\section{Preliminaries}
\subsection{Introduction}
Non-perturbative analysis of perturbation of eigenvalues embedded in the continuous spectrum has
been developed in the last decade and has been applied to the mathematical rigorous analysis of Hamiltonians  in quantum field theory.  Among other things, stability and instability of quantum mechanical  particle coupled to quantum fields
have  been  investigated from mathematical point of view.
In the previous paper \cite{hs08} enhanced binding is  shown for
the system of  $N$-nonrelativistic particles
  linearly  coupled to a massless scalar bose field.
This paper is then a  continuation of \cite{hs08} and we consider a relativistic version.
A quantum system of  Schr\"odinger particles linearly coupled to a scalar bose field
is the so-called   Nelson model \cite{nel64}.
In this paper replacing  the Schr\"odinger  particles with relativistic ones, we shall show
the existence of a ground state without assuming the existence of ground state of
 the zero-coupling Hamiltonian.
From purely mathematical point of view,  this problem  can be regarded as
the spectral analysis of a nonlocal operator coupled to a Gaussian random variables, or
non-perturbative analysis of  eigenvalues of nonlocal operator, which is  embedded in the continuous spectrum.

The existence of the ground state of the standard Nelson Hamiltonian was proven in e.g.,
\cite{bfs98,spo98,ger00,sas05} under some conditions.
The most basic assumptions for proving
the existence of a ground state are both
\begin{description}
\item[ (1)]
 infrared regular condition,
 \item[ (2)]
 the existence of ground state of
 the zero-coupling Hamiltonian.
\end{description} 
On the contrary the enhanced binding
is the phenomena such that  $N$-particles are bound
whenever a sufficiently strong coupling to a scalar bose field is turned on,
and  a ground state appears 
whether  the zero-coupling  Hamiltonian has  ground state or not.
In the intuitive level we can expect  that  the strong coupling derives  attractive effective pair-potentials among  particles, under which $N$-particles are bound and the total system has a ground state.


In this paper we consider
the Hamiltonian   of the form
\begin{align}
\label{r1}
H^V &=H_0+\kappa H_{\rm I}.
\end{align}
The operator $H_0=H_0(\kappa)$ describes the zero coupling Hamiltonian and is given by
\begin{align*}
H_0&=   \sum_{j=1}^N
\left(
\sqrt{-\Delta_j+m_j^2}-m_j + V(x_j)
\right)+ \kappa^2  H_{\rm f},
\end{align*}
where $m_j>0$ is the mass of the $j$-th particle, $V(x)$  an external potential,
$H_{\rm f}$  the free field  Hamiltonian,
and $\kappa>0$ denotes  a scaling parameter.
The operator  $H_I$
describes  a  particle-boson linear interaction.
 We notice that there are no pair potentials in $H^V$ and $V$ is assumed to be independent of $j$ for simplicity.
Introducing  a dressing transformation $e^{iT}$ to derive an effective potential $V_{{\rm eff}}$,
we transform $H^V$ as
\eq {t1}
 e^{-iT} H^V e^{iT}=
 h_{\rm eff}^V+ \kappa^2  H_{\rm f}+H_{\rm R}(\kappa),
\end{equation}
where
 $h_{\rm eff}^V $ is the effective particle Hamiltonian given by
 \begin{equation}{\label{yu1}}
h_{\rm eff}^V= \sum_{j=1}^N \left(
\sqrt{-\Delta_j+m_j^2}-m_j+V(x_j)
\right) +  V_{{\rm eff}}(x_1,...,x_N)
\end{equation}
and
$H_{\rm R}(\kappa)$  a remainder term to be regarded as a perturbation of  $h_{\rm eff}^V+\kappa^2H_{\rm f}$.
Compensating for deriving $V_{{\rm eff}}$ through the dressing transformation, we have the remainder term $H_{\rm R}(\kappa)$ which
is unfortunately no longer  linear
and
is the complicated form:
$$H_{\rm R}(\kappa)=\sum_{j=1}^N \left(
\sqrt{\left(
-i\nabla_j-\frac{1}{\kappa}
A_j(x_j)\right)
^2+m_j^2}-\sqrt{-\Delta_j+m_j^2}\right),$$
where
$A_j$ denotes some quantum vector field.
Nevertheless
 it  turns to be
a {\it small} perturbation for sufficiently large $\kappa$ in some sense.



We are interested in the existence of a ground state of $H^V$, equivalently that of $e^{-iT} H^V e^{iT}$.
We do not however assume the existence of ground states of
$H_0$.
As will be shown below,
the enhanced binding is well grasped
by  the transformed Hamiltonian \eqref{t1} rather than  $H^V$ itself.
Since we consider a  massless boson,
the bottom of the spectrum of $H^V$
is the edge of the continuous spectrum and the  regular perturbation
is not applied.
Then it is not clear whether $e^{-iT} H^V e^{iT} $ has a ground state even when $h_{\rm eff}^V$  has a ground state.


The conventional approach is to assume an infrared cutoff in the form factor $\hat \lambda$ in $H_{\rm I}$
by setting $\hat \lambda(k)\lceil_{|k|>\sigma}$,   $H^V$ with cutoff  $\hat \lambda\lceil_{|k|>\sigma}$ is denoted by $H_\sigma^V$,
and to show the existence of a ground state $\Phi_\sigma}%{\varphi_{\rm g}$ of $H_\sigma^V$.
The vector $\Phi_\sigma}%{\varphi_{\rm g}$ is called
an infrared-regularized ground state.
Then one is left to show that the sequence of ground states $\Phi_\sigma}%{\varphi_{\rm g}$ has a non-zero weak limit $\Phi$ as $\sigma\to 0$, which is the desired ground state  of
$ H^V $.
We  show in this paper:
\begin{description}
\item[(A)]
 the stability condition for $H^V$ is satisfied
 (Lemma  \ref{P4.2}),
\item[(B)]
   infrared-regularized ground states $\Phi_\sigma}%{\varphi_{\rm g}$ has exponential decay uniformly with respect to the infrared cutoff parameter $\sigma$
   (Lemma \ref{expdecay}),
\item[(C)]
we  prove that (1) stability condition and (2) exponential decay   imply  the existence of a ground state of $H^V$ (Appendix \ref{appa}).
\item[(D)]
 we show that
for an arbitrary fixed $\kappa$ large enough, there exists $\alpha_c(\kappa)>0$ such that
for $\alpha$ with $\alpha_c<|\alpha|<\alpha_c(\kappa)$, $H^V$ has a ground state
(Theorem \ref{main}).
Here $\alpha_c$ is
given by $V$ and cutoff functions, and
note that
when $H_{\rm p}$ has no ground state, $H_0(\kappa)=H_{\rm p}+\kappaH_{\rm f}$ has also no ground state for arbitrary $\kappa>0$.
\end{description} 

We explain more technical improvement of this paper.

{\bf (Reduction to the stability condition of $h_{\rm eff}^V$)}
 The stability condition is introduced in \cite{gll01}  to show the existence of ground state of the non-relativistic quantum electrodynamics.
 The key ingredient is that we reduce the stability condition of $H^V$ to that of $h_{\rm eff}^V $ in Lemma \ref{P4.2}.
Namely we show that
the stability condition of $h_{\rm eff}^V $ implies that of $H^V$.
These are proven by energy comparison inequality derived by functional integration of the heat semigroup
generated by \kak{t1}
(Lemma \ref{enebound1}) and a simple variation principle (Lemma \ref{enebound2}),
hence  we focus on showing the stability condition on $h_{\rm eff}^V $.

{\bf (Uniform exponential localization by functional integrations)}
Our method is a minor but nontrivial modification of \cite{hs08} and a mixture of
\cite{ger00,gll01}.
We  do not assume the compactness condition on particle Hamiltonian, which is  entered in \cite{ger00}.
Instead of this  we show  an  exponential localization of infrared-regularized ground states, $\|\Phi_\sigma}%{\varphi_{\rm g}(x)\|_{\mathscr F}\leq C_\delta e^{-\delta|x|}$, which  is
derived through functional integrations
in
Lemma \ref{expdecay}.   The crucial  point is to show  that this localization  is uniform on $\sigma>0$,i.e.,
$C_\delta$ and $\delta$ are independent of $\sigma>0$.

{\bf (Scaling parameter)}
The scaling parameter introduced in this paper can be obtained  by replacing  the annihilation operator $a$
and the creation operator $a^{\ast}$ with $\kappa a$ and $\kappa a^{\ast}$, respectively.
This scaling is the so-called weak coupling limit \cite{dav77,dav79}.
Roughly speaking at least in the nonrelativistic domain $H_{\rm p}\cong -\frac{1}{2m}\Delta+V$, and
then
$$H^V=\kappa^2(\kappa^{-2}H_{\rm p}+H_{\rm f}+\kappa^{-1} H_{\rm I})$$ with
$$\kappa^{-2}H_{\rm p}\cong -\frac{1}{2m\kappa^2}\Delta+\frac{1}{\kappa^2}V.$$
Thus our result may be interpreted that enhanced binding of $H^V$ occurs
 when  sufficiently heavy particle mass
and shallow external potential are assumed.
Alternate explanation of the scaling parameter is the tool to derive a Markov process
 from $e^{-tH^V}$.
Although the scalar product $(f\otimes \Omega, e^{-tH^V}g\otimes\Omega)$ does not define a Markov process,  
$(f,e^{-t(h_{\rm eff}-E_{\rm diag})}g)$ does   with generator $h_{\rm eff}-E_{\rm diag}$.
This can be obtained by
the scaling limit:  
$$(f\otimes \Omega, e^{-tH^V}g\otimes\Omega)\to(f,e^{-t(h_{\rm eff}-E_{\rm diag})}g)$$ as $\kappa\to\infty$.





