{\ijk}{\hat{\lambda}_i(k')\hat{\lambda}_j(k')}
\newcommand{\ij}{\la_i\la_j}




\newcommand{\bea}[1]
{\begin{array}{#1}}
\newcommand{\ena}{\end{array}}









\newcommand{\eq}[1]{\begin{equation}
\label{#1}}
\newcommand{\en}{\end{equation}}
\newcommand{\kak}[1]{(\ref{#1})}


\newcommand{\bt}[1]{\begin{theorem}
\label{#1}}
\newcommand{\et}{\end{theorem}}

\newcommand{\bc}[1]{\begin{corollary}
\label{#1}}
\newcommand{\ec}{\end{corollary}}

\newcommand{\bl}[1]{\begin{lemma}
\label{#1}}
\newcommand{\el}{\end{lemma}}
\newcommand{\bp}[1]{\begin{proposition}
\label{#1}}
\newcommand{\ep}{\end{proposition}}
\newcommand{\bi}{\begin{description}}
\newcommand{\ei}{\end{description}}

\newcommand{\bog}{\Sigma_{\rm B}}
\newcommand{\bogg}{\Sigma_{\rm B real}}






\makeatletter
\@addtoreset{equation}{section}
\makeatother
\def\theequation{\arabic{section}.\arabic{equation}}



\begin{document}
\setlength{\baselineskip}{18pt}
\pagestyle{myheadings}
\markboth{Observable effect}{Observable effect}
\title
{Observable effects and parametrized 
scaling limits  of a model in nonrelativistic quantum electrodynamics}
\author{ Fumio Hiroshima \thanks{
Department of Mathematics and Physics, 
Setsunan University, 
Ikeda-naka-machi 17-8, 572-8508, Osaka, Japan.}}
\maketitle
\begin{abstract}
Scaling limits of the Hamiltonian $H$ of  a system of $N$ charged particles  
coupled to a quantized radiation field 
are considered. 
Ultraviolet cutoffs, $\la_1,....,\la_N$,  are 
imposed on the radiation field and the Coulomb gauge is taken.  
It is so called the Pauli-Fierz model in nonrelativistic quantum electrodynamics. 
We mainly consider two cases: 
(i) all the ultraviolet cutoffs are identical, $\la_1=\cdots=\la_N$,  
(ii)  supports of ultraviolet cutoffs  have no intersection,  
${\rm supp}\la_i\cap{\rm supp}\laj=\emptyset$, $i\not= j$. 
The Hamiltonian acts on $\LR\otimes\fff$, where $\fff$ is a symmetric Fock space and 
has   the form 
$H=\ele\otimes\I+B+\I\otimes\fq$.
Here $\ele$ denotes a particle  Hamiltonian, 
$\fq$ a quadratic field operator,  
and $B$ an interaction term. 
The scaling is introduced as 
$H(\k)=\ele\otimes\I+\k^l B+\k^2\I\otimes\fq$, where $\k$ is a scaling parameter 
and 
$l\leq 2$ a parameter of the scaling. 
Performing a mass renormalization we consider 
the scaling limit  of $H(\k)$ as $\k\rightarrow \infty$ 
in the strong resolvent sense. 
Then effective  Hamiltonians $\eff$ in $\LR$ 
infected with  reaction of effect of the radiation field is derived. 
In particular 
(1)  effective Hamiltonians with an effective potential for $l=2$, and 
(2)  effective  Hamiltonians with an observed  mass for $l=1$,  are obtained. 
\end{abstract}

\newpage 
\tableofcontents
\newpage 
\section{Preliminaries}
\subsection{Observable effects and scaling limits}
In this paper we consider the Hamiltonian $H$ of a system of $N$  nonrelativistic 
charged particles interacting with a quantized radiation field. 
Ultraviolet cutoffs are imposed on the radiation field and recoil of the particles and the radiation field (photons) are excluded, i.e., the dipole approximation 
is made. 
Taking scaling limits of the Hamiltonian, we derive effective Hamiltonians. 
The main idea is to diagonalize $H$ by a  canonical transform derived from 
a symplectic structure. 


The \sc Hamiltonian of one charged particle with external potential $V$ is of the form 
\eq{sc}
\frac{1}{2m}p^2+V,
\en 
where $p$ denotes the momentum operator and 
$m$ the bare mass of the particle. 
To interpret the Lamb shift \cite{bet} intuitively,  
T. A. Welton \cite{we} formally derived an effective \sc Hamiltonian 
taking into account of effects of a radiation field. 
He expected that the position fluctuation of the  particle by 
the radiation field will effectively modify the external potential $V$. 
The  fluctuation was thought of as a Gaussian random variable $\Delta x$, 
then the effective potential is formally given by the mean value of 
$V(x+\Delta x)$; 
$$\veff(x):=\langle V(x+\Delta x)\rangle_{\rm AVE}
=(2\pi)^{-3/2}\int_{\RR^3}\hat{V}(k)e^{ikx}\langle e^{ik\Delta x}
\rangle_{\rm AVE}$$
$$=(2\pi C)^{-3/2}
\int_{\RR^3} V(y)e^{-|x-y|^2/2C}dy,$$
with a certain positive constant $C$. 
Then \kak{sc} effectively turns out to be 
the effective Hamiltonian, Welton's Hamiltonian,   given by 
\eq{v}
\frac{1}{2m}p^2+\veff.
\en
He gave an interpretation of the Lamb shift as the difference of the spectrum 
between the original Hamiltonian and the effective one. 


On the other hand it was found in \cite{pafi} 
that photons enhances the bare mass
$m$. 
Then $m$ amounts to the observed mass  
$m+\delta m$, $\delta m>0$,   through the coupling to a radiation filed. 
Then 
\kak{sc} formally turns out to be 
\eq{m}
\frac{1}{2(m+\delta m)}p^2+V.
\en




In this paper 
performing a mass renormalization to eliminate  a divergence contribution, 
we  present a rigorous derivation of   
\kak{v} and \kak{m} 
by  a parametrized scaling limits of  Hamiltonians  
describing an interaction between  particles and radiation fields. 
Thus we interpolate \kak{v} and \kak{m}. 
We suppose that $N$  particles move in the $d$-dimensional space 
and a radiation field has the helicity $d-1$. 
Throughout this paper we assume 
$$d\geq 3.$$
$\fff$ denotes  the Boson Fock space over $W:=\oplus^{d-1}\LL$, 
which is  defined by 
$$\fff:=\oplus_{n=0}^\infty [\otimes_s^nW],$$
where $\otimes_s^nW$ denotes  the $n$-fold symmetric tensor product of $W$, and 
$\otimes_s^0W:=\BC$. 
The bare vacuum $\Omega$ is defined by 
$$\Omega:=\{1,0,0,\cdots\}.$$
Hamiltonians under consideration are  of the form 
$$H:=\ele\otimes \I +B+\I  \otimes \fq$$ 
acting on the  Hilbert space 
$$\hhh:=\LR\otimes \fff. $$
Here 
$\ele$ describes the  particle Hamiltonian 
of the form 
$$\ele:=\jjj \mhalf  \vp^2+V.$$
Here 
$\vp:=-i\hbar \vec{\nabla}_j$ denotes 
the momentum operator of 
the $j$th particle canonically conjugate to the position operator $x_j$, 
$V:\BRN\rightarrow \RR$ an external potential, 
$\hbar$ the Planck constant divided by $2\pi$,   
$\mj$ the mass of the $j$th particle. 
$\fq$ a quadratic field operator in $\fff$, 
and $B$ an interaction term. 
We scale $H$ as follows: 
$$H(\k):=\ele\otimes \I  +\k^l B+\kt \I  \otimes \fq.$$
Davies \cite{da1,da2} considered the scaling limit $\k\rightarrow \infty$ of 
the case of 
$$l=1$$ for  certain simple models to derive 
$N$-body  
\sc Hamiltonians. 
He called this  scaling limit  {\it the weak coupling limit}. 
See also \cite{du,pa}. 
On the other hand Arai \cite{ar1} considered the case of 
$$l=2$$ 
for the one particle  Pauli-Fierz model in the dipole approximation without 
the self-interacting  term $A^2$. 
Then he derived effective Hamiltonians with an effective potential. 
In this paper we consider  scaling limits  
of 
$$l\leq 2$$
for the $N$-particle Pauli-Fierz model in the dipole approximation. 
In particular,  
performing a mass renormalization, 
subtracting from $H(\k)$ 
some energy reset, 
and 
taking $\k\rightarrow \infty$ in the resolvent sense, 
we derive, for $l=2$,  
Welton's Hamiltonians,  and 
for $l=1$, Hamiltonians with an observed mass.



\subsection{Nonrelativistic models}
Let $\add {}(f)$ and $\aaa {} (f)$, $f\in W$, 
 denote  the smeared creation and annihilation operators 
acting on $\fff$, respectively. $\ass {}$ stands for $a$ or $\add{}$.  
We set 
$$\ass r(f):=\add {}(
\underbrace{0\oplus0\cdots 
\oplus \stackrel{r_{\rm th}}{f}\oplus \cdots 0\oplus 0}_{d-1\ 
\mbox{{\footnotesize times}}}).$$
They   satisfy the canonical commutation relations, 
$$[\aaa s(g), \add r(f)]=\delta_{rs}\int_\BR f(k) g(k) dk,$$
$$[\add r(f),\add s(g)]=[\aaa r(f),\aaa s(g)]=0$$
on the finite particle subspace of $\fff$. 
We take the Coulomb gauge. 
The radiation field with the ultraviolet cutoff $\laj \in\LL$ 
is defined by 
$$A_{j\mu}(x_j):={A}_\mu (x_j,\laj ):=\sh \rrr 
\lkk \add r(\emr e^{-ikx_j}\widetilde{\la}_j )+\aaa r(\emr e^{ikx_j}\laj ) \rkk,$$
where $e^r(k)=(e^r_1(k),...,e^r_d(k))$ denotes polarization vectors satisying 
$e^r(k)\cdot e^s(k)=\delta_{rs}$ and $e^r(k)\cdot k=0$, 
$$\widetilde{\la}_j (k):=\laj (-k),$$
and $\hat{f}$ denotes the Fourier  transform of $f$. 
We assume 
$$\laj(k)=\laj(-k) $$ 
to ensure that $A_{j\mu}(x_j)$ is symmetric for each $x\in\BR$.  
The physically  reasonable choice of the ultraviolet cutoff  is 
$$\laj (k)=\frac{\sqrt{c}\hbar \hat{\rho}_j(k)}{\sqrt{(2\pi)^d \omega(k)}}, $$
where $\hat{\rho}_j$ denotes the Fourier transform of 
the  charge distribution $\rho_j$ of the $j$th particle and 
$c$ denotes the speed of the light. 
The dispersion relation $\omega(k)$ is given by 
$$\omega(k):=|k|.$$
The free Hamiltonian $\hf$ in $\fff$ is defined  by 
$$\hf:=\hbar c \rrr \int\omega(k)\add r (k)\aaa r(k) dk,$$
where $\add r(k)$ and $\aaa r(k)$ denote the  kernels of $\add r(f)$ and $\aaa r(f)$, 
respectively. 
The total  Hamiltonian  is given by 
$$H_{\rm total}:=\AAAx+\I\otimes \hf+V\otimes \I,$$
where 
$\al$'s are  non zero coupling constants. 
We take the dipole approximation, i.e., we replace $A_j(x_j)$ with 
$$A_j:=A_j(0).$$
Then the   Hamiltonian  in the dipole approximation is given by 
$$H:=\AAA+\I\otimes \hf+V\otimes \I.$$
We work with  the unit $c=\hbar =1$ in what follows. 
Since 
$$[\vp_\mu\otimes 1, 1\otimes \vA_\nu]=0,$$ 
we have 
$$H=\ele\otimes \I -\jjj \frac{\al}{\mj}\vp\otimes 
\vA+\I\otimes\lk \jjj \frac{\at}{2\mj}\vA^2+\hf\rk.$$
It is of mathematical interest to see  $A^2_j$-corrections in scaling limits. 
Then we define,  for $\e\geq 0$, 
$$\he:=\ele\otimes\I -\jjj\frac{\al}{\mj}\vp\otimes \vA+ \I\otimes\lk 
\jjj \frac{\e \at }{2\mj}A^2_j+\hf\rk$$
with the domain 
$$D(\he)=D(\ele\otimes\I)\cap D(\I\otimes\hf)\cap D(\I\otimes \jjj \frac{\epsilon\alpha_j^2}{2m_j} A_j^2).$$
Actually, for $\e<1$, $\he$ turns out to be 
an unphysical Hamiltonian for sufficiently large coupling constants, 
since in this case 
$\he$ is unbounded from below.  
See Remark \ref{999} for details.  
Let 
$$\k>1$$ 
be a  scaling parameter. 
We introduce the scaled Hamiltonian $\hk$ as follows:
$$\hk:=\ele\otimes\I -\k^l\jjj \frac{\al }{\mj}\vp\otimes
 \vA+\k^2 \I\otimes \lk \jjj  \frac{\e \at }{2\mj} A^2_j+\hf\rk,$$
where 
$$l\leq 2.$$
In particular we set 
$$H(\kappa):=H_1(\k).$$
Unless confusions arise we  omit $\otimes \I$ and $\I\otimes$ for simplicity 
in what follows. 
Throughout this paper we  suppose that  $V$ satisfies 
Hypothesis \vvvv:
\bi
\item[Hypothesis \vvvv:]
$V$ is infinitesimally small with respect to $\jjj\frac{p_j^2}{2m_j}$. 
\ei
First of all we have to establish self-adjointness of $\hk$. 
Note that if we assume that $\laj/\so,\laj,\omega\laj\in\LL$, then 
$$
\|\lk -\jjj\frac{\al}{\mj}\vp \vA+ 
\jjj \frac{\e \at }{2\mj}A^2_j\rk\Psi\|\leq a' \|\lk\jjjj+\hf\rk \Psi\|+b'\|\Psi\|$$
for $\Psi\in D(\hf)$ with some $a'$ and $b'$. 
Let 
$$\ddd:=\domain.$$
\bp{self6}
Let $\e=1$. 
We suppose that $\laj/\so,\laj,\omega\laj\in\LL$. 
(1) $H_{\rm total}$ and $H$ are self-adjoint on $\ddd$ and bounded from below. 
(2) Let $l=1$. Then $H(\k)$ is self-adjoint on $\ddd$ and bounded from below for all $\k$. 
\ep
\proof 
See \cite{hi3,hi4} for (1). 
In the case of $l=1$, $H(\k)$ is derived from the scaling 
$\al\rightarrow \k\al$ and $\omega\rightarrow \k\omega$ in $H$. 
Thus the self-adjointness follows from (1) 
\qed
\bp{eself6}
Let $\e\not=1$ and $V=0$. 
Suppose $\laj/\so,\laj,\omega\laj\in\LL$. Then 
$\hk$ is essentially self-adjoint on any core of $\xxx+\hf$. 
\ep
\proof 
The proof is due to Arai \cite[Theorem 2.2]{ar2}. 
Let $L:=\xxx+\hf+\I$. 
Then we see that 
$$\|\hk\Psi\|\leq c\| L\Psi\|,$$
and 
$$|(\hk\Psi, L\Psi)-(L\Psi,\hk\Psi)|\leq c'\|L^\han \Psi\|\|L^\han \Psi\|$$
with some $c$ and $c'$. 
Thus, by the Nelson commutator theorem \cite[Theorem X.36]{rs2},  
$\hk$ is essentially self-adjoint on any core of $L$. 
Thus the proposition follows. 
\qed
Self-adjointness of $\hk$ for $\e<1$ or $l\not=1$ is 
discussed later (Theorem \ref{self4}). 
We  denote the self-adjoint extensions of $H_\epsilon$ and $H_\epsilon(\k)$ 
by the same symbols. 




