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PACS: 05.30.Jp, 03.75.Fi, 67.40-w 
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Bose-Einstein Condensation, Superstable Potentials, One-Particle
Excitations
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\documentclass[twocolumn,showpacs,showkeys,preprintnumbers,prl,amsmath,amssymb]{revtex4}

%\documentclass[preprint,showpacs,showkeys,preprintnumbers,amsmath,amssymb]{revtex4}

\usepackage{graphicx}
\usepackage{bbm}

\newcommand{\N}{\mathbbm{N}}
\newcommand{\R}{\mathbbm{R}}
\newcommand{\D}{\Delta}
\renewcommand{\L}{\Lambda}
\newcommand{\eg}{\textit{e.g.}\ }
\newcommand{\ie}{\textit{i.e.}\ }

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}

\begin{document}

\preprint{Preprint-KUL-TF-2002/05}

\title{Proof of Bose-Einstein Condensation for Interacting Gases} 

\author{J. Lauwers}
\email{joris.lauwers@fys.kuleuven.ac.be}

\author{A. Verbeure}
\email{andre.verbeure@fys.kuleuven.ac.be}

\author{V. A. Zagrebnov}
\email{zagrebnov@cpt.univ-mrs.fr}
\altaffiliation[on leave of absence from ]{
Universit\'e de la M\'editerran\'ee and Centre de Physique Th\'eorique, 
CNRS-Luminy-Case 907, 13288 Marseille, Cedex 09, France
}

\affiliation{
Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven,   
Celestijnenlaan 200D, B-3001 Leuven, Belgium
}


\date{May 22, 2002}

\begin{abstract}
Using a specially tuned  mean-field Bose gas as a reference system, we
establish a positive lower bound on the condensate density for continuous Bose
systems with superstable two-body interactions and a finite gap in the one-particle
excitations spectrum, \ie we prove for the first time standard homogeneous
Bose-Einstein condensation for such interacting systems. 
\end{abstract}

\pacs{
05.30.Jp,	% Boson systems
03.75.Fi,	% Phase coherent atomic ensemble (Bose condensation)
67.40-w		% Boson degeneracy and superfluidity of 4He
}

\keywords{Bose-Einstein Condensation, Superstable Potentials, One-Particle
Excitations}

\maketitle

\section{introduction}


The long-standing problem of proving the existence of a Bose-Einstein
condensation (BEC) in non-ideal real Bose gases with a standard two-body 
interaction has recently been warmed up by the great success of observing this 
phenomenon in trapped gases. Notice that much earlier BEC has been observed in 
liquid $\mathrm{He^4}$, which however remained always under discussion.

In the present note we announce our result together with a sketch of the proof
about the existence of conventional BEC in realistic Bose gases with superstable 
two-body interactions and with gap in the one-particle excitation energy 
spectrum. To the best of our knowledge, this is a first proof of its sort for 
homogeneous systems. We are not using any scaling limits, (\eg type van der 
Waals limits \cite{lebowitz:1966,lieb:1966,buffet:1983b}) or truncation of 
particle interactions \cite{dorlas:1993,zagrebnov:2001}. We prove that BEC 
occurs by constructing a positive lower bound for the condensate density which 
is valid for low enough temperature and appropriate large density of particles.
 
We consider a gas of interacting Bosons in cubic boxes $\L = L
\times L \times L \subset \R^3$ with periodic boundary conditions. 
We look here in detail at the $3$ dimensional case, and comment later
on the case of other dimensions. Denote by $V= L^3$ the
volume of the box $\L$. The Hamiltonian for the volume $\L$ of this system reads
\begin{equation} \label{H-int}
H^\D_{\L, g} = T^\D_\L -\mu N_\L + g U_\L, \qquad g > 0,
\end{equation} 
where $T^\D_\L$ is the kinetic energy with gap $\Delta > 0$ in its spectrum,
\begin{equation}\label{T-Delta}
T^\D_\L = \sum_{k \in \L^*} \frac{\hbar^2k^2}{2m} a^\dagger_k a_k -\Delta
a^\dagger_0 a_0.
\end{equation}
The sum $k$ runs over the set $\L^*$, dual to $\L$, \ie $\L^* = \left\{ k
\in \R^3; k_\alpha = 2\pi n_\alpha /L; n_\alpha = 0,\pm 1,\ldots; \alpha
=1,2,3 \right \}$. 
 The operators $a^\dagger_k$ and $a_k$ are the Bose
creation and annihilation operators for mode $k$. As usual, the number 
operators are denoted by $N_k = a^\dagger_ka_k $, $N_\L = \sum_{k \in \L^*}N_k$
is the total number operator in the volume $\L$.