Finally we give some comments on related works on enhanced binding.
The enhanced binding is studied so far for the various kind of models in quantum field theory.
In \cite{hs01} the enhanced binding of
the Pauli-Fierz model with the dipole approximation is studied.
 In \cite{hss11} a  complement result of \cite{hs01} is established, i.e.,
the absence of ground state for sufficiently small coupling constant is shown.
See also \cite{ak03,blv05,bv04,ceh03,cvv03,hvv03} for the related works.
Furthermore since our Hamiltonian \kak{t1} has a similar form of the semi-relativistic Pauli-Fierz
model \cite{hir11}, it can be expected that our method can be also applied to show the existence
of ground state of the semi-relativistic Pauli-Fierz model.


The organization of this paper is as follows.

In the remainder of Section 1 we define the Nelson model with a relativistic kinetic term, and introduce a
 scaling parameter $\kappa>0$.

In Section 2 we introduce a dressing transformation, and mention the stability condition and uniform
exponential decay of $\Phi_\sigma}%{\varphi_{\rm g}(x)$.

In Section 3 we prove
   the stability condition in Section 3.1
   and uniform exponential decay in Section 3.2,
   and
   in Section 3.3 we show the enhanced binding.

In Appendix \ref{appa} we show that
the relativistic version of the stability condition also implies  the existence of the ground state.


In Appendix \ref{ess} we review fundamental properties of the bottom  of the essential 
spectrum of relativistic Schr\"odinger operator.

In Appendix \ref{appb} we give the functional integral representation of $e^{-tH^V}$
and show some inequality used in the proof of exponential decay of infrared regularized ground states.


 In Appendix \ref{appc}  we derive some energy comparison inequality of
 the translation invariant
 Hamiltonian
 $\sum_{j=1}^N (\sqrt{-\Delta_j+m_j}-m_j+V(x_j))$.
 


\subsection{Definition}
 We begin with giving the definition of the Nelson model with $N$-relativistic particles.
Throughout we assume $N\geq 2$ and the dimension of state space is $d\geq 3$.
The Nelson model can be realized as a self-adjoint operator on the tensor product of ${L^2(\RR^{dN})}$
and the boson Fock space ${\mathscr F}$ over ${L^2(\BR)}$,
 \eq{r2}
 {\ms H}={L^2(\RR^{dN})}\otimes{\mathscr F}.
  \end{equation}
 Here ${\mathscr F}$  is defined by
$
{\mathscr F}=\oplus_{n=0}^\infty L_{\rm sym}^2(\mathbb{R}^{dn})
$,
where $L_{\rm sym}^2(\mathbb{R}^{dn})$ is the  set of square integrable functions such that
$\Psi(x_1,\cdots,x_n)=\Psi(x_{\sigma(1)},\cdots,x_{\sigma(n)})$ for any $n$-degree permutation $\sigma$.
Vector $\Psi\in {\mathscr F}$ is written as
$\Psi=\{\Psi^{(n)}\}_{n=0}^\infty$
with $\Psi^{(n)}\in L_{\rm sym}^2(\BR)$, and the  Fock vacuum $\Omega\in{\mathscr F}$ is defined by
$\Omega=\{1,0,0,\ldots\}$.
$a(f)$ and $a^{\ast}(f)$, $f\in{L^2(\BR)}$,  denote the annihilation operator
and the creation operator in ${\mathscr F}$, respectively.
 They satisfy canonical commutation relations:
\begin{align}
 [a(f), a^{\ast}(g)]=(\bar f, g)\bbone ,\ \ \ [a(f), a(g)]=0=[a^{\ast}(f),a^{\ast}(g)]
\end{align}
and the adjoint relation
$a^{\ast}(f)=(a(\bar f))^\ast$ holds.
Throughout this paper $(F,G)_{\mathcal{K}}$ denotes the scalar product on Hilbert space $\mathcal{K}$, which is linear in $G$ and antilinear  in $F$.
We omit $\mathcal{K}$ until confusions arises. We informally write  as
$
 a^\#(f)=\int a^\#(k) f(k) dk$,
$a^\#=a, a^{\ast}$.
The second quantization
of the closed operator $A$ on ${L^2(\BR)}$ is  denoted by
${\rm d}\Gamma (A)$.
The free field Hamiltonian $H_{\rm f}$ is
the self-adjoint operator on ${\mathscr F}$,
which is given by the second quantization of
the multiplication operator
 $\omega(k)=|k|$ on ${L^2(\BR)}$:
\begin{equation}
H_{\rm f} = {\rm d}\Gamma (\omega).
\label{3}
\end{equation}


Next we introduce $N$-relativistic particles.
We suppose that the $N$-relativistic particles are governed by
the relativistic Schr\"odinger operator
 $H_{\rm p}$  of the form:
\eq{1}
H_{\rm p}=\sum_{j=1}^N  (\Omega_j+V_j)
\end{equation}
on ${L^2(\RR^{dN})}$,
where
\eq{2}
\Omega_j=\Omega_j({p_j} )=\sqrt{{p_j} ^2+m_j^2}-m_j,
\end{equation}
is the $j$-th particle Hamiltonian with momentum ${p_j} =-i\nabla_{x_j}$,
$m_j>0$, the mass of the $j$-th particle, and $V_j=V(x_j)$ denotes an external potential. Here we assume that there is no any pair interaction among particles for simplicity.


The Nelson Hamiltonian with the  relativistic kinetic term is
then defined by
\eq{10}
H^V = H_0+\kappa H_{\rm I},
\end{equation}
where
the zero-coupling  Hamiltonian $H_0$ is
given by
\eq{11}
H_0=H_{\rm p}\otimes \bbone + \kappa^2 \bbone \otimes  H_{\rm f}
\end{equation}
and $\kappa>0$ a scaling parameter.
$H_{\rm I}$ denotes the linear interaction
given by
\eq{6}
H_{\rm I}= \alpha\sum_{j=1}^N
 \int_{\mathbb{R}^{dN}}^\oplus
\phi_j(x_j) dX \end{equation}
under the identification:
$
\d {\ms H} \cong \int^\oplus_{\mathbb{R}^{dN}} {\mathscr F} dX$,
where $dX = dx_1\cdots dx_N$.
Here $\alpha\geq 0$ is a coupling constant, and the scalar field
$\phi_j(x)$ is given by
\eq{7}
\phi_j(x)=\frac{1}{\sqrt 2}\int_\BR
(a^{\ast}(k) \hat \lambda _j(-k) e^{-ikx}+a(k) \hat \lambda _j(k) e^{ikx} ) dk
\end{equation}
for each $x\in\BR$ with ultraviolet cutoff functions $\hat \lambda _j$.
The standard choice of the ultraviolet cutoff is $\hat \lambda_j(k)=(2\pi)^{-d/2}\bbone_{|k|\leq \Lambda}/\omega(k)$.
In this paper we do not however fix any  special cutoff function.

Throughout  we assume the following three conditions:
\begin{itemize}
\item[{(V)}]  $V(-\Delta+1)^{-1/2}$ is compact.
\item[ {(UV)}]
$\hat \lambda _j(-k)={\hat \lambda _j(k)}\geq 0$ and $\hat \lambda _j\in{L^2(\BR)}$ for  $j=1,...,N$.
\item[ {(IR)} ]   $\hat \lambda _j/\omega\in{L^2(\BR)}$ for  $j=1,...,N$.
\end{itemize}
Assumption (V) implies that $V$ is infinitesimally small with respect to $\sqrt{-\Delta+m^2}-m$
for all $m\geq 0$.
Hence, by the Kato-Rellich theorem,  $H_\mathrm{p}$ is self-adjoint on
$D(\sum_{j=1}^N\Omega_j)$ and essentially self-adjoint on any core for $\sum_{j=1}^N\Omega_j$,
where  $D(A)$ denotes the domain of $A$.
(UV) implies that $H_{\rm I}$ is symmetric.
Then (V), (UV) and (IR) also imply that, for  arbitrary $\alpha\in\mathbb{R}$ and $\epsilon>0$,
it holds that
\begin{align*}
  \|H_{\rm I}\Psi\|\leq \epsilon \|H_0\Psi\|+b_\epsilon
\|\Psi\|,\ \ \ \Psi\in D(H_0).
\end{align*}
Therefore, by the Kato-Rellich theorem, $H^V$ is self-adjoint on $D(H_0)$ for all $\kappa>0$.
Finally the nonnegativity  $\hat \lambda_j(k)\geq 0$ in (UV) is used in Lemma \ref{ene}.