\subsection{Problems}
For the case of $N\geq 2$ we have to take care of 
the choice of ultraviolet cutoffs. 
For simplicity we mainly consider two  cases: 
identical case, $\la_1=\cdots=\la_N:=\la$, 
and   independent case,  
$\supp \laj\cap \supp \la_i=\emptyset $ for $i\not=j$.


Arai \cite{ar1} considered  the scaling limit of the 
case of $l=2$,  $\epsilon=0$ and $N=1$. 
Moreover in \cite{hi1,hi2}  distinct scaling limits were  studied, 
but unfortunately  
$A^2$-corrections did not appear in scaling limits.  
In this paper we want to improve \cite{hi1,hi2},   and 
furthemore 
to see the relationship between Davies' scaling limit and Arai's one. 
As was seen  in subsection 1.1, 
 Davies' scaling limits correspond to $l=1$. 

We set 
$$\hz:=\he\lceil_{V=0}.$$
Since $\hz$ commutes with $p_{j\mu}$, $\he$ is decomposable in the spectrum of 
the particle momentum operators; 
$$\hz=\int_{\BRN}^\oplus \heq dq.$$
$\heq$ acts on $\fff$. Its ground state energy is denoted by 
$$E(q):=\inf\sigma(\heq).$$
Then we call 
$$m^\ast(\mu,\nu,j):=\lk\left.\frac{\partial^2}{\partial q_{j\mu}\partial q_{j\nu}}
E(q)\right\lceil_{p=0}\rk\f$$
an {\it observed mass}. 
In this paper 
we explicitly give ground state energy of $\hz$   
and its observed mass.  
  




Thus the  list of  problems considered in this paper is epitomized as follows. 
\begin{itemize}
\item[(1)] Explicit ground state energy,  $\inf\sigma(\hz)$.
\item[(2)] Observed mass,  $m^\ast(\mu,\nu,j)$. 
\item[(3)] Deriving effective Hamiltonians $\eff$ 
through scaling limits 
of  $\hk$ subtracting a divergent contribution  $\er(\k)$, 
$$s-\limk(\hk-\er(\k)-z)\f=(\eff-z)\f\otimes P_\e$$
for  $\zcr$, where $P_\e$ denotes a projection to the subspace spanned by ground states 
of 
$\xx+\hf$.
\item[(4)] What is a reasonable choice of 
$\er(\k)$?

\item[(5)] $l$-dependence of $\eff$. 
\item[(6)] $A^2_j$-dependence (i.e., $\e$-dependence) of $\eff$. 
\item[(7)] Ultraviolet cutoff  dependence of $\eff$. 
\end{itemize}
We give answers for (1) in Lemma \ref{form}, 
(2) in Corollary \ref{bbc}, 
(3) in Theorems \ref{m21}-\ref{m24},
(4) in \kak{we}, 
(5), (6) and (7) in \kak{star}.
 



This paper is organized as follows. 
In Section 2 we discuss  symplectic group structures of $\hz$. 
In Section 3 we 
construct the canonical transformation and diagonalize $\hz$. 
In Section 4 we derive  effective Hamiltonians by scaling limits. 
In Section 5 we give a remark on $g$-factor shift. 
Sections 6 and 7 are appendixes. 
\section{Symplectic group structures}
\subsection{Unitary representations}
Let $B(W)$ denote the set of bounded operators on $W$. 
We define for $K\in B(W)$  
$$\ov{K}f:=\ov{K\ov{f}}.$$
We introduce a notation. 
For $A,B,C,D\in B(W)$ we define 
$$
\mat A B C D
\v{a(f)}{\add {}(g)}
:=
\v{a(Af)+\add{}(Bg)}{a(Cf)+\add{}(Dg)}.$$
Set 
$$
\Tmat A B C D:=\mat A C B D.$$
\begin{definition}
Let $S,T\in B(W)$ and set 
$$\xi:=\mat{S}{\ov T}{ T}{\ov S}:W\oplus W\rightarrow W\oplus W.$$
\begin{itemize}
\item [(1)] We say that  $\xi$ is of 
 symplectic group $\Sigma$ if 
\eq{xi}
\xi^\ast J \xi=\xi J\xi^\ast =J,
\en
where 
$$\xi^\ast=\mat{S^\ast}{T^\ast}{\ov{T}^\ast}{\ov{S}^\ast},\ \ \ 
J:=\mat{I}{0}{0}{-I}.$$
\item[(2)] 
We say that $\xi$ is of $\bog$ if 
(i) $\xi\in\Sigma$ and  
(ii) $T$ is a Hilbert-Schmidt operator. 
\end{itemize}
\end{definition}
$\bog$ is  a subgroup of $\Sigma$. 
We define 
$$\hm:=L^2(\RR^m)\otimes\fff\cong \int_\rrm^\oplus \fff dq.$$
Suppose that 
$$\xi=\mat {S} {\ov{T}}{ T} {\ov{S}}\in\bog,$$
and 
$$L=(L_1,...,L_m)\in\oplus^m W.$$
For $f\in W$ we let 
$$\v{b(f)}{\bdd(f)}:=
\Tmat {S} {\ov{T}}{ T} {\ov{S}}
\v{a(f)}{\add{}(f)}$$
and 
$$B(f):=\I\otimes b(f)+ \nnn (\L, f) p_l\otimes\I,$$
$$\Bdd(f):=\I\otimes \bdd(f)+\nnn (\ov{\L},f) p_l\otimes\I,$$
where $p_l$, $l=1,...,m$,  denote the momentum operators in $L^2(\rrm)$. 
It follows that 
$$[b(f),\bdd(g)]=(\ov f, g),$$
$$
[b(f),b(g)]=[\bdd(f),\bdd(g)]=0.$$
Moreover since  
$$\xi\f=J\xi^\ast J=\mat {S^\ast} {-T^\ast} {-\ov{T}^\ast} {\ov{S}^\ast}, $$
we have 
\eq{aa}
\v{a(f)}{\add {}(f)}
=
\Tmat {S^\ast} {-T^\ast} {-\ov{T}^\ast} {\ov{S}^\ast} 
\v{b(f)}{\bdd(f)}
\en 
and 
\eq{aaa}
\v{\I\otimes a(f)}{\I\otimes \add {}(f)}
=
\Tmat {S^\ast} {-T^\ast} {-\ov{T}^\ast} {\ov{S}^\ast}
\v{B(f)}{\Bdd(f)}
-\nnn \v{
(S \L-\ov T \ov \L, f) p_l\otimes\I}
{(\ov S \ov \L- T \L, f) p_l\otimes \I}. 
\en 
Define 
$$\R:=T\L-\ov{S\L},$$
and 
$$\theta(q,\xi,L):=\exp\lkk\nnn q_l\lkk \add {}(R_l)-a(\ov R_l)\rkk \rkk,\ \ \ q=(q_1,\cdots,q_m)\in\rrm.$$
Furthermore set 
$$\th:=\int_\rrm^\oplus \theta(q,\xi,L)dq=
\exp\lkk\nnn p_l\otimes\lkk \add {}(R_l)-a(\ov R_l)\rkk \rkk.$$
\bp{sym}
Suppose $\xi=\mat{S}{\ov{T}}{T}{\ov{S}}\in\bog$. 
Then there  exists a  unitary operator $U(\xi)$ 
of $\fff$ such that for all $f\in W$  
\begin{enumerate}
\item[(1)] $ U(\xi)\f\bss (f)U(\xi)=\ass{}(f)$,  
\item[(2)] $U(\xi)\f \th \f \Bss(f)\th U(\xi)=\ass{}(f).$
\end{enumerate}
\ep
\proof 
See \cite{be,kmp, ru1} for details 
\footnote{We  easily see that 
for 
$\xi_1:=\mat {S} {\ov{T}} {T} {\ov{S}} \in\bog$ and 
$\xi_2:=\mat U {\ov{V}} V {\ov{U}} \in\bog$, 
$$U(\xi_2)U(\xi_1) a(f) U(\xi_1)\f U(\xi_2)\f=
a\lk(U S+\ov{V}T)f\rk+\add{}\lk(VS+\ov{U}T)f\rk,$$
$$U(\xi_2)U(\xi_1) \add{}(f) U(\xi_1)\f U(\xi_2)\f=
a\lk(U\ov{T}+\ov{V}\ov{S})f\rk+\add{}\lk(V\ov{T}+\ov{U}\ov{S})f\rk,$$
and $$\xi_2\cdot \xi_1=\mat
{US+\ov V T}{U\ov T +\ov V \ov S}{VS+\ov U T}{V\ov T+\ov U \ov S}.
$$
Then 
$U(\xi_2)U(\xi_1)=U(\xi_2\xi_1).$ 
Hence the map 
$\bog\ni\xi\mapsto U(\xi)$ 
defines  a  unitary representation of $\bog$.}. 
\qed
We set 
$$\cu(q,\xi,L):=\theta(q,\xi,L) U(\xi):\fff\rightarrow\fff,$$ 
and 
$$\cu(\xi,L):=\th U(\xi):\hhh_m\rightarrow\hhh_m.$$ 
We call $\cu(\xi,L)$ the canonical  transformation  associated with 
$\{\xi, L\}\in\bog\times \oplus^m W$. 
\subsection{Average of  fluctuations}
Define 
$$\pi(f):=i\lkk \add{ }(\bar{f})-a(f)\rkk,\ \ \ f\in W.$$
We want to obtain 
the average of fluctuation $F_{\xi L}$ of a  function $F:\RR^m\rightarrow \BC$;
$$F_{\xi L}:=\cu(\xi,L)\f F \cu(\xi,L)$$
$$= 
U(\xi)\f F(x_1+\pi(R_1),\cdots,x_m+\pi(R_m)) U(\xi),$$
with respect to the bare vacuum. 
Let $\Gamma$ be the $m\times m$ symmetric  matrix defined by 
$$\Gamma:=\lk ({L}_\mu,{L}_\nu)_W \rk_{1\leq \mu,\nu\leq m},$$
and $\ab{a,b}$ denote the scalar product on $\BC^m$. 
Suppose $${\rm Rank}\ \Gamma=n\leq m$$
and $T$ is a  unitary matrix such that 
$$T\Gamma T\f={\rm diag}
\{\mu_1,\cdots,\mu_n,0,\cdots,0\},\ \ \ \mu:=\mu_1\cdots\mu_n\not=0.$$
Let 
$$\ft(x):=F(T\f x),\ \ \ x\in \rrm.$$
We define 
$$\efff(x):=(2\pi \mu)^{-m/2}
\int_{\RR^n}\ft(y_1,\cdots,y_n,(Tx)_{n+1},\cdots,(Tx)_m)\times$$
$$\times 
e^{-\sum_{j=1}^n|(Tx)_j-y_j|^2/(2\mu)}dy.$$
In particular if ${\rm Rank}\ \Gamma=n$, then 
$$\efff(x)=
(2\pi{\rm det}\Gamma )^{-m/2}
\int_\rrm  F(y)e^{-|x-y|^2/(2{\rm det}\Gamma)} dy.$$
\bl{potential}
Let $\xi=
\mat {S} {\ov T} {{T}} {\ov{S}}
\in\bog$ and $(L_1,\cdots,L_m)\in\oplus^m W$. 
Suppose that $F$ is such that 
$$
\int_{\RR^n}|\ft|
(y_1,\cdots,y_n,(Tx)_{n+1},\cdots,(Tx)_m)e^{-\sum_{j=1}^n|(Tx)_j-y_j|^2/(2\mu)}dy
\in L_{\rm loc}^1(\rrm,dx).$$
Then for 
 $f\in L^2(\RR^m)$,  and $g\in L^2(\rrm)$  such that 
$\cu(\xi,L)g\otimes\Omega\in D(F)$, 
we see that $g\in D(\efff)$ and 
\eq{in}
(f\otimes \Omega, F_{\xi L} g\otimes \Omega)_\hm
=(f,\efff g)_{L^2(\rrm)}.
\en
\el
\proof 
We see that by \kak{aa}  
$$a(\ov \R )=b(S^\ast \ov \R )-\bdd (\ov{T}^\ast \ov \R ),$$
$$\add{}(\R)=-b(T^\ast \R )+\bdd(\ov{S}^\ast \R ).$$
Thus we have 
$$\add {}(\R)-a(\ov \R)=\bdd(\ov{S}^\ast \R +\ov{T}^\ast \ov \R )-
b(T^\ast \R +S^\ast \ov \R )$$
$$:=\bdd (\R')-b(\ov{\R'}).$$
It follows from $\xi^\ast J\xi=\xi J\xi^\ast$ that 
$$\R'=\ov{S}^\ast(T\L-\ov{S\L})+\ov{T}^\ast(\ov{T\L}-S\L)$$
$$=(\ov{S}^\ast T-\ov{T}^\ast S)\L+(\ov{T}^\ast \ov T-\ov{S}^\ast\ov S)\ov \L
=-\ov \L.$$
Hence 
$$\pi(\R)=i\lkk \add {}(\R)-a(\ov\R)\rkk =i\lkk b(\L)-\bdd(\ov \L)\rkk.$$
Then we have 
\eq{ll}
U(\xi)\f \pi(\R) U(\xi)=i\lkk a(L_l)-\add{}(\ov{L}_l)\rkk =-\pi(\L).
\en
Let 
$$F_N(x):=
\lkk \bea{cc}
F(x),&|x|<N,\\
0,&|x|\geq N.
\ena\right.$$
Moreover 
$$F_{N\e}:=\rho_\epsilon\ast F_N,\ \ \ \e>0,$$
where $\rho_\epsilon(x):=\rho(x/\epsilon)/\epsilon^m$, 
$\rho(x)\geq 0$, ${\rm supp}\rho\subset\{x\in\rrm||x|\leq 1\}$, and 
$\int_\rrm \rho(x)dx=1$.  
\kak{ll}
leads to 
$$(f\otimes \Omega, (F_{N\e})_{\xi L}  g\otimes \Omega)$$
$$=
(2\pi)^{-m/2}\int_\rrm dk 
e^{ikx}\check{F}_{N\e}(k) 
(f\otimes \Omega,  U(\xi)\f e^{i\ab{k, x+\pi(R)}} U(\xi)  g\otimes \Omega)$$
$$=(2\pi)^{-m/2}\int_\BR dk 
e^{ikx}\check{F}_{N\e}(k) 
(f\otimes \Omega,  e^{i\ab{k, x-\pi(\ov L)}} g\otimes \Omega)$$
$$ 
=(2\pi)^{-m/2}\int_\rrm dk 
e^{ikx}\check{F}_{N\e}(k) 
\ov{f(x)}g(x) e^{-\ab{k,\Gamma k}/2}$$
$$=(f,(F_{N\e})_{\rm eff} g).$$
Taking $\epsilon\rightarrow 0$, 
we have $(F_{N\e})_{\rm eff}(x)\rightarrow (F_N)_{\rm eff}(x)$, 
 and $(F_{N\e})_{\xi L}\rightarrow (F_N)_{\xi L}$ strongly. Then the lemma follows for $F_N$. 
Let $f\in C_0^\infty(\rrm)$. 
Then taking $N\rightarrow\infty$, 
we also have $(f,(F_N)_{\xi L} g)\rightarrow (f, F_{\xi L} g)$ and 
$(f, (F_N)_{\rm eff} g)\rightarrow (f,\efff g)$ by the dominated convergence theorem. 
Hence the lemma follows. 
\qed
\begin{remark}
$\efff$ is independent of $\xi$. 
\end{remark}
\subsection{Physical vacuum}
Let $\xi=\mat {S}{\ov T}{T}{\ov S}\in\bog$ and $L\in \oplus^{dN} W$. 
We define 
$$\Omega(q,\xi,L):=\cu(q,\xi, L)\Omega.$$
Let $K$ be a Hilbert-Schmidt operator. 
Then there exist complete orthonormal systems (CONS) 
$\{\psi_i\}$, $\{\phi_i\}$,  and a positive sequence 
$\{\mu_i\}$ such that 
$$Kf=\sum_{i=1}^\infty \mu_i (\psi_i,f)\phi_i,\ \ \ f\in\hhh_m,$$
with 
$$\sumi \mu_i^2=\|K\|^2_{\rm HS},$$
where $\|\cdot\|_{\rm HS}$ denotes the Hilbert-Schmidt norm. 
We define for a finite particle vector $\Psi$  
$$\ab{\add{ }|K|\add{ }}\Psi:=s-\limn \sum_{i=1}^n 
\mu_i\add{}(\ov\psi_i)\add{}(\phi_i)\Psi.$$
The above strong limit actually exists. It is well known that 
if 
$$(1)K\  \mbox{is a Hilbert-Schmidt operator}, 
\ \ \ (2)\ov K^\ast=K,\ \ \ (3)\|K\|<1,$$
then 
$$U_0(K):=s-\limN \sum_{n=1}^N \frac{1}{n!}\lk-\half \ab{\add{} |K|\add{}}\rk^n\Psi$$
exists for a finite particle vector $\Psi$. See \cite{ar9}. 
\bl{pv}
We have 
$$\Omega(q,\xi,L)={\rm det}(\I-(TS\f)^\ast TS\f)^{1/4}\theta(q,\xi,L)U_0(TS\f)\Omega.$$
\el
\proof 
See \cite{hi1} for details.
\qed