We assume \textit{a priori} the presence of a gap $\D$ in the one-particle
excitations spectrum, isolating the lowest energy level. In this letter we make 
no comments about the origin of this spectral gap, however its presence can be 
realised in several ways, a gap can be created using attractive boundary 
conditions \cite{robinson:1976, landau:1979} or part of the interparticle 
interaction could have caused this gap and as such being effectively 
incorporated \cite{zagrebnov:2001}.

Of course $\mu$ is the chemical potential and
the interaction between the particles is modelled by the two-body interaction
term
\begin{equation}\label{U-Lambda}
U_\Lambda =  \frac{1}{2}\int_{\Lambda^2}\!\mathrm{d}x\mathrm{d}y\ a^\dagger (x)
a^\dagger (y) v( x - y) a(y)a(x),
\end{equation}
where 
$a^\dagger (x),a^\dagger (y)$ and $a(y),a(x)$ are the creation and annihilation 
operators for the Bose particles at $x,y \in \R^3$. The interaction potential 
$v$ is assumed to be 
spherically symmetric,  
superstable \cite{ruelle:1969}, \ie it satisfies the inequality
\begin{equation}\label{s-stable}
\sum_{1 \leq i < j \leq n}v(x_i - x_j) \geq 
\frac{A}{2V}n^2 - B n
\end{equation}
for some constants $A > 0$, $B \geq 0$, and all $n \in \N$, $x_i \in \L$.
Consequently, the interaction term (\ref{U-Lambda}) satisfies
\begin{equation}\label{sstab2}
U_\L \geq \frac{A}{2V}N_\L^2 - B N_\Lambda. 
\end{equation}
This superstability property is, together with the spectral gap (\ref{T-Delta}), 
the physical foundation of our proof.  Intuitively, one might understand that 
condensation in the groundstate (\ie $k=0$), which is energetically isolated by 
a gap $\Delta$ can survive the switching-on of a non-catastrophic interaction, 
and that fluctuations must be of a macroscopical size to overcome this gap 
and lift particles out of the isolated groundstate.
More technical ingredients of the proof are the convexity
properties of thermodynamical potentials, such as the pressure, and the
use of an optimal choice for the constants $A,B$ in the superstability
criterion (\ref{s-stable}). Indeed, it was proved \cite{ruelle:1969} that 
continuous $L^1$-functions of positive type $v : \R^3 \to \R$ are superstable 
potentials if and only if
\begin{equation}\label{vhat-0}
 \hat{v}(0) = \int_{\R^3}\! \mathrm{d}x\ v(x) > 0.
\end{equation}
Moreover, Lewis, Pul\`e, and de Smedt \cite{lewis:1984} proved the existence of 
optimal constants $A = \hat{v}(0)(1 - \epsilon)$ and $B = v(0)/2$ in 
(\ref{s-stable}) for this type of potentials. Here, $\epsilon > 0$ is an arbitrarily
small positive constant. This optimal choice is of determining 
importance in our proof of Bose-Einstein condensation.