\section{Existence of a ground state}

\subsection{Dressing transformation}
To derive the effective particle Hamiltonian we  introduce the so-called dressing transformation
$e^{-iT}$,
where
$T=\frac{\alpha}{\kappa}\sum_{j=1}^N \pi_j$
and
  $$
\pi_j = \int_{\mathbb{R}^{dN}}^\oplus
dX
\left(
 \frac{i}{\sqrt2} \int
 \left(
a^{\ast}(k) e^{-ikx_j} \frac{\hat \lambda _j(-k)}{\omega(k)}  - a(k) e^{ikx_j} \frac{\hat \lambda _j(k)}{\omega(k)}
\right) dk
\right).$$
By (IR), $\pi_j$ is self-adjoint on ${\ms H}$ and then $T$
is unitary.
\bl{P4.1}
The unitary operator $e^{iT}$ maps $D(H^V)$ onto itself and
\eq{sa2}
 e^{-iT} H^V e^{iT}
 = h_{\rm eff}^V \otimes \bbone+\kappa^2 \bbone \otimes H_{\rm f}+H_{\rm R}(\kappa),
\end{equation}
where the  effective Hamiltonian is defined by
\begin{align}
 h_{\rm eff}^V = \sum_{j=1}^N (\Omega_j+V_j) + V_{{\rm eff}},
\end{align}
with the effective pair potential
\begin{align}
 V_{{\rm eff}}  &= \alpha^2 \sum_{1\leq i<j\leq N} W_{ij}(x_i-x_j),\\
 W_{ij}(x) &= -  \int_{\BR} \frac{\hat{\lambda}_i(-k)\hat \lambda _j(k)}{\omega(k)} e^{-ikx} dk.
\end{align}
Here $H_{\rm R}(\kappa)$ is the remainder term given by
\begin{align*}
H_{\rm R}(\kappa) & = \sum_{j=1}^N \left(
\Delta\Omega_j -
   \frac{\alpha^2}{2} \|\hat \lambda _j/\sqrt\omega\|^2 \right),\\
 \Delta\Omega_j&=\Omega_j\left(  {p_j}  +\frac{\alpha}{\kappa}A _j  \right)
 - \Omega_j\left( {p_j} \right)
\end{align*}
with a vector field
$$A_j = A _j(x_j) =
\int_{\mathbb{R}^{dN}}^\oplus\Bigg(\frac{1}{\sqrt 2}\int
 k \left( a^{\ast}(k) e^{-ikx_j}
\frac{\hat \lambda _j(-k)}{\omega(k)}+
a(k) e^{ikx_j} \frac{\hat \lambda_j(k)}{\omega(k)}\right) dk \Bigg) dX  . $$
\end{lemma}
{\noindent \it Proof:\ }
We directly see that
\begin{align*}
e^{-iT} {p_j} e^{iT}& = {p_j}+\frac{\alpha}{\kappa}A_j,\\
e^{-iT} \phi_j e^{iT}  &= \phi_j - \frac{\alpha}{\kappa} \sum_{i=1}^N
    \int_\BR \frac{\hat\lambda_i(k)\hat\lambda_j(-k)}{\omega(k)}e^{-ik(x_j-x_i)}dk, \\
 e^{-iT} H_{\rm f} e^{iT}  &= H_{\rm f} - \frac{1}{\kappa} H_{\rm I} +
     \frac{\alpha^2}{2\kappa^2}  \sum_{i,j=1}^N \int_{\BR}
   \frac{\hat\lambda_i(-k)\hat\lambda_j(k)}{\omega(k)}e^{-ik(x_i-x_j)}dk.
\end{align*}
Together with them, the lemma follows.
\qed
(UV) and (IR) imply that
$V_{{\rm eff}}$ is bounded.
Therefore $H^V_{\rm eff}$ is a self-adjoint operator on $D(\sum_{j=1}^N\Omega_j)$.
For notational convenience we set
\eq{sa5}
E_{\rm diag}=
\frac{\alpha^2}{2}\sum_{j=1}^N  \|\hat \lambda _j/\sqrt\omega\|^2.
\end{equation}

\subsection{Main results}
We state the main results in this paper.
\bt{main}
{\bf (Existence of ground state)}
Assume (V), (UV) and  (IR).
We suppose that
 there exists $\alpha_c>0$ such that
$\inf \sigma(H^V_{\rm eff}) \in \sigma_{\rm disc}(H^V_{\rm eff})$
for all $|\alpha|>\alpha_c$,
we fix a sufficiently large $\kappa>0$.
Then there exists $0<\alpha_c(\kappa)$ such that for $\alpha$
with $\alpha_c< |\alpha|<\alpha_c(\kappa)$,
$H^V$ has the ground state and it is unique.
\end{theorem}
Theorem \ref{main} gives an example of the enhanced binding.
 We introduce an assumption.
\begin{itemize}
 \item[{ (EN)}]
 \begin{itemize}
  \item[(1)]
  $\d \inf_{x\in\BR}
   V(x)>-\infty$ and $\d\liminf_{|x|\to\infty} V(x)=0$;
\item[ (2)]
 the operator $\sqrt{-\Delta}+N V$ in ${L^2(\BR)}$ has a negative energy ground state;
\item[(3)] $V$ is $d$-dimensional relativistic Kato-class, i.e.,
$$\lim_{t\downarrow 0}\sup_{x\in\BR}{\mathbb E}_{\rm P}^x
\left[
\int_0^t V(X_s) ds\right]=0,$$
where ${\mathbb E}_{\rm P}^x$ denotes the expectation on
 a probability space $({\cal D},{\cal B},{\rm P}^x)$, and
 $\pro X$ denotes the $d$-dimensional L\'evy process
with the characteristic function
${\mathbb E}_{\rm P}^x[e^{iuX_t}]=e^{-t(\sqrt{u^2+m^2}-m)}e^{iux}$.
\end{itemize}
\end{itemize}
We use (EN)(1) for showing spatial exponential decay of the infrared regularized ground state $\Phi_\sigma}%{\varphi_{\rm g}$, and
(EN)(2) in \kak{yy3}.
(EN)(3) is used to show
 the continuity of ground state energy of a translation invariant
  Hamiltonian in Lemma \ref{p=0}.


\bt{enhanced}
{\bf (Enhanced binding)}
Assume (V),  (EN), (UV) and  (IR).
Fix a sufficiently large $\kappa>0$.
Then there exist  constants $0< \alpha_c<\alpha_c(\kappa)$
such that  for $\alpha$
with $\alpha_c< |\alpha|<\alpha_c(\kappa)$,
$H^V$ has the  ground state, and it is unique.
\end{theorem}
\begin{remark}
\label{glll}
{\rm
In Theorem \ref{enhanced} we do {\it not} assume the existence of a ground state of
$H_{\rm p}$, i.e.,  the zero-coupling  Hamiltonian $H_0$ does not necessarily have a ground state.
}\end{remark}
\begin{example}
{\rm
We give examples of $\sqrt{-\Delta+1}-1+V$ satisfying (V) and (EN), but $\sqrt{-\Delta+1}-1+V$ has no ground state
in the dimension  $d\geq 3$.
Suppose that $\tilde V$ satisfies  $|\tilde{V}(x)|\leq c(1+|x|)^{-\epsilon}(c, \epsilon>0)$. Then (V) is satisfied with
$V=\delta\tilde{V}$ for all $\delta >0$.
Let  $\tilde{V}\not\equiv 0$, $\tilde{V}\leq 0$ and $\tilde{V}\in L^d(\BR)\cap L^{d/2}(\BR)$.
 Let $\delta>0$ be a sufficiently small constant and set
\eq{hiro}
H_\delta=\sqrt{-\Delta+1}-1+\delta \tilde{V}.
\end{equation}
Let $E(\cdot)$ be the spectral measure  of $H_\delta$.
By the relativistic version of the Lieb-Thirring bound \cite{dau83}, we have
$$ {\rm dim}{\rm Ran}E((-\infty,0])
\leq c_1 \int_\BR |\delta \tilde{V}(x)|^d dx + c_2 \int_\BR |\delta \tilde{V}(x)|^{d/2}dx
$$
where $c_1,c_2$ are constants.
Hence $H_\delta$ has no ground state for sufficiently small $\delta$.
Let $N$ be sufficiently large. Then
$\inf \sigma(\sqrt{-\Delta}+N\delta \tilde V)<0$
but the bottom of the essential spectrum of $\sqrt{-\Delta}+N\delta \tilde V$ is $0$.
Therefore $\sqrt{-\Delta} +N\delta \tilde V$ has a negative eigenvalue for sufficiently large $N$.
Thus $\sqrt{-\Delta}+N\delta\tilde V$ has a ground state.
}\end{example}