\section{Canonical transformations} 
\subsection{Fixed  particle-momentum Hamiltonians}
We define the fixed particle-momentum Hamiltonian 
by 
$$\heq:=\jjj \mhalf q_j^2-\jjj \frac{\al}{\mj} q_j\cdot A_j+
\jjj \frac{\e \at}{2\mj}A^2_j+\hf.$$
Here $q$ is a  fixed particle-momentum: 
$$q=(q_1,\cdots,q_N)\in\BRN.$$
\bp{self}
Suppose that 
$\laj/\sqrt{\omega}, \laj, \omega\laj\in\LL$ 
for $j=1,...,N$. 
Then 
for all $\aa$ and all $q\in\BRN$, 
$\heq$ is self-adjoint on $D(\hf)$ and bounded from below. 
\ep
\proof 
The  proof is similar to Proposition \ref{eself6}. 
Let $L:=\hf+\I$. 
Then we easily have 
$$\|\heq\Psi\|\leq c\|L\Psi\|,$$
and 
$$|(\heq\Psi,L\Phi)-(L\Psi,\heq\Phi)|\leq c'\|L^\han \Psi\|\|L^\han \Phi\|$$
with some constants $c$ and $c'$ for $\Psi,\Phi\in  D(\hf)$. 
Hence 
$\heq$  is essentially self-adjoint on 
$D(\hf)$ by the Nelson commutator theorem. 
Moreover it is seen that 
$$c''\lk\|\hf \Psi\|^2+\|\jjj\lk \frac{q^2_j}{2\mj}-\frac{\al}{\mj}q_j\cdot A_j+\frac{\e \al}{2\mj}A_j^2\rk\Psi\|^2\rk
\leq \|\heq\Psi\|^2+c'''\|\Psi\|^2$$
with some $c''$ and $c'''$. Then $\heq$ is closed on $D(\hf)$. 
Thus $\heq$ is self-adjoint on $D(\hf)$. 
\qed
By a canonical transformation  $\cu(q)$, we will 
diagonalize $\heq$ as 
\eq{h1}
\cu(q)\f\heq \cu(q)=E(q)+\hf
\en
with some constant $E(q)$. In \cite{hi1}, 
\kak{h1} was established for $N=1$, in \cite{hi2} for $N\geq1$ 
but sufficiently small $|\al|$'s.  
In this section we improve it as follows:
\bi
\item[(1)] we extend \kak{h1} to $N\geq1$ and arbitrary $\al$'s,
\item[(2)] we derive an explicit form of $E(q)$, 
\item[(3)] we show analytic properties of $E(q)$ in each $\al$'s.
\ei










\subsection{Assumptions  
on ultraviolet cutoffs and  coupling constants}
In this subsection we introduce assumptions on $\laj$ and  $\al$. 
We suppose the following assumptions on $\laj$'s.
\begin{itemize}
\item[(A.1)] $\laj$ is rotation invariant, i.e., 
there exits a function $\varphi_j:[0,\infty)\rightarrow \RR$ 
 such that 
$\laj (k)=\varphi_j(|k|)$, 
\item[(A.2)] $\laj/\omega,\omega\laj  \in\LL$, 
\item[(A.3)] $\tvp(s):=\varphi_i(\sqrt{s})\varphi_j(\sqrt{s})
s^{(d-1)/2}\in L^\delta([0,\infty),ds)$ for some 
$0<\delta<1$, and there exists $0<C_{ij}<1$ such that, for arbitrary $s,t\geq 0$, 
$$|\tvp(s)-\tvp(t)|\leq C_{ij}|s-t|.$$ 
\item[(A.4)] $\|\laj \omega ^{(d-2)/2}\|_\infty<\infty$ and 
$\|\laj \omega ^{d/2}\|_\infty<\infty$, where $\|g\|_\infty:={\rm sup}_{x\in\BR}|g(x)|$. 
\end{itemize}
Above assumptions are sufficient conditions to construct a canonical transformation. 
Let 
$$m_{ab}:=\lkk
\begin{array}{cc}
m_j, & a=b=j,\\ 
0,& a\not=b.
\end{array}
\right.
$$
Define 
$$D_{ij}(k):=\mij -\epsilon  \aij \dd\lime  \int_\BR\frac{\omega(k')\ijk}
{\omega(k)^2-\omega(k')^2+i\eta }dk'$$
\eq{D}
=\mij-\epsilon \aij \frac{ \dd}{2} S_{d-1}\lk \lime
\int_{|\omega(k)^2-x|>\eta, x>0}
\frac{\tvp(x)}{\omega(k)^2-x}dx-2\pi i \tvp(\omega(k)^2)\rk,
\en
where $S_{d-1}$ denotes the volume of the  $(d-1)$-dimensional unit sphere and 
$$\dd:=\frac{d-1}{d}.$$
We define 
$$D:=D(k)=(D_{ij}(k))_{1\leq i,j\leq N}:\oplus^N \LL\rightarrow\oplus^N\LL. $$
We introduce  two assumptions of ultraviolet cutoffs,  
\rIl\  and \rIIl,   as follows: 
\bi
\item[Hypothesis \rIl:] 
$\la_1=\cdots=\la_N:=\la$ and $\Im \la(k)\not= 0$ for all $k\in\BR$. 
\item[Hypothesis \rIIl:] Each $\laj$ has a compact support such that 
$\supp \laj\cap \supp \la_i=\emptyset $ for $i\not=j$, 
and 
$\left|\frac{\laj(k)}{D_{jj}(k)}\right|\rightarrow 0$ as 
$k$ converges to the boundary of ${\rm supp}\laj$.  
\ei
In what follows we mainly study above two cases. 
The next assumptions are  needed to see the boundedness of the Hamiltonians from below. 
\bi
\item[Hypothesis $\ie$:] 
Ultraviolet cutoffs satisfies  \rIl, and 
$\aa$ satisfies that 
$$\lkk \bea{ll}
\jjj \atmm j<\frac{1}{\oo}\frac{1}{1-\e},& 
0\leq \e<1,\\
\ &\\
\aa\in\RR^N,& \e\geq 1.
\ena\right.$$
\item[Hypothesis $\iie$:]
Ultraviolet cutoffs satisfies  \rIIl, and 
$\aa$ satisfies that 
$$\lkk \bea{ll} 
\atmm j  \leq \frac{1}{\ooj}\frac{1}{1-\e},\ \ \ j=1,...,N,&
0\leq \e<1,\\
\ &\\
\aa\in\RR^N,& \e\geq 1.\ena\right.
$$
\ei


























\subsection{Diagonalization of $\heq$}
Since 
$$D_{ij}(0)=m_{ij}+\e\aij\frac{\dd}{2}S_{d-1}\int_0^\infty \frac{\varphi_{ij}(x)}{x}dx,$$
we easily see that $D(0)\f$ exists as an $N\times N$ matrix for \rIl and \rIIl. 
Set 
$$
\lk\bea{c}
R_1\\
\vdots\\
R_N
\ena
\rk
:= \so D(0)\f
\lk\bea{c}
\la_1\\
\vdots\\
\la_N
\ena
\rk.
$$
We define
$$\theta(q):=\exp\lk \jjj\rrr\mmm \frac{\al q_{j\mu}}{\sqrt{2}}
\lkk a^r\lk\ov{
\frac{\emr R_j}{\omega^{3/2}}}\rk
-\add{r}
\lk \frac{\emr R_j}{\omega^{3/2}}\rk\rkk\rk.$$
\bp{dia}
\sup. 
\bi
\item[(1)] There exist $\W\in\bog$ and $Q_j$, $j=1,...,N$, such that 
$$  \theta(q, \W,L)=\theta(q),$$
where 
$L=(L_1^1,...,L_1^d,...,L_N^1,...,L_N^d)\in\oplus^{dN}W$ with  
$$L_j^\mu:=\frac{\al}{\sqrt{2}}\oplus_{r=1}^{d-1}\frac{\emr Q_j}{\omega^{3/2}}.$$
\item[(2)]
Let 
 $$\cu(q):=\theta(q) U(\W).$$  
Then $\cu(q)$  maps $D(\hf)$ onto itself, 
and 
$$  \cu(q) \f \heq\cu(q) =E(q)+\hf$$
with some $E(q)$. 
\item[(3)]
 \ $E(q)$ can be analytically continued to a neighborhood ${\cal O}$ of $\RR$ in each 
$\alpha_j$'s.
\ei
\ep
\proof 
See Appendix. 
\qed




\subsection{Ground state energy}
In this subsection we show an explicit form of  $E(q)$ for both cases \rIl\  and 
\rIIl. 
In the case of $N=1$ it is shown in \cite{hisp}. 
For the case of $N\geq 2$ one needs a slight modification of \cite{hisp}.
We do not, however,  need to go into the minor details for the moment. 
So a  proof of the following lemma is taken up  in Appendix. 
\bl{form}
(1) 
Suppose  \rIl. 
Then we have 
$$E(q)=\half\jjj\frac{q_j^2}{m_j}
-\half \eI\left|\jjj\frac{\al}{\mj}q_j\right|^2 
+\gI(m)$$
where 
$$\eI:= \frac{\dd \|\la/\so\|^2}{1+
\epsilon\rho \|\la/\so\|^2},$$
$$\gI(m):=\frac{d }{2\pi}\int_{-\infty}^\infty
\frac{\epsilon \rho 
s^2\|\so\la/(s^2+\omega^2)\|^2}{1+\epsilon\rho
\|\so\la/\sqrt{s^2+\omega^2}\|^2}ds,$$
$$\rho:=\dd \jjj\atmm j.$$
(2) 
Suppose \rIIl. 
Then we have 
$$E(q)=\half\jjj\frac{q_j^2}{m_j}
-\half\jjj\frac{q_j^2}{\mj} \eII^j
+\gII(m),$$
where $$\eII^j:= 
\frac{\rj\|\laj/\so\|^2}{1+\e \rj \|\laj/\so\|^2},$$
$$\gII(m):=
\jjj\frac{d}{2\pi}\int_{-\ii}^\ii\frac{\epsilon\rj s^2\|\so\laj/(s^2+\omega^2)\|^2}
{1+\epsilon \rj \|\so\laj/\sqrt{s^2+\omega^2}\|^2}ds,$$
$$\rj:=\dd\atmm j.$$
\el
\proof 
See Appendix. 
\qed
It is an interesting case where $\epsilon=1$. 
\begin{corollary} 
(1) Suppose \rIIl  and  
$\epsilon=1$. 
Then  
$$E(q)
=\jjj \frac{1}{2\mass}q_j^2
+
\jjj\frac{d}{2\pi}\int_{-\ii}^\ii\frac{\rj s^2\|\so\laj/(s^2+\omega^2)\|^2}
{1+ \rj \|\so\laj/\sqrt{s^2+\omega^2}\|^2}ds,$$
where 
$$\mass:=m_j+ \at \dd \|\laj/\so\|^2.$$
(2) Suppose $N=1$ and $\epsilon=1$. 
Then 
$$E(q)=\frac{1}{2\masss}q^2+
\frac{d}{2\pi}\int_{-\ii}^\ii
\frac{ \dd \alpha_1^2 s^2\|\so\la_1/(s^2+\omega^2)\|^2}{m+ \dd \alpha_1^2 \|\so\la_1 
/(s^2+\omega^2)\|^2}ds,$$
where 
$$\masss:=m_1+\alpha_1^2 \dd \|\la_1/\so\|^2.$$
\end{corollary}
\bl{ii}
(1) Suppose $\ie$.  Then for all $q\in\BRN$ 
$$ \jjjjq-\half \eI\left|\jjj \frac{\al}{\mj}q_j\right|^2>0.$$
(2) Suppose $\iie$. Then for all $q\in\BRN$ 
$$ \jjjjq-\half \jjj \frac{q_j^2}{\mj} \eII^j>0.$$
\el
\proof
By the Schwartz inequality we have 
$$\half \eI\left|\jjj \frac{\al}{\mj}q_j\right|^2\leq \frac{\rho \oo}{1+\e \rho \oo}\jjjjq.$$
Hence 
$$ \jjjjq-\half \eI\left|\jjj \frac{\al}{\mj}q_j\right|^2\geq 
\half\lk 1-\frac{\rho\oo}{1+\e\rho\oo}\rk \jjjjq.$$
Since Hypothesis  $\ie$ implies that 
$$1-\frac{\rho\oo}{1+\e\rho\oo}>0,$$
(1) follows.
(2) is similarly proven. 
\qed


\subsection{Observed mass}
We define the observed  mass 
by 
$$\ms(\mu,\nu,j):=\left. 
\lk \frac{\partial^2}{\partial q_{j\mu}\partial q_{j\nu}}E(q)\right
\lceil_{q=0}\rk\f.$$
Put 
$$\ms(\mu,\mu,j):=\ms({\mu,j}).$$
By Lemma \ref{form}  we have the following corollaries. 
\bc{bc}
(1) Suppose  \rIl. Then 
$$m^\ast(\mu,\nu, j)=\delta_{\mu\nu}
{\mj}\lk\frac{1+\dd (\e \sum_{i=1}^N\atmm i-\atmm j)\|\la/\so\|^2}
{1+\e \dd\jjj \atmm j\|\la/\so\|^2}\rk\f.
$$
In particular for the case of $\e=1$,  
$$m^\ast(\mu,\nu, j)=\delta_{\mu\nu}
{\mj}\lk\frac{1+\dd\sum_{i\not=j}\atmm i\|\la/\so\|^2}
{1+\dd\jjj \atmm j\|\la/\so\|^2}\rk\f.
$$
(2) Suppose \rIIl. Then 
$$ m^\ast(\mu,\nu, j)=\delta_{\mu\nu}
\frac{
m_j+\e \dd \al^2\|\laj/\so\|^2}{
1+(\e-1)\dd \atmm j\|\laj/\so\|^2}.
$$
In particular for the case of $\e=1$, 
$$ m^\ast(\mu,\nu, j)=\delta_{\mu\nu}
\lk 
m_j+\dd \al^2\|\laj/\so\|^2\rk.$$
\ec
\proof 
It is a direct calculation. 
\qed
It is interesting to compare the sizes of observed masses for \rIl and \rIIl. 
\bc{bbc}
Suppose $\e=1$. Let $\ms_{\rm I}(\mu,j)$ 
be the observed  mass for \rIl\ 
and ${\ms_{\rm II}(\mu,j)}$ for \rIIl. 
Suppose that 
$\|\la/\sqrt{\omega}\|\leq \|\laj/\sqrt{\omega}\|$ for  $j$. 
Then 
$${\ms_{\rm I}(\mu,j)}< {\ms_{\rm II}(\mu,j)}.$$
\end{corollary}
\proof 
Note that 
$$f(x)=\frac{x}{a+x},\ \ \ a\geq 0,$$ 
is a monotonously  increasing function of $x$. 
Directly we have 
$$\frac{1}{{\ms_{\rm I}(\mu,j)}}=
\frac{1}{\mj}\lk\frac{1+\dd\sum_{i\not=j}\atmm i\|\la/\so\|^2}
{1+\dd\jjj \atmm j\|\la/\so\|^2}\rk
$$
$$> \frac{1}{m_j}\lk \frac{1}{1+\dd\atmm j \|\la/\so\|^2}\rk
\geq \frac{1}{m_j}\lk \frac{1}{1+\dd\atmm j \|\la_j/\so\|^2}\rk$$
$$=
\frac{1}{\ms_{\rm II}(\mu,j)}.
$$
Hence the corollary follows. 
\qed

\subsection{Diagonalization of $\hz$ and its self-adjointness}
We define 
$$\theta:=
\exp\lk \jjj\rrr\mmm \frac{\al }{\sqrt{2}}
p_{j\mu} \otimes \lkk a^r\lk\ov{
\frac{\emr R_j}{\omega^{3/2}}}\rk
-\add{r}
\lk \frac{\emr R_j}{\omega^{3/2}}\rk\rkk\rk$$
and 
$$\cu:= \theta U(\W).$$
Note that $\cu$ maps $\ddd$ onto itself:
$$\cu:\ddd\rightarrow \ddd.$$
Moreover we define 
$E(p)$ by the pseudo differential operator with the kernel $E(q)$ in $\LR$. 
\bl{dia1}
Suppose \rIl or \rIIl. 
Then for $\Phi\in \ccc\widehat{\otimes} D(\hf)$, 
\eq{lll}
\cu\f\hz\cu \Phi =(E(p)\otimes \I+\I\otimes \hf)\Phi.
\en
\el
\proof
Let 
$\Phi=g\otimes\psi$ and $\Psi=f\otimes \phi$, $g,f\in\ccc$, $\psi,\phi\in D(\hf)$. 
We note that $\cu \Phi\in \ddd$. 
Then the left hand side of \kak{lll} is well defined. 
We have by Proposition \ref{dia}
$$(\Psi, \cu\f \hz \cu \Phi)_\hhh$$
$$=\int_{\BRN} dq (\widehat{\cu \Psi}(q), 
\widehat{\hz \cu \Phi}(q))_\fff=
\int_{\BRN} dq (\cu(q)\widehat{\Psi}(q), \heq \cu(q) \widehat{\Phi}(q))_\fff$$
$$
\int_{\BRN} dq \ov{\widehat{f}}(q) \widehat{g}(q) 
(\cu(q){\psi}, \heq \cu(q) {\phi})_\fff
=
\int_{\BRN} dq \ov{\widehat{f}}(q) \widehat{g}(q) 
({\psi}, (E(q)+\hf){\phi})_\fff$$
$$
=(\Psi, (E(q)\otimes\I+\I\otimes \hf)\Phi)_\hhh.$$
Thus the lemma follows.
\qed
\bl{self2}
Suppose $\ie$ or $\iie$. 
Then 
\eq{sp}
\cu\f\hz\cu=E(p)\otimes \I+\I\otimes \hf
\en
holds on $\ddd$. In particular 
$\he$ is self-adjoint on $\ddd$ and bounded from below. 
\el
\proof
By the assumption on $\aa$, $E(p)$ is nonnegative. Then  
$E(p)+\hf$ is self-adjoint on $\ddd$ and bounded from below. 
For $\Phi\in\ddd$, we can take a sequence $\Phi_n\in 
\ccc\widehat{\otimes}D(\hf)$ 
such that 
$\Phi_n\rightarrow\Phi$ and $(E(p)\otimes\I+\I\otimes \hf) \Phi_n\rightarrow 
(E(p)\otimes\I+\I\otimes \hf)\Phi$ strongly as $n\rightarrow \infty$. 
Since $\he$ is closed, $\he\cu\Phi_n\rightarrow 
\he\cu\Phi$ as $n\rightarrow\infty$.
Thus \kak{sp} holds. 
Since $\cu$ maps $\ddd$ onto itself, 
$\he$ is self-adjoint on $\ddd$ and bounded from below. 
\qed
\begin{remark}
\label{999} 
Let  $\e<1$. 
Then $E(p)$ is unbounded from below for sufficiently large coupling constants. 
Actually in the case of $N=1$, for 
$$\alpha_1^2\geq m_1 \lk \|\la_1/\so\|^2(1-\e)\rk\f,$$
$E(p)$ is unbounded from below.  
Hence $\hz$ 
is unbounded from below for sufficiently large coupling constants.
We should not overlook that 
the approximation of neglecting  the self-interaction  $A^2$ in $H$ is not reasonable for 
sufficiently large coupling constants.  
\end{remark}
\bt{self4}
Suppose $\ie$ or $\iie$. 
Then 
$\he$ is self-adjoint on $\ddd$ and bounded from below. 
\et
\proof
By Lemma \ref{self2} and the closed graph theorem there exists a constant $c$ 
such that 
$$\|(\jjjj+\hf)\Psi\|\leq c(\|\he \Psi\|+\|\Psi\|),\ \ \ \Psi\in\ddd.$$
Thus 
$$\|V\Psi\|\leq ac (\|\he \Psi\|+\|\Psi\|)+b\|\Psi\|$$
with  some sufficiently small $a$, and some $b$, 
the Kato-Rellich theorem leads to  the lemma.
\qed
\bc{gs}
Suppose \rIl. Then 
$$\inf \sigma(\hz)
=\gi(m)=\frac{d }{2\pi}\int_{-\infty}^\infty
\frac{\epsilon \dd \jjj \frac{\al^2}{\mj}
s^2\|\so\la/(s^2+\omega^2)\|^2}{1+\epsilon
\dd \jjj \frac{\al^2}{\mj}
\|\so\la/\sqrt{s^2+\omega^2}\|^2}ds.$$
Suppose  \rIIl. Then  
$$\inf \sigma(\hz)=\gii(m)=
\jjj\frac{d}{2\pi}\int_{-\ii}^\ii\frac{\epsilon
\dd  \frac{\al^2}{\mj}
s^2\|\so\laj/(s^2+\omega^2)\|^2}
{1+\epsilon \dd  \frac{\al^2}{\mj}\|\so\laj/\sqrt{s^2+\omega^2}\|^2}ds.
$$
\ec
\proof 
It follows from Lemmas \ref{form} and \ref{dia1}.
\qed
















\section{Effective Hamiltonian}
\subsection{Scaled Hamiltonians}
Here we recall the scaled Hamiltonian $\hk$; 
$$\hk:=\ele\otimes\I -\k^l\jjj \frac{\al }{\mj}\vp\otimes
 \vA+\k^2 \I\otimes \lk \jjj  \frac{\e \at }{2\mj} A^2_j+\hf\rk.$$
$\hk$ is derived from  
$\he$ scaled as 
\eq{1}\omega\rightarrow \k^2\omega,
\en
\eq{2}\al\rightarrow \k^l\al,
\en
\eq{3}\epsilon\rightarrow \k^{2-2l}\epsilon.
\en
Replacing  $\omega,\al,\epsilon$ as in \kak{1}--\kak{3}, we see 
that 
\eq{i2}
\theta\longrightarrow \exp\lk \k^{l-2}\jjj \mmm \frac{\al}{\sqrt{2}}
p_{j\mu}\otimes 
\lkk a^r\lk\ov{\frac{\emr R_j}{\omega}}\rk-\add{r}\lk\frac{\emr R_j}{\omega}\rk\rkk\rk,
\en 
$$
-\half \eI\left|\jjj\frac{\al}{\mj}q_j\right|^2 
\longrightarrow 
-\k^{2l-2}
\half \eI\left|\jjj\frac{\al}{\mj}q_j\right|^2,  
$$
$$
-\half\jjj\frac{q_j^2}{\mj} \eII^j
\longrightarrow 
-\k^{2l-2}
\half\jjj\frac{q_j^2}{\mj} \eII^j,$$
$$ \gI(m)\rightarrow  \gI(m/k^2),$$
$$ \gII(m)\rightarrow  \gII(m/k^2).$$
\bl{i1}
$U(\W)$ leaves invariant under the scaling 
\kak{1},\kak{2} and \kak{3}.
\el
\proof 
See Lemma \ref{bog} in Appendix. \qed
Let 
\eq{we}
\er(p):=\lkk\bea{ll}
-\half 
\eI\left|\jjj\frac{\al}{\mj}p_j\right|^2,&\   {\rm for} \ {\rm I_\lambda},\\
\  & \\
-\half\jjj\frac{p_j^2}{\mj} \eII^j, &\ {\rm for}\   {\rm II_\lambda}.
\ena\right.
\en
Moreover we set 
$$g(m):=\lkk
\bea{ll}
\gi(m),& {\rm for}\  {\rm I_\lambda},\\
\ & \\
\gii(m),& {\rm for}\  {\rm II_\lambda}.
\ena\right.$$
We define $\cu_\k$ by $\cu$ with $\omega$,$\al$ and $\e$ replaced 
as in \kak{1},\kak{2}, and \kak{3}, respectively. 
\bl{dia2}
Suppose ${\rm I}_\lambda^\alpha(\k^{2l-2}\e)$ or 
${\rm II}_\lambda^\alpha(\k^{2l-2}\e)$. 
Then 
$$\cu_\k\f \hk\cu_\k=
\half\jjj\frac{p_j^2}{m_j}
+\k^{2l-2}\er(p)+\kt \hf + g(m/\kt)+\cu_\k\f V\cu_k.$$
In particular $\hk$ is self-adjoint on $\ddd$ and bounded from below. 
\el
\proof 
It follows from Lemma \ref{self2}.
\qed
\bl{gro}
Suppose \rIl or \rIIl. 
Then there exists the unique ground state $\Omega_\e$ 
of $\xx+\hf$.
Moreover 
\eq{gp}
P_\e:=U(\W)P_\Omega U(\W)\f
\en
is the projection to the eigenspace spanned by $\Omega_\e$. 
 \el
\proof 
Set $q=0$ in Lemma \ref{dia} (2). Then 
$$\cu(0)\f\lk \epsilon\jjj \frac{\al^2}{2\mj}A^2_j +\hf\rk\cu(0)= \hf+g(m).$$
The right hand side has the unique ground state $\Omega$. 
Thus $\cu(0)\Omega=U(\W)\Omega=\Omega_\e$ is the unique ground state of 
$\xx+\hf$. 
Hence the  lemma follows.
\qed
\bl{scaling}
(1) Suppose  $l=2$. Then $\cu_\k$ is  independent of $\k$. 
(2) Suppose $l<2$. Then  
$s-\limk \cu_\k=U(\W).$
\el
\proof 
It follows from \kak{i1} and \kak{i2}.
\qed
We write as 
$$\cu_\ii:=\lkk\bea{ll}
\cu,&l=2,\\
\ & \\
U(\W),&l<2.
\ena \right.
$$

\subsection{Effective potentials}
Define $\veff$ acting on $\LR$  by 
$$(f, \veff g)_{\LR}:
=(f\otimes \Omega, \lk\cu_\infty\f V \cu_\infty\rk g\otimes \Omega)_\hhh.$$ 
Note $$D(\veff)\supset D(\jjjj).$$
We define 
the $N\times N$ correlation matrix 
$\tlm=\lk\tlm_{ij}\rk_{1\leq i,j\leq N}$ by 
$$\tlm_{ij}:=\half \aij\dd
\lk  \frac {Q_i}{\omega^{3/2}}, 
\frac {{Q_j}}{\omega^{3/2}}\rk.$$
Note that 
$$
 \lk 
{L}_i^\mu,{{L}_j^\nu}\rk=\frac{\aij}{2}\dd \delta_{\mu\nu}
\lk  \frac {Q_i}{\omega^{3/2}}, 
\frac {{Q_j}}{\omega^{3/2}}\rk=\delta_{\mu\nu}\tlm_{ij}.
$$
Suppose that 
${\rm Rank}\tlm=M$
and  $T=T_\Delta $ is an $N\times N$ unitary matrix such that 
$$T\tlm T\f={\rm diag}\{\mu_1,\cdots,\mu_M,0,\cdots,0\},
\ \mu=\mu_1\cdots\mu_M\not=0.$$
Let 
$$\TT:=\lk\bea{ccc}
T_{11}\I_d&\cdots&T_{1N}\I_d\\
\vdots&\ &\vdots\\
T_{N1}\I_d&\cdots&T_{NN}\I_d
\ena\rk.$$
Here $\I_d$ denotes the $d\times d$ identity matrix. 
\bl{eff2}
Let  $l=2$. 
Suppose 
$$\int_{\BRM} 
|V_{\TT\f}|(y_1,\cdots,y_M,
(\TT x)_{M+1},\cdots,(\TT x)_N) 
e^{-\sum_{j=1}^K|(\TT x)_j-y_j|^2/(2\mu)}dy
\in L_{\rm loc}^1(\BRN).$$  
\bi
\item[(1)] 
Suppose that 
${\rm Rank}\tlm=N.$
Then we have 
$$\veff(x)=
(2\pi)^{-dN/2} ({\rm det}\tlm )^{-d/2}
\int_{\BRN} e^{-|x-y|^2/(2{\rm det}\tlm)}V(y) dy.$$
\item[(2)] 
Suppose that 
${\rm Rank}\tlm=M<N.$
Then we have 
$$\veff(x)=(2\pi)^{-dN/2} \mu^{-d/2}
\int_{\BRM} 
V_{\TT\f}(y_1,\cdots,y_M,
(\TT x)_{M+1},\cdots,(\TT x)_N) $$
$$\times e^{-\sum_{j=1}^K|(\TT x)_j-y_j|^2/(2\mu)}dy.$$
\ei
\el
\proof 
It follows from Lemma \ref{potential}. 
\qed
\bl{eff3}
Let $l<2$. 
Then 
$\veff=V.$
\el
\proof 
Since $\cu_\ii$ and $V$ is commutative, the lemma follows. 
\qed