\subsection{Stability condition and exponential decay}
In order to prove Theorems \ref{main} and \ref{enhanced} we investigate the stability condition.
First of all we introduce  cluster Hamiltonians.
Let $C_N=\{1,2,\cdots,N\}$.
For each $\beta \subset C_N,(\beta\neq \emptyset)$,
we define
\begin{align}
 H^0(\beta) &= \sum_{j\in\beta} (\Omega_j+\kappa\alpha \phi_j)+\kappa^2H_{\rm f}, \\
 H^V(\beta) &= H^0(\beta) + \sum_{j\in\beta} V_j,
\end{align}
acting on $L^2(\mathbb{R}^{d|\beta|})\otimes {\mathscr F} $,
where
$
 \phi_j = \int_{\mathbb{R}^{d|\beta|}}^\oplus \phi_j(x_j)dX_\beta$, $X_\beta= (x_j)_{j\in\beta}$.
 Clearly $H^V=H^V(C_N)$.
Let
\eq{s1}
E^0(\beta)=\inf\sigma(H^0(\beta)),\quad
E^V(\beta)=\inf\sigma(H^V(\beta)).
\end{equation}
For the case of $\beta=\emptyset$, we set $E^0(\emptyset )=E^V(\emptyset)=0$.
The lowest two cluster threshold is defined as the minimal energy of systems such that only the particles involved in $\beta$  are bound by the origin
but others are sufficiently remote  from the origin.
It is defined by
\begin{align}
 \Sigma^V= \min\{E^V(\beta)+E^0(\beta^c)|\beta\subsetneqq C_N \} \label{threshold}
\end{align}
The gap between the ground state energy $E^V$ and the lowest two cluster threshold $\Sigma^V$ is related to  the existence of ground state by the proposition below.
Let $H_\sigma^V$ be defined by $H^V$ with $\hat \lambda_j$ replaced by $\hat \lambda_j(k)\bbone_{|k|>\sigma}$. \begin{proposition}
{\label{P3.1}}
\begin{description}
\item[(Case $\sigma>0$)]
Suppose that $  E^V < \Sigma^V$.
Then $H_\sigma^V$ has the unique  ground state.
We denote the ground state by $\Phi_\sigma}%{\varphi_{\rm g}$.
\item[(Case $\sigma=0$)]
Suppose that $  E^V < \Sigma^V$ and there exists $0<\delta$ independent of $\sigma$ such that
$\sup_{0<\sigma<{\bar\sigma}}
\|(e^{\delta|X|}\otimes\bbone)\Phi_\sigma}%{\varphi_{\rm g}\|_{\ms H}<\infty$ with some ${\bar\sigma}>0$.
Then $H^V$ has a ground state.
\end{description} 
\end{proposition}
{\noindent \it Proof:\ }
 The proof is a  minor modification of \cite{ger00,gll01},
and it is given in Appendix \ref{massive} for the case $\sigma>0$,
and  in Appendix \ref{massless} for the case $\sigma=0$.
\qed
The condition $\Sigma^V>E^V$ is called the stability condition.
For our model the uniform exponential decay of $\|\Phi_\sigma}%{\varphi_{\rm g}(x)\|_{\mathscr F}$ may be derived from the stability condition,
  but we do not check it. So we need not only stability  condition but also uniform exponential decay.

\section{Proof of the main theorem}
In order to show Theorems \ref{main} and \ref{enhanced}, by Proposition \ref{P3.1} it is enough to show
both (1) stability condition  and
(2) the uniform exponential decay of $\|\Phi_\sigma}%{\varphi_{\rm g}(x)\|_{\mathscr F}$.
\subsection{Stability condition}
It is however not straightforward  to show the stability condition, so we will make a detour and the discussion will be reduced to that of effective particle Hamiltonian $h_{\rm eff}^V$.
Let us define the lowest two cluster threshold of $h_{\rm eff}^V$ in a similar way to $H^V$ and
we shall compare it with
 $\Sigma^V$.
For $\beta \subset C_N$, we define effective cluster Hamiltonians
by \begin{align}
  h_{\rm eff}^0(\beta)  & = \sum_{j\in\beta} \Omega_j
   -\alpha^2 \sum_{i,j\in\beta, i<j} W_{ij}(x_i-x_j),\\
    h_{\rm eff}^V(\beta) & = h_{\rm eff}^0(\beta)+\sum_{j\in\beta}V_j.
\end{align}
We set
\begin{align}
 \mathcal{E}^0(\beta) = \inf\sigma(h_{\rm eff}^0(\beta)), \qquad
  \mathcal{E}^V(\beta) = \inf\sigma(h_{\rm eff}^V(\beta))
\end{align}
and   $\mathcal{E}^V = \mathcal{E}^V(C_N)$.
Then the lowest two cluster threshold of $h_{\rm eff}^V$ is defined by
\begin{equation}{\label{sa3}}
 \Xi^V
 = \min\{ \mathcal{E}^V(\beta) + \mathcal{E}^0(\beta^c)| \beta\subsetneqq C_N\}.
\end{equation}
Constants $c^V$ and $d^V$ are such that
$
   \|\sum_{j=1}^N \Omega_j \Psi \|
 \leq c^V \|h_{\rm eff}^V \Psi \|  + d^V  \|\Psi\|$ and set
\eq{poisson}
 {\mathcal G}(t) =
 \left(   \sum_{j=1}^N
 \|{\hat \lambda _j}/{\omega}\|
 \|\hat \lambda_j\|
 \right) t^2+
 \left(   \sum_{j=1}^N
\sqrt 2 m_j
 \|{\hat \lambda _j}/{\omega}\|
\right) t
 +  \sqrt{2}
N     \left(  c^V |\mathcal{E}^V| + d^V \right).
     \end{equation}
     The next lemma  is a key ingredient of this paper.
\begin{lemma}\label{P4.2}
We assume that
$\Xi^V-\mathcal{E}^V>0$, and
$\alpha$ and $\kappa$ satisfy
$\Xi^V-\mathcal{E}^V> {\mathcal G}(\alpha/\kappa)$.
Then the stability condition
$ \Sigma^V-E^V >0$ holds.
\end{lemma}
In order to prove Lemma \ref{P4.2}, we prepare two lemmas.
\begin{lemma} \label{enebound1}
 For all $\beta  \subset C_N$,
it follows that
\begin{align}
E^\#(\beta)   & \leq \mathcal{E}^\#(\beta)   +    \frac{\alpha^2}{2} \sum_{j\in\beta}
 \|\hat \lambda _j/\sqrt{\omega} \|^2,\quad \#=0,V.
\end{align}
In particular
it holds that
$\Xi^V\leq \Sigma ^V+E_{\rm diag}$.
\end{lemma}
{\noindent \it Proof:\ }
See Proposition \ref{yui}
 in Appendix \ref{appb}.
 \qed
\begin{lemma}{\label{enebound2}}
For all $\kappa>0$
it follows that
$  E^V   \leq \mathcal{E}^V  + {\mathcal G}(\alpha/\kappa) - E_{\rm diag}$,
\end{lemma}
{\noindent \it Proof:\ }
For arbitrary $\epsilon>0$, we can choose a normalized vector
$v \in C_0^\infty({\mathbb {R}^{dN}  })$ such that
$
  \|(h_{\rm eff}^V - \mathcal{E}^V) v\| \leq \epsilon$.
Set $\Psi=v\otimes\Omega$.
Then, by Lemma \ref{P4.1}, we have
\begin{align*}
 E^V
 \leq  {\mathcal E}^V + \epsilon + \left(  \Psi,
 \left(
- E_{\rm diag}+\sum_{j=1}^N
 \Delta \Omega_j\right)
\Psi \right) .
\end{align*}
Since $\pi_j$ commutes with $p_i$, $i\neq j$, by setting $T_j=\alpha\pi_j/\kappa$,
we can see that $\Delta\Omega_j = e^{-iT_j}\Omega_j e^{iT_j}-\Omega_j$ and
\begin{align*}
 |\left(  \Psi,
 \Delta\Omega_j\Psi \right) |
= |\left(  (e^{iT_j}-1)\Psi, \Omega_j e^{iT_j} \Psi \right) + \left(  \Psi, \Omega_j(e^{iT_j}-1)\Psi \right)  |.
\end{align*}
Hence
we have
\begin{align*}
 |\left(  \Psi,
 \Delta\Omega_j
 \Psi \right) |
\leq
  \frac{\alpha}{\kappa} \|\pi_j \Psi \| \cdot \|\Omega_j e^{iT_j}\Psi\|
          + \frac{\alpha}{\kappa} \|\pi_j \Psi \| \cdot \|\Omega_j \Psi\|.
          \end{align*}
          The right-hand side above is identical with
$$\d
 = \frac{\alpha}{\sqrt{2}\kappa} \|\hat \lambda _j/\omega\|
 \left(\left(  \Psi, \left({p_j}  + \frac{\alpha}{\kappa}A _j\right) ^2 \Psi \right) ^{1/2}
                     + \left(  \Psi, {p_j} ^2 \Psi \right) ^{1/2} \right).$$
Then
we have
\[
  |\left(  \Psi,  \Delta\Omega_j \Psi \right) |
\leq  \frac{\alpha}{\sqrt{2}\kappa} \|\hat \lambda _j/\omega\|
 \left( 2 \| \Omega_j \Psi \|  + 2m_j
 +  \frac{\sqrt{2}\alpha}{\kappa} \| |k|\hat \lambda _j/\omega \|  \right)
 \]
and
\begin{align*}
 E^V&
\leq
\mathcal{E}^V +\epsilon
 + \sum_{j=1}^N \frac{\alpha}{\sqrt{2}\kappa}
\|\hat \lambda_j/\omega\|
\left(
2m_j+\frac{\sqrt{2}\alpha}{\kappa}\| |k|\hat \lambda _j/\omega\|
\right)
\\
&\quad  + \sum_{j=1}^N  \frac{\sqrt{2}\alpha}{\kappa}
\|\hat \lambda _j/\omega\|
     \left(  c^V (|\mathcal{E}^V| + \epsilon) + d^V\right)
     -E_{\rm diag}.
\end{align*}
Since $\epsilon>0$ is arbitrary, we prove the lemma.
\qed
\textit{Proof of Lemma \ref{P4.2}:}

By Lemmas \ref{enebound1} and \ref{enebound2}, we have
\eq{saigo}
 \Sigma^V -E^V \geq \Xi^V - \mathcal{E}^V - {\mathcal G}(\alpha/\kappa)>0 .
 \end{equation}
Then
the lemma is proven.
\qed


\subsection{Exponential decays}
It is proven that the functional integration is a strong tool to show an exponential localization of bound state in quantum mechanics.
That can be also applied  in quantum field theory.