\subsection{Scaling limits}
Let 
$$V_\k:=\cu_\k\f V \cu_\k.$$
We define 
$$\ren:=\lkk\bea{ll}
\hk-\k^{2l-2} \er(p)-g(m/\kt),& 1<l\leq 2,\\
\ &\\
\hk-g(m/\kt),&l\leq <1,
\ena
\right.
$$
and 
\eq{star}
\eff:=\lkk 
\bea{ll}
\jjjj+\veff,&l=2,\\
& \\
\jjjj+V,& 1<l<2,\\
& \\
\jjjj+\er(p)+V,&l=1,\\
& \\
\jjjj+V,&l<1,
\ena\right..
\en
\bl{self3}
Let $\k$ be sufficiently large. 
Define 
$$\effk:=
\lkk 
\bea{ll}
\jjjj +\k^2\hf+V,&   1<l\leq 2,\\
 & \\
\jjjj +\k^{2l-2}\er(p)+\k^2\hf+V, &   l\leq 1, \\
\ena\right..
$$
(1) Suppose \rIl and \rIIl,  and $l\not=1$. Then 
\eq{p}
\cu_\k\ren \cu_\k =\effk.
\en
In particular, for all $\aa$, $\ren$ 
is self-adjoint on $\ddd$ and bounded from below. 
(2) Suppose $\ie$ or $\iie$, and $l=1$. 
Then \kak{p} holds. In particular 
$\ren$ is self-adjoint on $\ddd$ and bounded from below. 
\el
\proof 
By Hypothesis \vvvv  we see that for $\Psi\in\ddd$ 
$$
\|V_\k\Psi \|\leq a\|\jjjj \Psi\|+b\|\Psi\|.
$$
Thus 
by the Kato-Rellich theorem $\effk$ for $1<l\leq 2$ is self-adjoint on $\ddd$ and bounded from below. For $l<1$, since $\k$ is sufficiently large, we have $\jjjj+\k^{2l-2}\er(p)$ is positive and 
$$\|(\jjjj+\er(p))f\|\geq \e(\all) \|\jjjj f\|$$ 
with some $\e(\all)>0$ independent of large $\k$. 
Then 
\eq{vv}
\|V_\k\Psi\|\leq 
\frac{a}{\e(\all)}\|(\jjjj+\er(p)) \Psi\|+b\|\Psi\|. 
\en
Hence 
 $\effk$ for $l\not=1$ is self-adjoint on $\ddd$ and bounded from below. 
By \kak{dia1}, \kak{p} holds on $\ccc\widehat{\otimes}D(\hf)$, 
and $\cu_\k$ maps $\ddd$ onto itself. 
Then \kak{p} holds on $\ddd$ by a limiting argument. 
Thus (1) follows. 
For $l=1$, $\xxx+\er(p)$ is 
a nonnegative self-adjoint operator. 
Thus $\effk$ is self-adjoint on 
$\ddd$ and bounded from below. Thus (2) follows similarly.
\qed
\bl{self5}
(1) Suppose $\ie$ or $\iie$, and $l=1$. 
Then $\eff$ is self-adjoint 
on $D(\xxx)$ and bounded from below.
(2) Let $l\not=1$. 
Then $\eff$ is self-adjoint on $D(\xxx)$ and bounded from below. 
\el
\proof 
It is clear. We omit it.\qed
Let $P_\e$ denotes the projection to the subspace spanned by the unique ground state of 
$\jjj\frac{\e \al^2}{2\mj}A_j^2+\hf$.
Note $P_0:=P_\Omega$ is the projection to $\Omega$. 
\bt{m21}
Let $l=2$. 
 Suppose \rIl or \rIIl. 
Moreover suppose 
$|\veff|\in L_{\rm loc}^1(\BRN)$.  Then 
for all $\aa$ and $\zcr$, 
$$s-\!\limk (\ren-z)\f
=
\theta\lkk 
\lk
\eff
-z\rk\f\otimes \P\rkk \theta\f.$$
\et
\bt{m22}
Let $1<l<2$. 
Suppose \rIl or \rIIl. 
Then for all $\aa$ and $\zcr$, 
\eq{ren}
s-\!\limk (\ren-z)\f=\lk \eff
-z\rk\f \otimes \P.
\en
\et
\bt{m23}
Let $l=1$. 
Suppose $\ie$ or $\iie$. 
Then \kak{ren} holds. 
\et
\bt{m24}
Let $l<1$.  
Suppose  \rIl or \rIIl. 
Then for all $\aa$ and $\zcr$, \kak{ren} holds. 
\et
\begin{table}
\arrayrulewidth=0.8pt
\def\arraystretch{3}
\begin{center}
 \begin{tabular}{|c||c|c|}
\hline 
\ & $\ren+g(m/\kt)$ &  $\eff$\\
\hline
$l=2$, \rIl  & $\hk+\frac{\kt}{2} \eI \left|\jjj \frac{\al}{\mj}p_j\right|^2
$ & $\mmhalf p^2+\veff$ \\
\hline
$l=2$, \rIIl  & $\hk+\frac{\kt}{2} \jjj \frac{p_j^2}{\mj}\eII^j
$ & $\mmhalf p^2+\veff$ \\
\hline
$1<l<2$, \rIl  & $\hk+\frac{\k^{2l-2}}{2} \eI \left|\jjj \frac{\al}{\mj}p_j\right|^2
$ & $\mmhalf p^2+V$ \\
\hline
$1<l<2$, \rIIl  & $\hk+\frac{\k^{2l-2}}{2} \jjj \frac{p_j^2}{\mj}\eII^j
$ & $\mmhalf p^2+V$ \\
\hline 
$l=1$, \rIl & $\hk$ & $\jjjj
-\half \eI \left|\jjj \frac{\al}{\mj}p_j\right|^2+V$\\
\hline
$l=1$,  \rIIl & $\hk$ & $\jjjj-\half \jjj \frac{p_j^2}{\mj}\eII^j+V$\\
\hline 
$l<1$ & $\hk$ & $\mmhalf p^2+V$ \\
\hline 
\end{tabular}
\caption{Effective Hamiltonians }
\label{table2}
\end{center}
\end{table}
Before proving Theorems \ref{m21}--\ref{m24} we show a lemma. 
\bl{eff}
Suppose $\zcr$. 
\bi
\item[(1)] Let $l=2$. 
Suppose $|V|_{\rm eff}\in L_{\rm loc}^1(\BRN)$, and \rIl or \rIIl.  Then 
for all $\aa$ 
\eq{ffe}
s-\limk\lk \effk-z\rk\f=\lk\eff -z\rk\f\otimes P_\Omega.
\en
\item[(2)] Let $1<l<2$. 
Suppose \rIl or \rIIl. Then for all $\aa$, \kak{ffe} holds. 
\item[(3)] Let $l=1$. 
Suppose $\ie$ or $\iie$. Then \kak{ffe} holds.
\item[(4)] Let $l<1$. 
Suppose \rIl or \rIIl. Then for all $\aa$, \kak{ffe} holds. 
\ei
\el
\proof
By an abstract theorem in \cite[p.2654, (i),(ii),(iii)]{ar1} 
it is enough to prove that 
\begin{itemize}
\item[(1)] $D(V_\k)\supset D(\eff)$ and $V_\k(\eff-z)\f$ is a bounded operator for sufficiently large $z$ with $\lim_{z\rightarrow\infty}\|V_\k(\eff-z)\f\|=0$ uniformly in $\k$, 
\item[(2)] $V_\k(\eff-z)\f$ is strongly continuous in $\k$ and 
$s-\limk V_\k(\eff-z)\f=\veff(\eff-z)\f$. 
\end{itemize}
By the definition of $\veff$ and Hypothesis \vvvv , we have 
$$\|\veff f\|\leq a'\|\jjjj f\|+b'\|f\|$$
with some  positive $a'$  and $b'$. 
Together with \kak{vv} we get 
$$\|V_\k f\|\leq a''\|\eff  f\|+b''\|f\|$$
with some  positive $a''$  and $b''$. 
Moreover 
$$\|V_\k(\eff-z)\f f\|\leq a''\|\eff(\eff-z)\f f\|+b''\|(\eff-z)\f f\|.$$
Hence $V_\k(\eff-z)\f f\rightarrow 0$ as $z\rightarrow \infty$. 
Then (1) follows. 
Since  $\cu_\k$ is continuous in $\k$ and 
$s-\limk \cu_k=\cu_\ii$, 
we have 
$$\|(V_\k-V_{\kappa'}) f\|
\leq 
\|\cu_\k\f V\cu_k f-\cu_\k\f V\cu_{\kappa'} f\|+
\|\cu_\k\f V\cu_{\kappa'} f-\cu_{\k'}\f V\cu_{\kappa'} f\|$$
$$\leq 
\|V (\cu_\k- \cu_{\k'}) f\|+
\|(\cu_\k\f-\cu_{\kappa'}\f) V\cu_{\k'} f\|$$
$$
\leq a \| (\cu_\k- \cu_{\k'}) \jjjj f\|+
b \| (\cu_\k- \cu_{\k'}) f\|+
\|(\cu_\k\f-\cu_{\kappa'}\f) V\cu_{\k'} f\|.$$
Here in the third inequality we use that $\cu_k$ and $p_j, j=1,...,N$ 
are commutative. 
Then the first statement of (2)  follows. 
The second statement is also proven similarly.
\qed
{\it A proof of Theorems \ref{m21}--\ref{m24}}\\
Suppose $l=2$. Then $\cu_\k=\cu$ is independent of $\k$. 
We have by \kak{p}
$$(\ren-z)\f=
\cu(\effk-z)\f\cu\f.$$
Taking $\k\rightarrow\infty$, we have 
$$\limk (\ren-z)\f=
\cu\lkk \lk\eff -z\rk\f\otimes P_\Omega\rkk \cu\f$$
$$=\theta \lkk \lk\eff-z\rk\f\otimes U(\W)P_\Omega U(\W)\f \rkk \theta\f.$$
Since $U(\W)P_\Omega U(\W)\f=P_\e$, Theorem \ref{m21} follows. 
Next suppose $l=1$. 
$$(\ren-z)\f=
\cu_\k\lk\effk -z\rk\f \cu_\k\f.$$
Since 
$$s-\limk \cu_\k=U(\W),$$
taking $k\rightarrow \infty$,  we have 
$$\limk (\ren-z)\f=U(\W)\lkk \lk\eff -z \rk\f\otimes P_\Omega \rkk U(\W)\f.$$
Hence Theorem \ref{m23} follows. 
Theorems \ref{m22} and \ref{m24}  are proven similarly. 
\qed

 
 





\subsection{Examples}
In this subsection we assume that 
$|\veff|\in L_{\rm loc}^1(\BRN)$. 
A typical example is for $N=1$ and $\e=1$. 
Put $A_1=A$, $m_1=m$, $p_1=p$, $\alpha_1=\alpha$. 
\bc{v3} 
Let $N=1$ and $\e=1$. 
Suppose \rIl or \rIIl. 
Then for all $\alpha\in\RR$ and $\zcr$  
\bi
\item[($l=2$)]
$$s-\!\limk \lk 
H(\k)+\kt\frac{1}{2(m+\dm)}\frac{\dm}{m}p^2-g(m/\kt)-z\rk\f$$
$$=
\theta\lkk 
\lk
\frac{1}{2m}p^2+\veff-z\rk\f \otimes P_1 \rkk \theta\f.$$
\item[($1<l<2$)]
$$s-\!\limk 
\lk H(\k)+\k^{2l-2}\frac{1}{2(m+\dm)}\frac{\dm}{m}p^2-g(m/\kt)-z\rk\f $$
$$=
\lk
\frac{1}{2m}p^2+V
-z\rk\f \otimes P_1.$$
\item[($l=1$)]
$$s-\!\limk \lk H(\k)-g(m/\kt)-z\rk\f =
\lk \frac{1}{2(m+\dm)}p^2 +V-z\rk\f \otimes P_1.$$
\item[($l<1$)]
$$s-\!\limk\lk H(\k)-g(m/\kt)-z\rk\f =\lk \mmhalf p^2+V-z\rk\otimes P_1.$$
\ei
\ec
By virtue of Corollary \ref{v3} we see that 
in the case of $l=2$ the Welton Hamiltonian dose appear and in the case of $l=1$ 
the Hamiltonian with observed mass does. 








\begin{table}
\arrayrulewidth=0.8pt
\def\arraystretch{3}
\begin{center}
 \begin{tabular}{|c||c|c|}
\hline 
\ & $H^{\rm ren}(\k)+g(m/\kt)$ &  $\eff$\\
\hline
$l=2$ & $H(\k)+\kt\frac{1}{2(m+\dm)}\frac{\dm}{m}p^2$ & $\mmhalf p^2+\veff$ \\
\hline 
$1<l<2$ & $H(\k)+\k^{2l-2}\frac{1}{2(m+\dm)}\frac{\dm}{m}p^2$ & $\mmhalf p^2+V$\\
\hline 
$l=1$ & $H(\k)$ & $\frac{1}{2(m+\dm)}p^2 +V$ \\
\hline 
$l<1$ & $H(\k)$ & $\mmhalf p^2+V$ \\
\hline 
\end{tabular}
\caption{Effective Hamiltonians for $N=1$ and $\epsilon=1$}
\label{table1}
\end{center}
\end{table}
\bc{v1}
Suppose \rIl and $l=2$. 
 Moreover we assume that 
$\alpha_1=\cdots=\alpha_N:=\alpha$ and $m_1=\cdots= m_N:=m$.
Then 
$$\veff=
(2\pi C)^{-d/2}\int_{\BR} V_{\TT\f}(y,(\TT x)_2,\cdots,(\TT x)_N))
e^{-|(\TT x)_1-y|^2/2 C}dy, $$
where 
$$C:=\frac{N\alpha^2}{2}\frac{d-1}{d}\|
\frac{Q}{\omega^{3/2}}\|^2,
$$
\ec
\proof 
Since the correlation matrix is given by 
$$\Delta=\frac{\alpha^2}{2}\|\frac{Q}{\omega^{3/2}}\|^2\dd
\lk\bea{ccc}
1&\cdots&1\\
\vdots&\ &\vdots\\
1&\dots&1
\ena
\rk,$$
where 
$$Q=\so D\f\lk\bea{c}
\la\\
\vdots\\
\la
\ena\rk,$$
we have  ${\rm Rank}\ \Delta=1$. 
Thus the corollary follows. 
\qed
\bc{v2}
Suppose \rIIl and $l=2$.  Then 
$$\veff=
(2\pi C)^{-d/2}\int_\BRN  V(y)e^{-|x-y|^2/2 C}dy, $$
where 
$$C:=\Pi_{j=1}^N \lk 
\frac{d-1}{2d} \frac{\at}{\mj}
\int_\BR\frac{|\laj(k)|^2}{\omega(k)^2(\mj-\epsilon \delta C_j)}dk\rk, 
$$
$$
\delta C_j:=\frac{d-1}{2d}\at \int_\BR\frac{\omega(k)|\laj(k')|^2}
{\omega(k)^2-\omega(k')^2+i0}dk'.
$$
\ec
\proof
Since the correlation matrix is given by 
$$\Delta=
{\rm diag}\{
\half \alpha_1^2 \dd \|\frac{Q_1}{\omega^{3/2}}\|^2,\cdots,
\half \alpha_N^2 \dd \|\frac{Q_N}{\omega^{3/2}}\|^2\},
$$
where 
$$Q_j=\frac{\so \laj}{D_{jj}},$$
the corollary follows. 
\qed

\section{Concluding Remarks ($g$-factor shift)}
For simplicity we set $\e=1$,  $N=1$, and $d=3$. 
Put $A_1=A$, $m_1=m$, $p_1=p$, $\alpha_1=\alpha$. 
We suppose that  spin $1/2$  is included in the particle and 
an external vector potential $a(x)$ is imposed. 
Thus the Hamiltonian is read 
$$H_{\rm spin}:=\frac{1}{2m}\lkk \sigma\cdot\lk p-a(x)-\alpha A\rk\rkk^2+V+\hf$$
acting on the Hilbert space 
$$\BC^2\otimes L^2(\RR^3)\otimes\fff.$$
Here $\sigma=(\sigma_1,\sigma_2,\sigma_3)$ denotes the Pauli matrices: 
$$
\sigma_1:=\half \mat0 1 0 1,\ \ \ 
\sigma_2:=\half \mat{0}{-i}{i}{0},\ \ \ 
\sigma_3:=\half \mat{1}{0}{0}{-1}.$$
We have 
$$H_{\rm spin}=\frac{1}{2m}(p-\alpha A)^2+\hf+V-\frac{g}{2m}\sigma\cdot\curl a +\hi,$$
where 
$$g=2$$
and 
$$\hi:=-\frac{1}{2m}\lkk(p-\alpha A)\cdot a-a\cdot(p-\alpha A)
\rkk+\frac{1}{2m}a^2-\frac{\alpha}{2m}\sigma\cdot \curl A,$$
and 
$$\curl A:=
\frac{-i}{\sqrt{2}}\sum_{r=1}^2 \lkk\add r\lk (k\times e^r)_\mu^r \tilde{\la}\rk
+a^r\lk (-k\times e^r)_\mu \la\rk\rkk.$$
For simplicity we neglect $\hi$. Then our Hamiltonian is defined by 
$$H=
\frac{1}{2m}(p-\alpha A)^2+\hf+V-\frac{g}{2m}\sigma\cdot\curl a.$$
Let $l=1$. Then the scaled Hamiltonian is given by 
$$H(\k):=
\frac{1}{2m}(p-\k \alpha A)^2+\kt \hf+V-\frac{g}{2m}\sigma\cdot\curl a.$$
\bc{spin}
Suppose that $a$  is sufficiently smooth and that  $\curl a$ is infinitesimally small with respect to $p^2$. 
Then $H(\k)$ is self-adjoint on $\ddd$ and bounded from below. Furthermore
for $\zcr$ 
$$s-\limk (H(\k)-g(m/\kt)-z)\f=(\eff-z)\otimes P_1.$$
Here 
$$\eff=
\frac{1}{2(m+\dm)}p^2+\frac{g_{\rm eff}}{2(m+\dm)}\sigma\cdot \curl  a+V,$$
and 
$$g_{\rm eff}:=\frac{m+\dm}{m}g>2.$$
\ec
\proof
The self-adjointness follows from Proposition \ref{self6}. 
Since $\sigma_j$ and $\cu_k$ are commutative, 
by Theorem \ref{m23},  we have 
$$s-\limk (H(\k)-g(m/\kt)-z)\f
=\lk\frac{1}{2(m+\dm)}p^2-\frac{g}{4m}\sigma\cdot \curl a+V-z\rk\f\otimes P_1.$$
Hence the corollary follows. 
\qed


\section{Appendix}
\subsection {A proof of Proposition  \ref{dia}}
\subsubsection{Construction of canonical transformations}
We recall 
$$
D_{ij}:=
\mij-\epsilon \aij \frac{ \dd}{2} S_{d-1}\lk \lime
\int_{|\omega(k)^2-x|>\eta, x>0}
\frac{\tvp(x)}{\omega(k)^2-x}dx-2\pi i \tvp(\omega(k)^2)\rk.$$
\bl
{ij}
Suppose \rIl or \rIIl. Then  $\|D_{ij}\|_\ii<\infty$.
\el
\proof
Set 
$$
\ttvp(s):=\lime\int_{|s-x|>\eta, x>0}
\frac{\tvp(x)}{s-x}dx.$$
Then by  (A.3)  and \cite{ti} we see that for $s,t\geq 0$ 
$$|\ttvp(s)-\ttvp(t)|\leq C_{ij}|s-t|.$$ 
Hence  $\ttvp$ is continuous. 
Since 
$$\lim_{|s|\rightarrow \infty}\ttvp(s)=0,$$
it follows that  
$\|D_{ij}\|_\infty<\ii.$
\qed
\bl{lambda}
Suppose \rIl. 
Then there exists 
$$Q=\lk \bea{c}Q_1\\\vdots\\
Q_N\ena\rk\in\oplus^N\LL$$ such that 
(1) $DQ=\so\lk 
\bea{c}
\la_1\\
\vdots\\
\la_N
\ena\rk,$
(2) $\|\omega^{(d-1)/2}Q_j\|_\infty<\infty$ and $\|\omega^{(d-3)/2}Q_j\|_\infty<\infty$, 
(3) $Q_j/\omega^{3/2}\in \LL.$
\el 
\proof
We set 
$$D(k):=M-\e\dd g(k) A(\alpha_1,\cdots,\alpha_N),$$
where 
$$M:= {\rm diag}\{m_1,\cdots,m_N\},$$
$$A(\alpha_1,\cdots,\alpha_N):=
\lk\begin{array}{ccc}
\alpha_1\alpha_1&\cdots&\alpha_1\alpha_N\\
\vdots& \ &\vdots \\
\alpha_N\alpha_1 &\cdots & \alpha_N\alpha_N
\end{array}
\rk,$$
and 
$$g(k):=\lim_{\eta\downarrow 0}\int_\BR\frac{\omega(k')|\la(k')|^2}{\omega(k)^2-\omega(k')^2+i\eta}dk'.$$
$A$ is  a positive definite  matrix and we  denote its eigenvalues  by 
\eq{aaaa}
\{a_1,\cdots,a_M,0,\cdots,0\},\ \ \ a_j>0,\ \  \ j=1,...,M,
\en
with some $M\leq N$. 
$A_M$ denotes the $M$-dimensional space spanned by eigenvectors with eigenvalues 
$a_1,...,a_M$, and its orthogonal complement by $A_M^\perp$. 
Set 
$m=\min\{m_1,\cdots,m_N\}$, $a=\min \{a_1,\cdots,a_M\}$. 
For $z\in A_M^\perp$ we have 
$$(z,D(k)z)=(z,Mz)\geq m|z|^2.$$
For $z\in A_M$, we have 
$$|(z,D(k)z)|\geq \e \dd |\Im g(k)||(z,Az)|\geq \e a \dd |\Im g(k)|  |z|^2.$$
Hence for $\e_0<|k|<\e_1$ with $0<\e_0$ and $0<\e_1$ there exists $d'=d'(\e_0,\e_1)>0$ 
such that 
$$
|(z,D(k)z)|\geq d'|z|^2.$$  
For $k=0$ it follows that  
\eq{44}
(z,D(0)z)=(z,Mz)-\e \dd g(0) (z, Az)\geq m |z|^2,
\en
and as $|k|\rightarrow \infty$ 
\eq{45}
(z,D(k)z)\rightarrow (z,Mz)\geq m|z|^2.\en
Combining \kak{44} with  \kak{45} we see that 
for $|k|\leq \e_0$ or $|k|\geq\e_1$  with 
some sufficiently small $\e_0$ and sufficiently large $\e_1$ there exists $d''$ such that 
$$|(z,D(k)z)|>d''|z|^2.$$
Hence for all $k\in\BR$ 
$$|(z,D(k)z)|>\min\{d',d''\}|z|^2.$$
In particular the inverse $D\f$ of $D$ exists as a bounded operator of $\oplus^N\LL$.  
Putting $Q:=\sqrt{\omega} D\f 
\lk \bea{c}
\la_1\\
\vdots\\
\la_N
\ena\rk,$
we see that $Q$ satisfies (1).
It is seen that 
$$\|\omega^n Q_j\|_\infty \leq 
\|D\f\|\|\omega^{n+(\han)}\laj\|_\infty.$$
From $\|\laj\omega^{(d-2)/2}\|_\infty<\infty$ 
and $\|\laj \omega^{d/2}\|_\infty<\infty$ (see (A.4)), 
(2) follows. 
Moreover 
form 
$$\|\omega^n Q_j\|_\LL \leq 
\|D\f\|\|\omega^{n+(\han)}\laj\|_\LL,$$
(3) follows, since $\la/\omega\in\LL$ (see (A.2)). 
Thus the lemma  follows.
\qed
\bl{lambda2}
Suppose  \rIIl. 
Then  there exists $Q$ such that 
(1)--(3) in Lemma \ref{lambda} are satisfied. 
\el
\proof
Note that 
$$D_{jj}(k)\not =0,\ \mbox{for}\ k\in
\{k\in\BR|\laj(k)\not =0\}.$$
Hence we can define  $Q=(Q_1,\cdots,Q_N)$ by 
$$Q_j(k):=\lkk 
\bea{ll}
\frac{\sqrt{\omega(k)} \laj(k)}{D_{jj}(k)},& k\in 
\{k\in\BR|\laj(k)\not =0\},\\
0,& 
k\not \in  {\rm supp }\laj.
\ena\right.$$
Since 
$\left|\frac{\laj(k)}{D_{jj}(k)}\right|\rightarrow 0$ as 
$k$ converges to the boundary of ${\rm supp}\laj$, 
$Q(k)$ is well defined for all $k\in\BR$,  and 
we can easily check that $Q$ satisfies (1)--(3). 
Thus the lemma follows.
\qed
Let 
$$R_j:=\so D(0)\f\lk\bea{c}
\la_1\\
\vdots\\
\la_N
\ena\rk.$$
Define 
$$Gf(k):=\lime 
\int_\BR\frac{f(k')}{(\omega(k)^2-\omega(k')^2+i\eta)
(\omega(k)\omega(k'))^{(d-2)/2}}dk'.$$
$G$ is a skew symmetric bounded operator of $\LR$. 
Let 
$$\tmn f:=
\delta_{\mu\nu}f +
\e \jjj \at    Q_j \omega^{(d-2)/2}G\omega^{(d-2)/2}\dmn \sqrt{\omega}\laj  f.$$
\bl{qs}
Suppose \rIl or \rIIl. Then the following algebraic relations hold: 
\bi
\item[(1)]
$\sum_{\beta=1}^d 
T_{\mu\beta}^\ast d_{\beta \nu}(1/\omega^2) Q_j=
d_{\mu\nu}{R_j}/{\omega^{3/2}}.$
\item[(2)]
$[\omega^2,\tmn^\ast]f=
-\jjj \al^2\lk Q_j, f \rk_\LL \dmn\so\laj.$
\item[(3)]$\tmn\so\laj=\delta_{\mu\nu}m_jQ_j.$
\item[(4)]$\tmn^\ast d_{\nu\beta}T_{\beta \gamma}=d_{\mu\gamma}.$
\item[(5)]$e_\mu^r \tmn d_{\nu\beta}T_{\beta\gamma}e_\gamma^s=\delta_{rs}.$
\ei
\el
\proof 
By (1) of Lemmas \ref{lambda} and \ref{lambda2}, it is proven. 
See \cite[Lemma 2.6]{hi2} for details. 
\qed
We define the bounded operator  
$W_\pm=(\wpmrs)_{1\leq r,s\leq d-1}:W\rightarrow W$ by 
$$\wprs f:=\half \emr\lk \omega^{-\han}\tmn^\ast\omega^\han+\omega^\han\tmn^\ast\omega^{-\han}\rk\ens f,$$
$$\wmrs f:=\half \emr\lk \omega^{-\han}\tmn^\ast\omega^\han-\omega^\han\tmn^\ast\omega^{-\han}\rk\widetilde{\ens f}.$$
It is  checked by (A.4), 
Lemmas \ref{lambda} and \ref{lambda2} that 
$$\|\wpmrs f\|\leq c\|f\|,$$
where 
$$ c= 1+\half\lk  
\|  \omega^{(d-3)/2} Q_j\|_\ii\|\omega^{d/2}\laj\|_\ii 
+\|\omega^{(d-1)/2}Q_j\|_\ii 
\|\omega^{(d-2)/2}\|_\ii\rk 
\|G\|.$$
We define 
$$\W:=\mat{W_+}{\ov{W_-}}{W_-}{\ov{W_+}}.$$
\bl{bog} 
Suppose \rIl or \rIIl. Then 
$\W\in\bog.$
Moreover $\W$ is invariant under the scaling $\omega\rightarrow \kt \omega$, $\al\rightarrow \k^l\al$, and 
$\e\rightarrow \k^{2l-2}\e$. 
\el
\proof 
It follows from Lemma \ref{qs} (4) and (5). 
\qed
We define 
$$\v{b(f)}{\bdd(f)}:=
{ }^T\W\v{a(f)}{\add {}(f)}.$$
Let 
$$\vl:=\jjj \mmm q_{j\mu}{L}_j^\mu\in W$$
with 
$${L}_j^\mu :=\frac{\al }{\sqrt{2}} \oplus_{r=1}^{d-1} 
\frac{e_\mu^r Q_j}{\omega^{3/2}}.$$
$L_j^\mu$ is well defined by Lemmas \ref{lambda} and \ref{lambda2}. 
We define 
$$\v{B_q(f)}{\Bdd_q(f)}:=
\v{b(f)}{\bdd {}(f)}+
\v{(\vl,f)_W}{(\ov{\vl},f)_W}. $$
By \kak{aa}  we have 
$$\v{a(f)}{\add{}(f)}={ }^T(\W\f) 
\v{B_q(f)}{\Bdd_q(f)} 
-\v{ \lk W_+{\vl}-\ov{W_-}\ov{\vl}, 
f\rk_W}{
\lk \ov{W_+{\vl}-\ov{W_-}\ov{\vl}}, f\rk_W}. $$
By Proposition \ref{sym} 
there  exists a unitary operator $U(\W)$ such that 
$$U(\W)\f \bss(f)U(\W)=\ass{}(f).$$
Recall 
$$\theta(q):=\exp\lk \jjj\rrr\mmm \frac{\al q_{j\mu}}{\sqrt{2}}
\lkk a^r\lk\ov{
\frac{\emr R_j}{\omega}}\rk
-\add{r}
\lk \frac{\emr R_j}{\omega}\rk\rkk\rk$$
and 
$$\cu(q):=\theta(q)U(\W).$$  


{\it A proof of Proposition \ref{dia} (1)}


By Lemma \ref{qs} (1), 
we have 
$$W_-\vl-\ov{W_+\vl}=-\jjj \mmm q_{j\mu}\frac{\al}{\sqrt{2}}\oplus_{r=1}^{d-1}
\frac{\emr R_j}{\omega}.$$
Hence 
$\theta(q,\W,L)=\theta(q)$ follows. 
\qed