Let $\pro X= (X_t^1,\dots,X_t^N)_{t\geq 0}$
 be the $N$  independent
 $d$-dimensional L\'evy processes
 on a probability space $({\cal D}, {\cal B}, {\rm P}^x)$, $x\in{\mathbb {R}^{dN}  }$,  with
  the characteristic function
\eq{char}
{\mathbb E}_{\rm P}^0[e^{-iu\cdot X_t}]=
 e^{-t\sum_{j=1}^N (\sqrt{u_j^2+m_j^2}-m_j)},\quad u=(u_1,...,u_N)\in{\mathbb {R}^{dN}  }.
 \end{equation}
 Here and in what follows ${\mathbb E}_m^x[\cdots]$ denotes the expectation  with respect to a path measure $m^x$ starting  from $x$.
Let $W_{\rm eff}=W_{\rm eff}(x_1,..,x_N)=\sum_{j=1}^N V(x_j)+V_{{\rm eff}}(x)$.
 \bp{b2}
There exists $\sigma_0>0$ such that for all $\sigma\leq \sigma_0$,
\eq{l777}
\|\Phi_\sigma}%{\varphi_{\rm g}(X)\|_{\mathscr F}
\leq
e^{t( E^V + E_{\rm diag}+\epsilon(\sigma))}
\left(
{\mathbb E}_{\rm P}^X \left[
e^{-2\int_0^t W_{\rm eff}(X_s) ds} \right]
\right)^{1/2}
 \|\Phi_\sigma}%{\varphi_{\rm g}\|_{\ms H}
\end{equation}
for each $X\in{\mathbb {R}^{dN}  }$, where $\epsilon(\sigma)>0$ satisfies $\lim_{\sigma\to0}\epsilon(\sigma)=0$.
\end{proposition}
{\noindent \it Proof:\ } See Proposition \ref{ito}.
\qed
From Proposition \ref{b2} it suffices to
estimate
$
e^{t( E^V + E_{\rm diag})} {\mathbb E}_{\rm P}^X \left[
e^{-2\int_0^t W_{\rm eff}(X_s) ds} \right]^{1/2}
$
for
the exponential decay of $\|\Phi_\sigma}%{\varphi_{\rm g}(X)\|_{\mathscr F}$.
To estimate this we divide $W_{\rm eff}$ into two
parts.
Let
$$B_R=\{x=(x_1,...,x_N)\in {\mathbb {R}^{dN}  }||x|\geq 2R \mbox{ and } \min\{|x_i-x_j|,i\not= j\}\leq |x|/2\}.$$
Define
$V_{\rm eff, \infty}^R=V_{{\rm eff}} \bbone_{B_R}$ and
$V_{\rm eff, 0}^R=V_{{\rm eff}} \bbone_{B_R^c}$.
Then
\eq{kosi6}
W_{\rm eff}=
V+V_{\rm eff, 0}^R+V_{\rm eff, \infty}^R.
\end{equation}
By the Riemann Lebesugue lemma $\lim_{|x|\to\infty}W_{ij}(x)=0$.
Then notice that
\begin{align*}
&\lim_{|x|\to\infty} (V(x)+V_{\rm eff, 0}^R(x) )=0,\\
 &\|V_{\rm eff, \infty}^R\|_\infty
\leq \frac{\alpha^2}{2}
\sum_{i\not = j} \int \frac{\hat \lambda_i(k)\hat \lambda_j(-k)}{\omega(k)}dk.
\end{align*}
The L\'evy measure  $\nu_j(dx)=\nu_j(x) dx $ associated with
 the L\'evy  process $\pro {X^j}$ is given by
\eq{l1}
\nu_j(x)=2\left(\frac{m_j}
{2\pi}\right)^{\frac{d+1}{2}}
\frac{1}{|x|^{\frac{d+1}{2}}}
\int _0^\infty \xi^{\frac{d-1}{2}}
e^{-\frac{1}{2}(\xi+\xi^{-1})m_j|x|}
d\xi,\quad x\in\BR.
\end{equation}
We note that
$
\nu(x)\leq
C e^{-c |x|}$
with some constants $C, c \geq 0$.
\bp{l4}
There exist $\eta>0$,   $C_1>0$ and $C_2>0$ such that
\eq{l5}
{\rm P}^0\left( \sup_{0\leq s\leq t}|X_s|>a\right)
\leq
C_1e^{-\eta a}e^{C_2 t}
\end{equation}
for all $a>0$.
\end{proposition}
{\noindent \it Proof:\ }
We see that
\begin{align*}
{\rm P}^0\left( \sup_{0\leq s\leq t}|X_s|>a\right)
=
{\mathbb E}_{\rm P}^0
\left[
\bbone_{\sup_{0\leq s\leq t}|X_s|-a>0}
\right]
\leq
e^{-\eta a}
{\mathbb E}_{\rm P}^0
\left[e^{\eta \sup_{0\leq s\leq t}|X_s|}
\right].
\end{align*}
It is known  that
${\mathbb E}_{\rm P}^0[e^{\eta (
\sup_{0\leq s\leq t}|X_s|)}]<C_1 e^{C_2t}$ for sufficiently small $0<\eta$ \cite{cms90}.
Hence the proposition follows.
\qed
We define
${\ms B}=\{X_s\in B_R^c \mbox{ for all } 0\leq s\leq t\}$.
Since $V_{\rm eff, \infty}^R(X_s)=0$ for on ${\ms B}$,
we have
\begin{align}
{\mathbb E}_{\rm P}^X\left[e^{-2\int_0^t W_{\rm eff}(X_s) ds}\right]
=
\label{kosi22}
{\mathbb E}_{\rm P}^X\left[\bbone_{\ms B} e^{-2\int_0^t (V+V_{\rm eff,0}^R)(X_s) ds}\right]+
{\mathbb E}_{\rm P}^X\left[\bbone_{{\ms B}^c}e^{-2\int_0^t W_{\rm eff}(X_s) ds}\right].
\end{align}
By the Schwartz inequality
\eq{kosi}
{\mathbb E}_{\rm P}^X\left[ \bbone_{{\ms B}^c}e^{-2\int_0^t W_{\rm eff}(X_s) ds}\right]
\leq
{\mathbb E}_{\rm P}^X\left[\bbone_{{\ms B}^c} e^{-4\int_0^t V_{\rm eff, \infty}^R(X_s) ds}\right]^{1/2}
{\mathbb E}_{\rm P}^X\left[\bbone_{{\ms B}^c} e^{-4\int_0^t (V+V_{\rm eff,0}^R)(X_s) ds}\right]^{1/2}.
\end{equation}
We will estimate terms in \kak{kosi22} and \kak{kosi}.
Set
\begin{align*}
&
W_a^R(x)=\inf\{
V(y)+V_{\rm eff,\infty}^R(y)
||x-y|<a\},\\
&
W_\infty^R=\inf_{x\in{\mathbb {R}^{dN}  }}
(V(x)+V_{\rm eff,\infty}^R(x)).
\end{align*}
\bl{l77}
Suppose (1) of (EN).
Let $R>0$ and $a>0$.
Then for all $X\in{\mathbb {R}^{dN}  }$ and $t>0$ it follows that
\eq{l8}
 {\mathbb E}_{\rm P}^X
 [e^{-2\int_0^t
(V(X_s)+V_{\rm eff,\infty}^R(X_s) )ds}]
 \leq
 e^{-2t W_a^R(x)}+
C_1 e^{-2tW_\infty^R }
e^{C_2t}e^{-\eta a},
\end{equation}
where $C_1,C_2$ and $\eta$ are given in \kak{l5}.
\end{lemma}
{\noindent \it Proof:\ }
Set $ A = \{
\sup_{0\leq s\leq t} | X_s| < a \}\subset{\cal D}$.
Since $\pro X$ under the probability measure ${\rm P}^X$ and $(X_t+X)_{t\geq0}$ under ${\rm P}^0$ are
identically distributed,
we have the identity:
$ {\mathbb E}_{\rm P}^X \left[
e^{-2\int_0^t
(V(X_s)+V_{\rm eff,\infty}^R(X_s) ) ds} \right]
=
{\mathbb E}_{\rm P}^0\left[
e^{-2\int_0^t
(V(X_s+X)+V_{\rm eff,\infty}^R(X_s+X) ) ds} \right]
$.
Then
we have
\begin{align*}
 {\mathbb E}_{\rm P}^0\left[
\bbone_{A}e^{-2\int_0^t
(V(X_s+X)+V_{\rm eff,\infty}^R(X_s+X) )
 ds} \right]
&\leq e^{-2t W_a^R(x)},\\
{\mathbb E}_{\rm P}^0\left[
\bbone_{A^c}e^{-2\int_0^t
(V(X_s+X)+V_{\rm eff,\infty}^R(X_s+X) ) ds}\right]&
\leq
e^{-2t W_\infty^R}
{\mathbb E}_{\rm P}^0\left[
\bbone_{A^c}
\right]
\leq
e^{-2t W_\infty^R}
C_1e^{C_2 t}e^{-\eta a}
\end{align*}
by
Proposition  \ref{l4}.
Then the lemma follows.
\qed
\bl{kosi2}
Let $X\in{\mathbb {R}^{dN}  }$ and  set  $R=|X|$.
Then it follows that
\eq{kosi3}
{\mathbb E}_{\rm P}^X[\bbone_{{\ms B}^c}e^{-4\int_0^t V_{\rm eff, \infty}^R(X_s) ds}]\leq
e^{4\|V_{\rm eff, \infty}\|_\infty t }C_1e^{C_2t}e^{-\eta R},
\end{equation}
where $C_1,C_2$ and $\eta$ are given in \kak{l5}.
\end{lemma}
{\noindent \it Proof:\ }
Since ${\mathbb E}_{\rm P}^X[e^{-4\int_0^t V_{\rm eff, \infty}^R(X_s) ds}]
\leq
{\mathbb E}_{\rm P}^X[e^{4\| V_{\rm eff, \infty}\|_\infty \int_0^t \bbone_{{B_R}}(X_s) ds}]$,
we can see that
\begin{align*}
&{\mathbb E}_{\rm P}^X[e^{-4\int_0^t V_{\rm eff, \infty}^R(X_s) ds}]
\leq
\sum_{n=0}^\infty \frac{(4\| V_{\rm eff, \infty}\|_\infty)^n}{n!}
\int_0^t ds_1\cdots\int_0^t ds_n {\mathbb E}_{\rm P}^X\left[
\bbone_{{\ms B}^c}
\prod_{j=1}^n  \bbone_{{B_R}}(X_{s_j})\right]\\
&=
{\mathbb E}_{\rm P}^X
\left[
\bbone_{{\ms B}^c}
\right]
+
\sum_{n=1}^\infty \frac{(4\| V_{\rm eff, \infty}\|_\infty)^n}{n!}
\int_0^t ds_1\cdots\int_0^t ds_n
{\mathbb E}_{\rm P}^{{0}}
\left[
\bbone_{{\ms B}^c}
\prod_{j=1}^n  \bbone_{B_R}(X+X_{s_j})\right]
\end{align*}
We see that
\eq{sala}
{\mathbb E}_{\rm P}^X
\left[
\bbone_{{\ms B}^c}
\right]
\leq {\rm P}^0(\sup_{0\leq s\leq t}|X_s+X|>2R)\leq
{\rm P}^0(\sup_{0\leq s\leq t}|X_s|>2R-|X|)=
{\rm P}^0(\sup_{0\leq s\leq t}|X_s|>R).
\end{equation}
By the definition of $B_R$ in a similar wat to above
we have
\begin{align*}
&
{\mathbb E}_{\rm P}^X[e^{-4\int_0^t V_{\rm eff, \infty}^R(X_s) ds}]\\
& \leq
{\rm P}^X({\ms B}^c)+
\sum_{n=1}^\infty \frac{(4\| V_{\rm eff, \infty}\|_\infty)^n}{n!}
\int_0^t\!\! ds_1\!\!\cdots\!\!\int_0^t\!\! ds_n {\rm P}^0(|X_{s_1}+X|>2R,\!\!\cdots\!\!,|X_{s_n}+X|>2R)\\
&\leq
{\rm P}^X({\ms B}^c)+
\sum_{n=1}^\infty \frac{(4\| V_{\rm eff, \infty}\|_\infty)^n}{n!}
\int_0^t ds_1\cdots\int_0^t ds_n {\rm P}^0(|X_{s_1}|>R,\cdots,|X_{s_n}|>R).
\end{align*}
By ${\rm P}^0(|X_{s_1}|>R,\cdots,|X_{s_n}|>R) \leq
{\rm P}^0\left( \sup_{0\leq s\leq t} |X_{s}|>R\right)$ and
Proposition \ref{l4}, we have
\begin{align*}
&{\mathbb E}_{\rm P}^X[e^{-4\int_0^t V_{\rm eff, \infty}^R(X_s) ds}]\\
&\leq
{\rm P}^0\left( \sup_{0\leq s\leq t} |X_{s}|>R\right)+\sum_{n=1}^\infty \frac{(4\||V_{\rm eff, \infty}\|_\infty)^n}{n!}
\int_0^t ds_1\cdots\int_0^t ds_n {\rm P}^0\left( \sup_{0\leq s\leq t} |X_{s}|>R\right) \\
&\leq
\sum_{n=0}^\infty \frac{(4\||V_{\rm eff, \infty}\|_\infty)^n}{n!}
t^n
C_1 e^{C_2 t} e^{-\eta R}\\
&=
e^{4\||V_{\rm eff, \infty}\|_\infty t}
C_1 e^{C_2 t} e^{-\eta R}.
\end{align*}
Hence the lemma follows.
\qed