{\it A proof of Proposition  \ref{dia} (2)}\\
It is seen that by Lemma \ref{qs} (2) and (3), for $f\in W$, 
$$[\heq, B_q(f)]=-B_q((\oplus^{d-1}\omega) f),$$
$$[\heq, \Bdd_q(f)]=\Bdd_q ((\oplus^{d-1}\omega) f)$$ 
on some dense domain. 
Using these commutation relations we can prove that 
\eq{h11}
e^{it\heq}B^\dagger (f)e^{-it\heq}=B^\dagger (e^{it\oplus^{d-1}\omega} f),
\en
\eq{h2}
e^{it\heq}B (f)e^{-it\heq}=-B (e^{it\oplus^{d-1}\omega} f).
\en
Then we have 
$$B(f)e^{it\heq}\cu(q)\Omega=0,$$
for all $f\in W$. 
It implies that 
$e^{it\heq}\cu(q)\Omega=\mbox{constant}\times \cu(q)\Omega$. 
Since $e^{it\heq}$ is one parameter unitary group, 
\eq{h3}
e^{it\heq}\cu(q)\Omega=e^{itE(q)}\cu(q)\Omega
\en
with some constant $E(q)$. 
Hence by \kak{h11},\kak{h2} and \kak{h3} we obtain that  
$$\cu(q)\f e^{it \heq} \cu(q)\add {}(f_1)\cdots \add{}(f_n)\Omega=
e^{it(\hf-E(q))}\add {}(f_1)\cdots \add{}(f_n)\Omega.$$
Then 
$$\cu(q)\f e^{it \heq}\cu(q)=e^{it (\hf-E(q))}.$$
Since $\heq$ is self-adjoint on $D(\hf)$, we see that $\cu(q)$ maps $D(\hf)$ onto itself and $\cu(q)\f \heq \cu(q)=\hf-E(q)$ holds by Stone's theorem.  
\qed
\subsubsection{Analytic continuation}
\bl{ana1}
Suppose \rIl. Let $f\in\LL$. 
Then $Q_jf$ can be strongly analytically continued to 
a neighborhood ${\cal O}$ of $\RR$ in each $\alpha_i$'s. 
\el
\proof
We fix $j$. 
We can find  a neighborhood ${\cal O}$ of $\RR$ such that, 
for $z_j\in{\cal O}$, 
$$D(z_j)=D(z_j,k):=M-\e \dd g(k) A(\alpha_1,\cdots,z_j,\cdots,\alpha_N)$$
is a strongly analytic bounded operator on $\oplus^N\LL$, 
and
$$\inf_{k\in\BR}|(w, D(z_j,k)w)|>\delta|w|^2,\ \ \ \mbox{for}\  w\in\BC^N,$$
with some positive $\delta$. 
Hence $D(z_j)\f$ exists as a bounded operator for $z_j\in{\cal O}$ and 
$$\|D(z_j)\f\|\leq\frac{1}{\delta}.$$
We can take a sufficiently small neighborhood ${\cal O}(z_j)\subset{\cal O}$ 
of $z_j$ such that 
\eq{iff}
\inf_{z_j\in{\cal O}(z_j)}\|D(z_j)\f\|<\infty.
\en
Since 
$$\frac{D(z_j)\f-D(z_j')\f}{z_j-z_j'}=D(z_j)\f\lk \frac{D(z_j')-D(z_j)}{z_j-z_j'}\rk D(z_j')\f,$$
by \kak{iff} we conclude that $D(z_j)\f$ is strongly differentiable in ${\cal O}$.
Thus $D(z_j)\f$ is a strongly analytic bounded operator. 
Hence the lemma follows.
\qed
\bl{ana2}
Suppose \rIIl. 
Let $f\in\LL$. Then $Q_jf$ can be strongly analytically continued to 
a neighborhood ${\cal O}$ of $\RR$ in each $\alpha_i$'s. 
\el
\proof 
Note 
$$Q_j(k)=\frac{1}{\e\dd\alpha_j^2}\frac{1}{\frac{m_j}{\e \dd \alpha_j^2}-g_j(k)}.$$
Here 
$$g_j(k):=
\lim_{\eta\downarrow 0}\int_\BR
\frac{\omega(k')|\laj(k')|^2}{\omega(k)^2-\omega(k')^2+i\eta}dk'.$$
It is seen that 
$$(1)\  g_j(0)<0,\ \ \ (2)\  \Im g_j(k)<0\ {\rm for}\ k\in {\rm supp}\laj,$$
$$
(3)\  g(k)=0\ {\rm for}\  k\not \in {\rm supp}\laj, \ \ \ 
(4)\  g_j\  \mbox{is continuous}.$$
Hence, for arbitrary  $\delta>0$,  
$${\rm dist}({\rm Ran}g_j,[\delta,\infty))>M$$
with some positive $M$. 
Thus $Q_j f$ can be analytically continued to 
$${\cal O}':=\{a+ib,a\in\RR,b\in\RR||a|>\delta,|b|<M\}$$
in $z_j:=m_j/(\e \dd \al^2)$. 
Thus $Q_j f$ can be  strongly analytically continued 
in a neighborhood ${\cal O}$ of 
$\RR$ 
in each 
$\al$. 
\qed
{\it A proof of Lemma \ref{dia} (3)}

Note that 
\eq{eq}
E(q)=\frac{(\Omega, H_\epsilon(q)\cu (q)\Omega)}{(\Omega, \cu(q)\Omega)}.
\en
It is proven in \cite{ar2,hi2} 
that
by Lemma \ref{pv} 
$E(q)$ can be expressed as 
$$E(q)=\jjj \frac{1}{2m_j}(q_j+\tilde{q}_j)^2+\Delta,$$
where 
$$\tilde{q}_{j\nu}:=\sum_{i=1}^N\rrr\sum_{s=1}^{d-1}\sum_{\mu=1}^d 
\half q_{\mu i}\aij \lk e_\mu^r R_j/\omega,(\I+W_-W_+\f)_{rs}e_\nu^s\lai\rk,$$
$$\Delta=\jjj\rrr\sum_{\mu=1}^d 
\frac{\al^2}{4m_j}\lk e_\mu^r \laj,(\I-W_-W_+\f)_{rs} e_\mu^s\laj\rk.$$
By Lemmas \ref{ana1} and \ref{ana2} we see that $\tmn$ can be strongly analytically 
continued to some ${\cal O}\subset \RR$ in each $\al$'s. Moreover 
$W_\pm$  also can be done. 
By the identities derived from the symplectic structure, 
$$W_+^\ast W_+ -\ov{W_-^\ast W_-}=I,$$
$$W_+ W_+^\ast  -\ov{W_- W_-^\ast}=I,$$
taking a sufficiently small neighborhood ${\cal O}'\subset {\cal O}$, 
we see that $W_+\f$ can be extended to $\al\in{\cal O}'$ as a bounded operator $W_+\f(\al)$.
Moreover it is seen that 
${\rm sup}_{\al\in{\cal O}(\al)}\|W_+\f(\al)\|<\infty$ 
for sufficiently small neighborhood ${\cal O}(\al)$ of $\al$. 
Thus $W_+\f(\al)$ is strongly differentiable in $\al$, 
and hence strongly analytic on ${\cal O}'$.
\qed



\subsection{A proof of Lemma \ref{form}} 
{\it A proof of Lemma \ref{form} (1)} \\
This proof is a slight extension of \cite{hisp} to the $N$ particle case 
as we have been suggested. 
We take 
a momentum lattice approximation. 
The momentum lattice approximated Hamiltonian \kak{ap} below  
can be identified with 
 harmonic oscillators. Then its ground state energy can be explicitly obtained. 
By a limiting argument we get $E(q)$. 
First we assume that $|\alpha_j|$'s are sufficiently small and 
$$\od(k):=\omega(k)+\delta,\ \ \ \delta>0.$$
Let $1/a>0$ be the width of the momentum lattice, and some $L>0$ be fixed. 
For 
$$l=(l^1,\cdots,l^d)
\in (2\pi {\Bbb Z}/a)^d$$ 
 we define 
$$\Gamma(l):= \left[l^1,l^1+\frac{2\pi}{a}\rk
\times\cdots\times
\left[l^d,l^d+\frac{2\pi}{a}\rk.$$
Let 
$|l|:=\max_{r}|l^r|$. 
Suppose 
$$|l|\leq 2\pi L.$$
Set $$D:=(2[aL]+1)^d,$$
where $[z]$ denotes the integer part of $z$. 
We named the lattice points of the rectangle with width 
$2[aL]$ centered at the origin 
as 
$$l_1,l_2,\cdots, l_{D}.$$
We set 
$$\qq:=\sh\frac{1}{\sqrt{\od (l)}}
\lkk \add{r}\lk{\chi_{\Gamma(l)}}\rk+
a^r\lk {\chi_{\Gamma(l)}}\rk
\rkk_{1\leq r\leq d-1,|l|\leq 2\pi L},$$
$$\pp:=\frac{i}{\sqrt{2}}
\sqrt{\od (l)}
\lkk \add{r}\lk 
{\chi_{\Gamma(l)}}\rk -
a^r\lk  {\chi_{\Gamma(l)}}\rk 
\rkk_{1\leq r\leq d-1,|l|\leq 2\pi L},$$
where $\chi_I$ denotes the characteristic function of $I\subset \BR$. 
Define 
$$
\heqq:=\jjj \lkk 
\frac{1}{2\mj}q_j^2-\frac{\al}{\mj}\mmm (q_{j\mu}\vv^j,\qq)+\epsilon 
\mmm \frac{\al^2}{2\mj}(\qq,\vv^j)(\vv^j,\qq)\rkk $$
\eq{ap}
+\half\lkk 
(\pp, \pp)+(\qq, A_0\qq)\rkk-\half \tr\sqrt{A_0} .
\en
Here $A_0$ is the $(d-1)D\times (d-1)D$ symmetric matrix defined by 
$$A_0:=\lk 
\I_{(d-1)}
\delta_{ll'}\od (l)^2\rk_{|l|,|l'|\leq 2\pi L}$$
\eq{az}
=
\lk \begin{array}{cccc}
\od (l_1)\I_{(d-1)}& \ &\ &\ \\
\ &\od (l_2)\I_{(d-1)}& \ &\ \\
\vdots &\ldots  &\ddots  &\vdots \\
\ & \ & \ & \od (l_{D})\I_{(d-1)}
\end{array}
\rk, 
\en
and 
$$\vv^j:=\vv:=\lk
\begin{array}{c}
\sqrt{\od (l_1)}\la(l_1)e_\mu^1(l_1)\\
\vdots \\
\sqrt{\od (l_1)}\la(l_1)e_\mu^{(d-1)}(l_1)\\
\vdots\\
\vdots\\
\sqrt{\od (l_D)}\la(l_D)e_\mu^1(l_D)\\
\vdots \\
\sqrt{\od (l_D)}\la(l_D)e_\mu^{(d-1)}(l_D)
\end{array}
\rk\in 
\RR^{(d-1)D}.$$
Here $\I_{(d-1)}$ denotes the $(d-1)\times (d-1)$ identity matrix. 
It is seen that 
\eq{uni}
(\heqq-z)\f 
\rightarrow (\he(q,\delta)-z)\f
\en
uniformly as 
$a\rightarrow \infty$ and then $L\rightarrow \infty$. 
Here $\he(q,\delta)$ denotes $\heq+\delta N$. 
We define the projection $P$ by 
$$P:=\epsilon \jjj \mmm \atmm j|\vv\rangle\langle \vv|.$$
Then 
$$\heqq=
\jjj \lkk \mhalf q_j^2-\frac{\al}{\mj}\mmm (q_{j\mu}\vv,\qq)\rkk 
+\half (\qq,  P\qq)$$
$$+
\half(\pp,\pp)+\half(\qq, A_0\qq)-\half\tr\sqrt{A_0}.$$
Set 
$$A:=A_0+P.$$
Since $\od (k)>\delta>0$, the inverse of $A_0$ exists, and hence 
$A$ also has  the inverse, and it is symmetric. 
Then we have 
$$=
\jjj \mhalf q_j^2-\jjj 
\mmm \frac{\al}{\mj}(AA\f q_{j\mu}\vv,\qq)+\half (\qq, A\qq)+
\half(\pp,\pp)-\half\tr\sqrt{A_0}$$
$$=
\jjj \mhalf q_j^2+\half((\qq-\ff), A(\qq-\ff))-\half(\ff,A\ff)+
\half(\pp,\pp)-\half\tr\sqrt{A_0}.$$
Here we put 
$$f:=\jjj \mmm \frac{\al}{\mj}A\f q_{j\mu}\vv.$$
By the commutation relation on $e^{is\pp}$ and $e^{it\qq}$:
$$e^{is\pp_{rl_n}}e^{it\qq_{r'l_n'}}=
e^{its \delta_{rr'}\delta_{l_nl_n'}}e^{it\qq_{r'l_n'}}e^{is\pp_{rl_n}},$$ 
and the 
von Neumann uniqueness theorem, we can identify 
$\pp$ and $\qq$ with the momentum and position operators, respectively, 
$$\pp_{t,l_n}\cong-i\frac{\partial}{\partial x_{r,n}},\ \ \ 
\qq_{r,l_n}\cong x_{r,n},$$
for $r=1,...,d-1$ and $n=1,...,D.$
Hence we identify $\heqq$ as a harmonic oscillator in $L^2(\RR^{(d-1)D})$. 
Generally for the harmonic oscillator 
$$H_T:=\half (\pp,\pp) +\half (\qq, T\qq)$$
with nonnegative symmetric matrix $T$, we have 
$$\inf \sigma(H_T)=\tr \sqrt{T}.$$ 
Then it follows that 
$$E(q)=\inf\sigma(\heqq)$$
$$=
\inf\sigma\lk \half(\pp,\pp)
+\half((\qq-\ff), A(\qq-\ff))\rk
+\jjj \mhalf q_j^2
-\half(\ff,A\ff)
-\half\tr\sqrt{A_0}$$
$$=
\jjj \mhalf q_j^2-\half (f,Af)+\half\tr(\sqrt{A}-\sqrt{A_0}).$$
Separately we calculate 
\bi
\item[(1)] $\jjj \mhalf q_j^2-\half (f,Af)$, 
\item[(2)] $\half\tr(\sqrt{A}-\sqrt{A_0})$. 
\ei 
First we calculate (1). 
We give a remark. 
Let 
$$\dmn(k):=\rrr e_\mu^r(k) e_\nu^r(k)=\lk \I-\frac{k}{|k|}
\otimes \frac{k}{|k|}\rk_{\mu\nu}=
\delta_{\mu\nu}-\frac{k_\mu k_\nu}{|k|^2}.$$
For rotation invariant functions $f$ and $g$ it follows that 
$$(\dmn f, g)=\delta_{\mu\nu}\dd(f,g).$$
We have 
$$(f,Af)=\sum_{i,j=1}^N\am i \am j q_{i\mu} q_{j\nu} (A\f v_\mu,v_\nu).$$
Note that we assume that $|\alpha_j|^2$ are suffieictly small. 
Then 
$$A\f=\sumn \ww A_0\f (PA_0\f)^{n-1}.$$
Hence 
$$(A\f v_\mu,v_\nu)= \sumn \ww (A_0\f 
(PA_0\f)^{n-1}v_\mu,v_\nu)$$
$$=\sumn\ww \sum_{j_1,...,j_{n-1}=1}^N\sum_{\mu_1,...,\mu_{n-1}=1}^{d} 
\epsilon^{n-1}\atm 1\cdots \atm {n-1}\times $$
$$
\times (v_\mu A_0\f\vvv 1)(\vvv 1,A_0\f \vvv 2)
(\vvv 2,A_0\f \vvv 3)\cdots 
(\vvv {n-2},A_0\f \vvv {n-1})
(\vvv {n-1},A_0\f v_\nu)$$
$$=\delta_{\mu\nu} \sumn\ww \sum_{j_1,...,j_{n-1}=1}^N\sum_{\mu_1,...,\mu_{n-1}=1}^{d} 
\epsilon^{n-1} \dd^n  \atm 1\cdots \atm{n-1}
\xi^n, $$
where 
$$\xi:=\sum_{|l|\leq 2\pi L}\frac{|\la(l)|^2}{\od (l)}.$$
Then 
it follows that 
$$= 
\delta_{\mu\nu}
\sumn\ww  \dd \xi\lkk \epsilon \dd \xi \jjj \atmm  j\rkk^{n-1}$$
$$=\delta_{\mu\nu} \frac{\dd\xi}{1+\epsilon \dd \xi \jjj \atmm j}.$$
We have 
$$(f, Af)=
\sum_{i,j=1}^N\delta_{\mu\nu}\am i\am j q_{i\nu} q_{j\mu}
\frac{\dd\xi}{1+\e \dd \xi \jjj \atmm j}$$
$$=
\frac{\dd\xi}{1+\e \dd \xi  \jjj \atmm j}\left| \jjj \am j q_j\right|^2.$$
Hence we have 
$$(1)=\half \jjj \frac{q_j^2}{\mj}-\half 
\frac{\dd\xi}{1+\e \dd \xi  \jjj \atmm j}\left| \jjj \am j q_j\right|^2.$$
Next we calculate (2). 
 Using the formula for nonnegative self-adjoint operator $B$,  
$$\sqrt{B}=\frac{1}{\pi}\int_{-\ii}^\ii B(s^2+B)\f ds,$$ 
we see that 
$$(2)=\frac{1}{2\pi}\int_{-\ii}^\ii\tr \lkk A(s^2+A)\f-A_0(s^2+A_0)\f\rkk ds.$$
Since $$(s^2+A)\f=(s^2+A_0)\f\sumn \ww \lkk P(s^2+A_0)\f\rkk^{n-1},$$
we have 
$$
A(s^2+A)\f-A_0(s^2+A_0)\f = \rI+\rII.
$$
Here we put 
\eq{222}\rI:=A_0(s^2+A_0)\f\sumn\www\lkk P(s^2+A_0)\f\rkk^n,
\en
\eq{12}
\rII:=P(s^2+A)\f.
\en
It follows that 
$$\tr \rI=\sumn\www \sum_{\varphi:\cons}
(\varphi, A_0\sa \lkk P\sa\rkk^n\varphi),$$
where $\sum_{\varphi:\cons}$ means to sum up vectors in a complete orthonormal system. 
Then 
$$=\sumn \www \sum_{\varphi:\cons}
 (\varphi, A_0\sa \underbrace{P\sa P\sa\cdots P
\sa}_{n\  \mbox{{\footnotesize times}}}\varphi)$$
$$=\sumn \www \sum_{\varphi:\cons}
\sum_{j_1,...,j_n=1}^N\sum_{\mu_1,...,\mu_n=1}^d 
\epsilon^n \atm 1\cdots \atm n \times $$
$$\times (\varphi, A_0\sa \vvv 1)(\vvv 1,\sa \vvv 2)\times \cdots$$
$$\cdots \times 
  (\vvv {n-1}, \sa \vvv n)
(\vvv n, \sa \varphi) .$$
We take as a CONS   
$$\lkk \varphi_1:=
\frac{\sa \vvv n}{\|\sa \vvv n\|},\varphi_2,\varphi_3,...,\rkk.
$$
Then 
$$=\sumn \www 
\sum_{j_1,...,j_n=1}^N\sum_{\mu_1,...,\mu_n=1}^d 
\epsilon^n \atm 1\cdots \atm n \times $$
$$\times (\vvv n, A_0(s^2+A_0)^{-2} \vvv 1)(\vvv 1,\sa \vvv 2)\times \cdots$$
$$\cdots   \times (\vvv {n-1}, \sa \vvv n)
\frac{(\vvv n, (s^2+A_0)^{-2} \vvv n)}{
\|\sa \vvv n\|^2}$$
$$=d \sumn \www 
\lk\jjj\atmm j\rk^n \epsilon^n \dd^n  \xi(s)^{n-1}\eta(s) ,$$
where 
$$\xi(s):=\sum_{|l|\leq 2\pi L}\frac{\od (l)|\la(l)|^2}{s^2+\od (l)^2},$$
and 
$$\eta(s):=
\sum_{|l|\leq 2\pi L}\frac{\od (l)^3|\la(l)|^2}{(s^2+\od (l)^2)^2}.$$
Hence 
$$=d \sumn \www \lk \e \dd \xi(s) \jjj\atmm j\rk^n\frac{\eta(s)}{\xi(s)} $$
$$=d\frac{-\e\dd \xi(s) \jjj \atmm j }
{1+\e \dd\xi(s) \jjj \atmm j }
\frac{\eta(s)}{\xi(s)} 
=d \frac{-\epsilon\rho\eta(s)}{1+\epsilon \rho \xi(s) }.$$
Next we see that 
$$\tr \rII=\sum_{\varphi:\cons}(\varphi, P(s^2+A)\f \varphi)$$
$$=\sumn \ww \sum_{\varphi:\cons}(\varphi, P\sa \lk P\sa\rk^{n-1} \varphi)$$
$$=\sumn \ww \sum_{\varphi:\cons}
\sum_{j_1,...,j_n=1}^N\sum_{\mu_1,...,\mu_n=1}^d 
\atm 1\cdots \atm n \epsilon^n\times $$
$$\times (\varphi, \vvv 1)(\vvv 1, \sa \vvv 2)\cdots (\vvv {n-1},\sa\vvv n)
(\vvv n,\sa \varphi)
.$$
Take as a CONS 
$$\lkk \varphi_1:=\frac{\sa \vvv n}{\|\sa\vvv n\|},\varphi_2,
\varphi_3,...,\rkk.$$
Then we have 
$$\tr \rII
=\sumn \ww 
\sum_{j_1,...,j_n=1}^N\sum_{\mu_1,...,\mu_n=1}^d 
\atm 1\cdots \atm n \epsilon^n\times $$
$$\times (\vvv n, \sa \vvv 1)(\vvv 1, \sa \vvv 2)\times \cdots $$
$$\cdots \times (\vvv {n-1},\sa\vvv n)
\frac{(\vvv n,(s^2+A_0)^{-2}\vvv n)}{\|\sa\vvv n\|^2}$$
$$=\sumn \ww \sum_{j_1,...,j_n=1}^N\sum_{\mu_1,...,\mu_n=1}^d 
\atm 1\cdots \atm n \epsilon^n\dd^n d \xi(s)^n$$
$$=d \sumn \ww  \lk \e \dd \xi(s) \jjj \atmm j \rk^n 
=d\frac{\epsilon \rho \xi(s) }{1+\epsilon \rho \xi(s) }.$$
Hence 
$$\tr (\rI+\rII)=\frac{d\epsilon \rho}{1+\epsilon \rho \xi(s)}(\xi(s)-
\eta(s)).$$
From 
$$\xi(s)-\eta(s)=\sum_{|l|\leq 2\pi L}\lkk \frac{|\la(l)|^2\od (l)}{s^2+\od (l)^2}-\frac{\od (l)^3|\la(l)|^2}{(s^2+\od (l)^2)^2}\rkk 
=\sum_{|l|\leq 2\pi L}\frac{s^2|\la(l)|^2\od (l)}{(s^2+\od (l)^2)^2}
:=
\eta'(s),$$ 
it follows that 
$$(2)=\frac{\epsilon }{2\pi}\int_{-\ii}^\ii\frac{\e \rho \eta'(s)}
{1+\epsilon\rho \xi(s)}ds.$$ 
As was mentioned in \kak{uni}, as  $a\rightarrow \infty$ 
and then $L\rightarrow \ii$, 
$\heqq$ 
uniformly converges to $\heq+\delta N$ in the resolvent sense. 
Thus as $a\rightarrow \ii$ and then $L\rightarrow \ii$, it follows that 
$$E(q, \delta, a, L)\rightarrow \inf\sigma (\he(q)+\delta N):=E(q,\delta).$$
Since $N\geq 0$, we have 
$$\liminf_{\delta\rightarrow 0}E(q,\delta)\geq E(q)$$
and 
the strong resolvent convergence of $\heq+\delta N$ to $H_\epsilon(q)$ 
as $\delta\rightarrow 0$ 
implies that 
$$E(q)\geq \limsup_{\delta\rightarrow 0} E(q,\delta).$$
Thus 
$$E(q,\delta)\rightarrow E(q)$$
as $\delta\rightarrow 0$. Thus the lemma follows for 
sufficiently small $|\alpha_j|$'s. By Proposition \ref{dia} (3), 
$\inf\sigma(\heq)$ can be analytically continued in each $\alpha_j$'s to ${\cal O}$, 
and clearly $E(q)$ also can be done. 
Hence the lemma follows for all $\alpha_j$'s in $\RR$.  



\qed
{\it A proof of Lemma \ref{form} (2)}\\
The proof is similar to Lemma \ref{form}. 
Define the projection operator  
$$P:=\epsilon\jjj\mmm\atmm j |\vv^j\rangle\langle \vv^j|,$$ and 
$$A:=A_0+P,$$
where $A_0$ is defined in \kak{az}. 
Here 
$$\vv^j:=\lk
\begin{array}{c}
\sqrt{\od (l_1)}\laj (l_1)e_\mu^1(l_1)\\
\vdots \\
\sqrt{\od (l_1)}\laj (l_1)e_\mu^{(d-1)}(l_1)\\
\vdots\\
\vdots\\
\sqrt{\od (l_D)}\laj (l_D)e_\mu^1(l_D)\\
\vdots \\
\sqrt{\od (l_D)}\laj (l_D)e_\mu^{(d-1)}(l_D)
\end{array}
\rk\in 
\RR^{(d-1)D}.$$
Let 
$$f:=\jjj\mmm\frac{\al}{\mj}A\f  q_{j\mu}\vv^j.$$
Similarly to Lemma \ref{form}
it is enough to calculate 
\bi
\item[(1)] $\half\jjj \frac{1}{\mj}q_j^2-\half (Af,f)$,
\item[(2)] $\half\tr(\sqrt{A}-\sqrt{A_0})$. 
\ei 
First we calculate  (1). 
We have 
$$(A\f\vv^i,{v}_\nu^j)=\delta_{\mu\nu}\delta_{ij} 
\sumn\ww 
\dd^n\epsilon^{n-1}\lk\atmm j\rk^{n-1}\xi_j^n$$
$$=
\delta_{\mu\nu}\delta_{ij}\frac{\dd\xi_j}{1+ \e \dd \xi_j  \atmm j },$$
where 
$$\xi_j:=\sum_{|l|\leq 2\pi L}\frac{|\laj(l)|^2}{\od (l)}.$$
Note that, for function $g$,   
$$(\laj, g \la_i)=\delta_{ij}(\laj, g\laj).$$
Then 
$$(Af,f)=\sum_{i,j=1}^N\sum_{\mu,\nu=1}^d  (A\f\vv^i, {v}_\nu^j)$$
$$=\sum_{i,j=1}^N\sum_{\mu,\nu=1}^d \delta_{\mu\nu}\delta_{ij}
 \am i\am j q_{i\nu}q_{j\mu}
\frac{\dd\xi_j}{1+\epsilon\dd \xi_j  \atmm j}$$
$$=\jjj \lk\am j\rk^2q_j^2 \frac{\dd\xi_j}{1+
\e \dd \xi_j   \atmm j}.$$
Then 
$$\half \jjj  \frac{q_j^2}{\mj}-\half (Af,f)=
\half \jjj\frac{q_j^2}{\mj}\lk \frac{\mj-(1-\epsilon)\dd \at \xi_j}{\mj+
\epsilon\dd \at \xi_j}\rk.$$
Next we calculate  (2). 
Let (I) and (II) be as in \kak{222} and \kak{12}, respectively. 
We define 
$$\xi_j(s):=\sum_{|l|\leq 2\pi L}\frac{\od (l)|\laj(l)|^2}{s^2+\od (l)^2},$$
$$\eta_j(s):=
\sum_{|l|\leq 2\pi L}\frac{\od (l)^3|\laj(l)|^2}{(s^2+\od (l)^2)^2},$$
and 
$$\eta_j'(s):=\sum_{|l|\leq 2\pi L}\frac{s^2|\laj(l)|^2\od (l)}{(s^2+\od (l)^2)^2}.$$
We have 
$$\tr \rI=d \jjj \sumn \www \lk\atmm j\rk^{n}\epsilon^n\dd^n  \xi_j(s)^{n-1}
\eta_j(s)$$
$$=-d\jjj \eta_j(s)\epsilon \atmm j \dd \frac{1}{1+\epsilon \xi_j(s) \dd \atmm j}
=-d\jjj\frac{\epsilon \rj  \eta_j(s)}{1+\epsilon \rj \xi_j(s)}.$$
Moreover 
$$\tr \rII=\jjj\sumn \ww \atmm j \epsilon^n\dd^n\xi_j(s)^n$$
$$=d \jjj\frac{\epsilon\rj  \xi_j(s)}{1+\epsilon \rj\xi_j(s)}.$$
Hence 
$$\rI+\rII=d \jjj\frac{\epsilon  \rj}{1+\epsilon \rj \xi_j(s)}(\xi_j(s)-\eta_j(s))
=
d \jjj\frac{\epsilon  \rj}{1+\epsilon \rj \xi_j(s)}\eta_j'(s).$$
Then 
$$
\half\tr(\sqrt{A}-\sqrt{A_0})=
\half \tr(\rI+\rII)=\jjj\frac{d}{2\pi} \int_{-\ii}^\ii\frac{\epsilon \rj 
\eta'_j(s)}{1+\epsilon \rj\xi_j(s) }ds.$$
The same limiting argument on $a,L,\delta$ 
as in Lemma \ref{form} 
leads to the desired results. 
\qed


{\bf Acknowledgements}\\
This work is supported by Grant-in-Aid  13740106 for Encouragement of Young Scientists from the Ministry of 
Education, Science, Sports and Culture. 








\begin{thebibliography}{99}



\bibitem{bet}
H. A. Bethe, 
The electromagnetic shift of energy levels, {\it Phys. Rev.} {\bf 72}, 
 (1947) 339-342.

\bibitem{we}
T. A.  Welton, Some observable effects of the quantum-mechanical 
fluctuations of the electromagnetic field, {\it Phys. Rev.}  
{\bf  74} (1948), 1157-1167.

\bibitem{pafi}
W.  Pauli and M.   Fierz,   Zur Theorie der Emission langwelliger Lichtquanten,   
{\it Nuovo Cimento} {\bf 15}  (1938),   167--188.   




\bibitem{da1}
\jmp{E. B. Davies}{particle-boson interactions and the weak coupling limit}{20}{1979}{345--351}.
\bibitem{da2}
\jfa{E. B. Davies}{Asymptotic analysis of some abstract evolution equations}{25}{1977}{81--101}.


\bibitem{du}
R.  D\"umcke, Convergence of multitime correlation functions in the weak and singular coupling limits, 
{\it J. Math. Phys.} {\bf 24}  (1983), 311-315. 


\bibitem{pa}
P. F. Palmer, 
The singular coupling and weak coupling limits, 
{\it J. Math. Phys.} {\bf 18}   (1977), 527-529. 



\bibitem{ar1}
\jmp{A. Arai}{An asymptotic analysis and its applications to the nonrelativistic limit of the Pauli-Fierz and a spin-boson model}  {31} {1990}{2653-2663}.



\bibitem{hi3} 
F. Hiroshima,
Essential self-adjointness of translation-invariant quantum field models for 
arbitrary coupling constants, 
{\it Commun.  Math. Phys.} {\bf 211}  (2000), 585-613. 

\bibitem{hi4} 
{F. Hiroshima} Self-adjointness of the Pauli-Fierz Hamiltonian for arbitrary coupling constants, preprints. mp-arc 01-97.


\bibitem{ar2}
A. Arai, 
Note on scattering theory in non-relativistic quantum electrodynamics, 
{\it J. Phys. A: Math. Gen.} {\bf 16} (1983), 49-70.


\bibitem{hi1}
\jmp{F. Hiroshima}{ Scaling limit of a model of quantum electrodynamics}
{ 34} {1993}{4478-4518}.


\bibitem{hi2}
\rmp {F. Hiroshima}{Scaling limit of a model of quantum electrodynamics with many nonrelativistic particles}{   9 } {1997}{ 201--225}. 


\bibitem{rs2}
M.  Reed and B.  Simon,   {\it Methods of Modern Mathematical Physics II},   
Academic Press,  1975.  



\bibitem{be}
E. A. Berezin, {\it The Method of second quantization},  Academic press 1966. 


\bibitem{kmp} 
P. Kristensen, L. Mejlbo and E. T.  Poulsen, 
Tempered distributions in infinitely many dimensions I. 
Canonical field operator, {\it Commun. Math. Phys.} {\bf 1}(1965), 175-214, 
II. 
Displacement operator, 
{\it Math. Scand.} {\bf 14} (1964), 129-150,  
III. 
Linear transformations of field operators, 
{\it Commun. Math. Phys.} {\bf 6}(1967), 29-48. 




\bibitem{ru1}
S.  N.   M.  Ruijsenaars, On Bogoliubov transforms. II. The general case. 
{\it Ann. Phys.}{\bf 116} (1978), 105-134. 


\bibitem{ar9}
A.  Arai,   
Noninvertible Bogoliubov transformations and instability of embedded eigenvalues,   
{\it J.  Math.  Phys.  }{\bf 32}  (1991),   1838--1846.   


\bibitem{hisp}
F.  Hiroshima and H.  Spohn,   
Enhanced binding through  coupling to a quantum field,   
to be published in {\it Anal. Henri Poincar\'e}. 












\bibitem{ti}
E. C. Titchmarsh, 
{\it Theory of Fourier Integrals}. 2nd ed., Oxford  Univ. Press, 1948. 



\end{thebibliography}


\end{document}
---------------0111050943986--