\bl{expdecay}
Let $\Phi_\sigma}%{\varphi_{\rm g}$ be the infrared regularized
ground state.
Suppose (1) of (EN)
and
$E^V+E_{\rm diag} <0$.
Furthermore we assume that
 $E^V+E_{\rm diag}+\epsilon(\sigma)<-\gamma$ with some $\gamma>0$ for $\sigma<{\bar\sigma}$,
  where $\epsilon(\sigma)$ is given in Proposition \ref{b2}.
Then there exist $\delta>0$ and $C_\delta>$ independent of $\sigma$ such that
\eq{l9}
\sup_{0<\sigma<{\bar\sigma}}
\|\Phi_\sigma}%{\varphi_{\rm g}(X)\|_{\mathscr F}\leq
{C_\delta}e^{-\delta \min \{\gamma,\eta\}|X|},
\end{equation}
where $\eta>0$ is given in Proposition \ref{l4}.
\end{lemma}
{\noindent \it Proof:\ }
We set  $\tilde E=E^V+E_{\rm diag}+\epsilon(\sigma)$.
It is enough to estimate
$e^{2t\tilde E} {\mathbb E}_{\rm P}^X \left [e^{-2\int_0^t W_{\rm eff}(X_s) ds}\right]$
by Proposition \ref{b2}.
Recall that
 $W_a^R(x)=\inf\{W^R(y)||x-y|\leq a\}$.
Then
\eq{ln1}
\lim_{|x|\to\infty}W_{|x|/2}^{|x|}(x)=0.\end{equation}
Hence there  exists
a positive constant $R^\ast$ such  that
$|W_{|X|/2}^{|X|}(X) |\leq |\tilde E|/2$ for all $X$ such that $|X|>R^\ast$.
Suppose that $|X|>R^\ast$ and
let
 $R=|X|$. We divide $W_{\rm eff}$ as in \kak{kosi6} for $R$.
We have
\begin{align*}
&e^{2t\tilde E}{\mathbb E}_{\rm P}^X \left[
e^{-\int_0^t W_{\rm eff}(X_s) ds}
\right]\\
&\leq
e^{2t\tilde E}
{\mathbb E}_{\rm P}^X \left[
\bbone_{{\ms B}}
e^{-2\int_0^t  (V+V_{\rm eff,0}^R)(X_s) ds}
\right]\\
&+
e^{2t\tilde E}
\left(
{\mathbb E}_{\rm P}^X \left[
\bbone_{{\ms B}^c}
e^{-4\int_0^t  (V+V_{\rm eff,0}^R)(X_s)ds}
\right]\right)
^{1/2}
\left(
{\mathbb E}_{\rm P}^X \left[\bbone_{{\ms B}^c}
e^{-4\int_0^t  (V+V_{\rm eff,\infty}^R)(X_s) ds}
\right]\right)^{1/2}
\end{align*}
Two terms
${\mathbb E}_{\rm P}^X\left[\bbone_{\ms B} e^{-2\int_0^t (V+V_{\rm eff,0}^R)(X_s) ds}\right]
$  and
${\mathbb E}_{\rm P}^X\left[\bbone_{{\ms B}^c} e^{-4\int_0^t (V+V_{\rm eff,0}^R)(X_s) ds}\right]
$
 can be estimated as
\begin{align}
&{\mathbb E}_{\rm P}^X\left[\bbone_{\ms B} e^{-2\int_0^t (V+V_{\rm eff,0}^R)(X_s) ds}\right]
\leq
 e^{-2t W_a^R(x)}+
C_1 e^{-2tW_\infty^R }
e^{C_2t}e^{-\eta a},
\\
&
{\mathbb E}_{\rm P}^x\left[\bbone_{{\ms B}^c} e^{-4\int_0^t (V+V_{\rm eff,0}^R)(X_s) ds}\right]
\leq
 e^{-4t W_a^R(x)}+
C_1 e^{-4tW_\infty^R }
e^{C_2t}e^{-\eta a}
\end{align}
by Lemma \ref{l77}.
Let us set $t=t(X)=\epsilon|X|$ and $a=|X|/2$.
Then we can  see that
$W_{|X|/2}^{|X|}(X) -\tilde E>-\tilde E/2>0$,
since $\tilde E<0$ by assumption.
Hence
\begin{align*}
e^{2t\tilde E}
{\mathbb E}_{\rm P}^X\left[\bbone_{\ms B} e^{-2\int_0^t (V+V_{\rm eff,0}^R)(X_s) ds}\right]
&\leq
e^{\epsilon \tilde E|X|}+C_2e^{\epsilon C_2|X|-\eta |X|/2-2\epsilon W_\infty^{|X|}|X|}\\
&\leq
e^{-\epsilon \gamma |X|}+C_2
e^{-(\eta/2 +2\epsilon W_\infty^{|X|} -\epsilon C_2)|X|}.
\end{align*}
Similarly
we have
\begin{align*}
&e^{4t\tilde E}
{\mathbb E}_{\rm P}^X\left[\bbone_{\ms B} e^{-4\int_0^t (V+V_{\rm eff,0}^R)(X_s) ds}\right]
\leq
e^{-2\epsilon \gamma |X|}+C_2
e^{-(\eta/2 +4\epsilon W_\infty^{|X|} -\epsilon C_2)|X|}.
\end{align*}
Finally by Lemma \ref{kosi2} we have
\begin{align*}
e^{4t\tilde E}
{\mathbb E}_{\rm P}^X\left[\bbone_{\ms B} e^{-4\int_0^t W_{\rm eff}(X_s) ds}\right]
&\leq
C_1e^{4\epsilon \tilde E +4\|V_{\rm eff,\infty}\|_\infty \epsilon  +C_2\epsilon-\eta)|X|}\\
&\leq
C_1e^{-(4\epsilon \gamma -4 \|V_{\rm eff,\infty}\|_\infty \epsilon - C_2\epsilon+\eta)|X|}.
\end{align*}
Note that $W_\infty^{|X|}\to 0$ as $|X|\to\infty$.
Take sufficiently small $\epsilon>0$ such  that
$\eta/2+(2W_\infty^{|X|}-C_2)\epsilon>0$, $\eta/2+(4W_\infty^{|X|}-C_2)\epsilon>0$ and
$(4\gamma -4 \|V_{\rm eff,\infty}\|_\infty  - C_2)\epsilon+\eta>0$, then
$\|\Phi_\sigma}%{\varphi_{\rm g}(X)\|_{\mathscr F}\leq D_1e^{-\min\{\eta,\gamma\}D_2|X|}$  follows.
 Then the lemma is proven.
 \qed

\bc{b3}
Suppose (1) of (EN). Then  \kak{l9} holds for  sufficiently small $|\alpha/\kappa|$.
\end{corollary}
{\noindent \it Proof:\ }
Notice that
 $E^V\leq {\cal E}^V+{\mathcal G}(\alpha/\kappa) -E_{\rm diag}$ in Lemma \ref{enebound2}.
Since ${\cal E}^V<0$ and $\lim_{t\to 0}{\mathcal G}(t)=0$, the corollary follows.
\qed






\subsection{Proof of Theorem \ref{main} and Theorem \ref{enhanced}}
\subsubsection{Proof of  Theorem \ref{main}}
{\it Proof of  Theorem \ref{main}}:

Note that $\mathcal{E}^V<\Xi^V$ is equivalent to $\inf\sigma(H^V_{\rm eff})\in \sigma_{\rm disc}(H^V_{\rm eff})$.
Hence
$\mathcal{E}^V<\Xi^V$ for all $|\alpha|>\alpha_c$.
By Lemma \ref{P4.2} and the fact that $\lim_{t\to 0}{\mathcal G}(t)=0$, there exists $\alpha_c(\kappa)>\alpha_c$ such that
for all $\alpha_c<|\alpha|<\alpha_c(\kappa)$, the stability condition $E^V<\Sigma^V$ holds.
Moreover uniform exponential decay
$\|\Phi_\sigma}%{\varphi_{\rm g}(x)\|_{\mathscr F} \leq C_\delta e^{-\delta|x|}$ is shown for sufficiently small $|\alpha/\kappa|$ in Lemma \ref{expdecay}.
  Therefore, by Proposition \ref{P3.1}, $H^V$ has a ground state.
\qed
\subsubsection{Proof of  Theorem \ref{enhanced}}
Now we show the enhanced binding.
   It is enough to show   $ \mathcal{E}^V <  \Xi^V$, since the uniform exponential decay $\|\Phi_\sigma}%{\varphi_{\rm g}(x)\|_{\mathscr F}<C_\delta e^{-\delta|x|}$ is established by Proposition \ref{P3.1}.
\begin{lemma} \label{ene}
 Let $\beta\subsetneqq {C_N} $ but $\beta\neq\emptyset$.
Then there exists  $\alpha_1>0$ such that,
for all $\alpha$ with $|\alpha|>\alpha_1$,
$   \mathcal{E}^0 < \mathcal{E}^V(\beta)+ \mathcal{E}^0(\beta^c)$.
In particular $\mathcal{E}^0<\Xi^V$ holds
for $|\alpha|>\alpha_1$.
\end{lemma}
{\noindent \it Proof:\ }
We have
\begin{align*}
 &   \mathcal{E}^0  = \alpha^2 \sum_{i<j} W_{ij}(0) + o(\alpha^2), \quad
 \mathcal{E}^V(\beta) = \alpha^2 \sum_{\atopb{i<j}{i,j\in\beta}}W_{ij}(0) + o(\alpha^2), \\
&   \mathcal{E}^0(\beta^c) = \alpha^2 \sum_{\atopb{i<j}{i,j\in\beta^c}}W_{ij}(0) + o(\alpha^2) .
\end{align*}
Since
$
\d \sum_{\atopb{i<j}{i\in\beta, j\in\beta^c}}W_{ij}(0)+
  \sum_{\atopb{i<j}{i\in\beta^c, j\in\beta}}W_{ij}(0)<0$,
  the lemma holds.
\qed
To see the enhanced binding we want to investigate the center of motion of $h_{\rm eff}^V$. Notice
that  $h_{\rm eff}^0$ commutes with the total momentum $P_{\rm tot}=\sum_{j=1}^N p_j$. Then it can be decomposable with respect to the spectrum of $P_{\rm tot}$.
Let
$
  \ms U = e^{ix_1\cdot \sum_{j=2}^N {p_j}  }$, which
  diagonalize $P_{\rm tot}$ as
  $\ms U P_{\rm tot} \ms U^{-1} =p_1$. Hence it also diagonalize $h_{\rm eff}^0$, and we obtain
 that
  \begin{align*}
&  \ms U h_{\rm eff}^0 \ms U ^{-1} = \Omega_1\left(
p_1-\sum_{j=2}^N{p_j}
\right) + \sum_{j=2}^N \Omega_j({p_j} )  + \sum_{j\geq 2} \alpha^2W_{1j}(x_j) +\!\! \sum_{2\leq i<j\leq N} \alpha^2 W_{ij}(x_i-x_j),\\
&  \ms U h_{\rm eff}^V\ms U ^{-1} = h_{\rm eff}^0  +V(x_1) + \sum_{j=2}^N V(x_1+x_j).
\end{align*}
Then we have
\begin{align*}
&  \ms U h_{\rm eff}^0\ms U ^{-1} =  \int_{\BR}^\oplus k(P)  dP, \\
&
k(P) = \Omega_1\left(
P-\sum_{j=2}^N{p_j}
\right)
 + \sum_{j=2}^N \Omega_j({p_j} )
       + \sum_{j\geq 2} \alpha^2 W_{1j}(x_j)  + \sum_{2\leq i<j\leq N} \alpha^2  W_{ij}(x_i-x_j).
\end{align*}

\bl{p=0}
It follows that ${\cal E}^0=\inf\!\sigma (k(0))$.
\end{lemma}
{\noindent \it Proof:\ }
Set $\inf\!\sigma (k(P))=E(P)$ for simplicity.
It can be seen
in Appendix \ref{appc} that
\eq{ebound}
E(0)\leq E(P)
\end{equation}
 holds
for all $P$, and that $E(P)$ is continuous in $P$.
Then it follows that
$\d (\Phi, H\Phi)=\int_\BR (\Phi(P), k(P)\Phi(P))dP\geq E(0)\|\Phi\|^2$ for $\Phi\in D(H)$. Then $E(0)\leq {\cal E}^0$.  On the other hand
let us set $\Phi_\epsilon =\int^\oplus_\BR  \Phi(P)
\bbone_{[0,\epsilon)}(P)dP$.
 We have
$\|\Phi_\epsilon\|^2
{\cal E}^0\leq (\Phi_\epsilon, H\Phi_\epsilon)\leq \sup_{|P|<\epsilon}E(P)\|\Phi_\epsilon\|^2$. Take $\epsilon\downarrow 0$ on both sides we have ${\cal E}^0\leq E(0)+\delta$ for arbitrary $\delta>0$, since $E(P)$ is continuous in $P$.
Hence $E(0)\geq {\cal E}^0$  and then ${\cal E}^0=E(0)$ follows.
\qed
\bl{gr}
There exists $\alpha_2(P)>0$ such that
$\inf\!\sigma (k(P))\in \sigma_{\rm disc}(k(P))$ for every $P\in\BR$ for $|\alpha|>\alpha_2(P)$.
 In particular $k(0)$ has a ground state for
 $|\alpha|>\alpha_2$ with some $\alpha_2>0$.
\end{lemma}
{\noindent \it Proof:\ }
Notice that $W_{ij}(0)<0$,  $W_{ij}(x) >  W_{ij}(0)$ for $x\not =0$, and $\lim_{|x|\to\infty}W_{ij}(x)=0$.
Set
 $\mathbf{X}=(x_2,\dots,x_N)$.
Let
$a=\{2,...,N\}$.
Let $\{\til{j}_\beta\}_{\beta \subset a }$ be the  Ruelle-Simon partition of unity \cite[Definition 3.4]{cfks87},
i.e., $\til j_\beta(\lambda \mathbf{X})=\til j_\beta(\mathbf{X})$ for all $\lambda>1$, $|\mathbf{X}|=1$, and there exists a constant
$C>0$ such that
${\rm supp}\til j_\beta\cap \{\mathbf{X}||\mathbf{X}|>1\}\subset\{\mathbf{X}||\mathbf{X}_i-\mathbf{X}_j|\geq C|\mathbf{X}|\mbox{for all }(ij)\not\subset \beta\}$.
 We set $j_\beta(\mathbf{X})= \til{j}_\beta(\mathbf{X}/R)$.
 Then
\begin{align}
  k(P) = j_{a}k(P)j_{a} +
            \sum_{\beta \subsetneq a} j_\beta k(P) j_\beta
            + o(\bbone),   \label{imsk}
\end{align}
where $o(\bbone)$ denotes a bounded operator such that $\lim_{R\to\infty} \norm{o(\bbone)}=0$.
We set
\begin{align*}
  &k_\beta = \sum_{j\in\beta}(\Omega_j(p_j)+\alpha^2 W_{1j}(x_j)) + \sum_{i,j\in\beta} \alpha^2 W_{ij}(x_i-x_j)\\
  &\bar{k}_{\beta^c} =  \sum_{j\in\beta^c} \Omega_j(p_j) + \sum_{i,j\in\beta^c} \alpha^2 W_{ij}(x_i-x_j)
\end{align*}
With the identification $L^2(\mathbb{R}^{d(N-1)}) \cong  L^2(\mathbb{R}^{d|\beta|})\otimes L^2(\mathbb{R}^{d|\beta|^c)})$,
we can write
\begin{align}
 j_\beta k(P) j_\beta
 =   j_\beta \Omega_1\left( P-\sum_{j=2}^Np_j\right)  j_\beta
      +  j_\beta ( k_\beta \otimes \bbone + \bbone \otimes \bar k_{\beta^c}) j_\beta + I_\beta j_\beta^2 \label{imsk2}
\end{align}
where
$
  I_\beta = \sum_{j\in\beta^c}\alpha^2 W_{1j}(x_j) + \sum_{\atopb{i\in \beta, j\in\beta^c}{i\in\beta^c,j\in\beta}}
                 \alpha^2 W_{ij}(x_i-x_j)$.
Hence, \eqref{imsk} and \eqref{imsk2} imply
\begin{align*}
 k(P) \geq E_0(k(P))j_{a}^2
          + \sum_{\beta \subsetneq a} j_\beta(k_\beta \otimes \bbone + \bbone \otimes \bar{k}_{\beta^c}
             + I_\beta) j_\beta + o(\bbone),
\end{align*}
where $E_0(A)$ denotes the ground state energy of $A$.
Note that $j_{a}^2$ and $I_\beta j_\beta^2$ are relatively compact with respect to $k(P)$.
Thus we have
\begin{align*}
 \inf \sigma_\mathrm{ess}(k(P)) \geq \max\{E_0(k_\beta) + E_0(\bar{k}_{\beta^c}) |
         \beta \subsetneq a\}.
\end{align*}
For all $\beta\subsetneq a$ it holds that
\begin{align}
& \lim_{\alpha\to\infty} \frac{E_0(k(P))}{\alpha^2} = \sum_{i<j}W_{ij}(0)
    <  \sum_{j\in \beta} W_{1j}(0) + \sum_{\atopb{i,j \in\beta }{i<j}}W_{ij}(0)
              + \sum_{\atopb{i,j\in \beta^c}{i<j}}W_{ij}(0) \\
&  =   \lim_{\alpha\to\infty} \frac{E_0(k_\beta)+E_0(\bar{k}_{\beta^c} ) }{\alpha^2}.\nonumber
\end{align}
Therefore there exist $\alpha_2(P)$ such that for all $\alpha>\alpha_2(P)$,
$
 \inf\sigma_\mathrm{eff}(k(P)) > E_0(k(P))$.
 \qed

\begin{lemma}{\label{delta}}
Let $|\alpha|>\alpha_2$, where $\alpha_2$ is given in Lemma \ref{gr},  and
 $u_\alpha$ be a normalized ground state of $k(0)$.
Then
$  |u_\alpha(x_2,\ldots,x_N)|^2 \to \delta(x_2)\cdots\delta(x_N)$ as $\alpha\to\infty$ in the
sense of distributions.
\end{lemma}
{\noindent \it Proof:\ }
It suffices to show that for all $\epsilon>0$,
\begin{align}
  \lim_{\alpha\to\infty} \int_{|{\bf X}}%\underbar {X} |>\epsilon} |u_\alpha({\bf X}}%\underbar {X} )|^2 d{\bf X}}%\underbar {X} =0, \label{2.2400}
\end{align}
where ${\bf X}}%\underbar {X} =(x_2,\cdots,x_N)$,
since
\eqref{2.2400} implies
that
$$\d\lim_{\alpha\to0}
 \int_{\mathbb{R}^{d(N-1)}}f({\bf X}}%\underbar {X} )|u_\alpha({\bf X}}%\underbar {X} )|^2 d{\bf X}}%\underbar {X}
 =  f(0)$$ for
all $f\in C_0^\infty(\mathbb{R}^{d(N-1)})$.
We
write
$k_\alpha(0)$ to
emphasize the $\alpha$ dependence of
$k(0)$.
Since $k_\alpha(0)/\alpha^2 \geq \sum_{i<j}W_{ij}(0)$ and
$
 \lim_{\alpha\to\infty} \inf\sigma(k_\alpha(0))  /\alpha^2 =\sum_{i<j}W_{ij}(0)$,
we have
\begin{align*}
 \sum_{i<j}W_{ij}(0) =& \lim_{\alpha\to0} \alpha^{-2}(u_\alpha, k_\alpha(0)u_\alpha)  \notag \\
 \geq& \liminf_{\alpha\to\infty} \Big(u_\alpha, \Big(\sum_{j\geq2} W_{1j}(x_j)  + \sum_{2\leq i<j\leq N}W_{ij}(x_i-x_j)\Big)u_\alpha\Big)
 \geq  \sum_{i<j}W_{ij}(0). \label{2.25-3}
\end{align*}
Then
\eq{sasa10}
 \sum_{i<j}W_{ij}(0) =
   \liminf_{\alpha\to\infty} \Big(u_\alpha, \Big(\sum_{j\geq2} W_{1j}(x_j)  + \sum_{2\leq i<j\leq N}W_{ij}(x_i-x_j)\Big)u_\alpha\Big)
   \end{equation}
   follows.
Suppose that
$
\d c_\epsilon=\liminf_{\alpha\to\infty} \int_{|{\bf X}}%\underbar {X} |>\epsilon} |u_\alpha({\bf X}}%\underbar {X} )|^2 \d{\bf X}}%\underbar {X}  >0$.
Then
\begin{align*}
\liminf_{\alpha\to\infty} \int_{\mathbb{R}^{d(N-1)}} \! \sum_{j\geq 2}(W_{1j}(x_j)-W_{1j}(0))
 |u_\alpha({\bf X}}%\underbar {X} )|^2 d{\bf X}}%\underbar {X}
  > c_\epsilon \sum_{j\geq 2}\sup_{|{\bf X}}%\underbar {X} |>\epsilon} (W_{1j}(x_j)-W_{1j}(0)) >0,
\end{align*}
which contradicts \eqref{sasa10}. Therefore \eqref{2.2400} holds.
 \qed

\textit{Proof of Theorem \ref{enhanced}:}


First we assume that $V \in C_0^\infty(\BR)$.
It is enough to show
$\mathcal{E}^V<\Xi^V$, since the uniform exponential decay $\|\Phi_\sigma}%{\varphi_{\rm g}(x)\|\leq C_\delta e^{-\delta|x|}$  is established in Lemma  \ref{expdecay} for sufficiently small $|\alpha/\kappa|$.
Assume $|\alpha| >\max\{\alpha_1,\alpha_2\}>0$.
Let $u_\alpha$ be a normalized ground state of $k(0)$.
By
$\Omega_1(a+b)\leq |a| + \Omega_1(b)$
 for $a,b\in\BR$,
we have
\begin{align}
 \ms U  h_{\rm eff}^0\ms U ^{-1}  \leq  \sqrt{-\Delta_1} + k(0). \label{dec5.2}
\end{align}
By (2) of (EN), there exists a normalized vector $v\in C_0^\infty(\BR)$ such that
\begin{align}
( v, (\sqrt{-\Delta} + NV) v ) < 0. \label{yy3}
\end{align}
We set
$\Psi(x_1,\cdots,x_N) = v(x_1)u_\alpha(x_2,\cdots,x_N)$.
Then, by \eqref{dec5.2}
\begin{align}
\label{yy4}  \mathcal{E}^V  & \leq  (\Psi, \ms U h_{\rm eff}^V\ms U ^{-1}\Psi)
 \leq  (  v, (\sqrt{-\Delta}+V)  v)  + {\cal E}^0
 + (\Psi,\sum_{j=2}^N V(x_1+x_j) \Psi).
\end{align}
Let
$
\d
  V_{j,\mathrm{smeared}}^\alpha(x_1) =
  \int_{\mathbb{R}^{d(N-1)}}
  V(x_j+x_1) |u_\alpha({\bf X}}%\underbar {X} )|^2d{\bf X}}%\underbar {X}
  $.
By Lemma~\ref{delta}, we have
$$
  \lim_{\alpha\to\infty}( \Psi, V(x_j+x_1) \Psi)
 =\lim_{\alpha\to\infty} ( v, V_{j,\mathrm{smeared}}^\alpha v)
  = ( v, Vv )$$ and then
by \kak{yy3} and \kak{yy4},
\begin{align}
\label{yyy4}  \mathcal{E}^V
 \leq  (  v, (\sqrt{-\Delta}+NV)  v)  +
 {\cal E}^0<{\cal E}^0
 \end{align}
follows for $\alpha>\alpha_3$ with some $\alpha_3>0$.
By this inequality and  Lemma \ref{ene}, we conclude that for $\alpha$ with
$|\alpha|>\alpha_c=\max\{\alpha_1,\alpha_2,\alpha_3\}$,
$\Xi^V - {\cal E}^V   \geq {\cal E}^0  -  {\cal E}^V > 0$.
Then the theorem  follows for $V\in C_0^\infty(\BR)$. 
For general $V$ we can prove the theorem by the same limiting argument as
\cite[Appendix]{hs08}. See Appendix \ref{ess}

\qed