\section{sketch of proof}

The main idea of the proof is to estimate the Bose condensate of the full
model (\ref{H-int}) by the condensate of a particularly chosen reference system
for which one knows the occurrence of condensation. The clever choice of this 
reference system is the subtle point of our proof. 
The reference system is a so-called mean-field Bose gas, an exactly solvable 
model of Bosons \cite{huang:1967, davies:1972, fannes:1980b, buffet:1983,
berg:1984,papoyan:1986, lewis:1988} (for a review see \cite{zagrebnov:2001}) 
defined by Hamiltonians as
\begin{equation}\label{H-MF}
H^{\Delta}_{\Lambda,g,\lambda} = T_\Lambda^\Delta -\mu N_\Lambda +
g\frac{\lambda}{2V}N_\Lambda^2.
\end{equation}
The kinetic energy operator $T_\Lambda^\Delta$  (\ref{T-Delta}) is as 
for the general interacting system (\ref{H-int}), but the interaction term
(\ref{U-Lambda}) is replaced by a mean-field interaction term.

The reference system (\ref{H-MF}), a mean-field Bose gas, emerges as 
the \textit{van der Waals} limit of the fully interacting system
\cite{lebowitz:1966,lieb:1966, buffet:1983b}.  In that case,
the constant $\lambda$ equals $\hat{v}(0)$ (\ref{vhat-0}), which means that the
van der Waals limit reduces the full interaction to the $\hat{v}(0)$
contribution. In our proof we tune the constant $\lambda$ in order to get the
best possible lower bound for the condensate density of the full interaction
model.


Remark that our reference system (\ref{H-MF}) does show Bose condensation for
large enough densities (\ie for $\mu$ large) at any given temperature. Moreover,
systems of the type of our reference system have better properties than the
ideal Bose gas which is in many ways a \textit{pathological} model, \eg in the
sense that there is no equivalence of ensembles \cite{cannon:1973,lewis:1974} 
and in the sense that the chemical potential has a zero upperbound in order to 
safeguard thermodynamical stability.  These are the reasons for our strategy of 
using the free of those pathologies reference system \eqref{H-MF}.

\textit{Thermodynamic properties of reference the system (\ref{H-MF}).} It is
well known that our reference system (\ref{H-MF}) is a soluble models. In the
case of vanishing gap $\D = 0$, the complete solution can be found at several
places in the literature \cite{huang:1967,davies:1972,fannes:1980b,buffet:1983,
berg:1984,papoyan:1986,lewis:1988,zagrebnov:2001}. In particular there is
condensation for all dimensions $D \geq 3$ at any temperature  for densities
large enough. It is a student exercise to work out now the case with gap 
$\D > 0$. There is one main difference with the gapless case, namely the
presence of the gap provokes a shift in the chemical potential and its
treshholds, and one gets condensation in all dimensions $D \geq 1$. One derives
straightforwardly that for the reference model (\ref{H-MF}) one gets
condensation for all values of the chemical potential satisfying,    
\begin{equation*}
\mu > g\lambda \rho^{P}(\beta,-\D) -\D.
\end{equation*}
where $\rho^{P}(\beta,-\D)$ is the total density of the perfect Bose Gas
(PBG) at inverse temperature $\beta$ and chemical potential equal to $-\D$. 
Moreover, the condensate density  
\begin{equation*}
\rho^{\D }_{0,g, \lambda}(\beta,\mu) = \lim_{V \to \infty}\frac{1}{V}
 \langle N_0 
\rangle_{H_{\L,g, \lambda}^\D}(\beta,\mu),
\end{equation*}
\ie the density at the zero mode in the thermodynamic limit 
$(V \to \infty)$ of 
the grand-canonical Gibbs states $\langle . \rangle_{H_{\L}}^{(\beta,\mu)}$, in 
volumes $\L$ for a certain choice of temperature and chemical 
potential $(\beta,\mu)$ and Hamiltonian $H_\L$, is
explicitly given by
\begin{equation}\label{r0-mf-D} 
  \rho^{\D}_{0,g,\lambda} = \frac{\mu + \D}{g\lambda}
  -\rho^{P}(\beta,-\D).
\end{equation}  
One also computes the total density for given $(\beta,\mu)$ to be
\begin{equation}\label{r-mf-D}
  \rho^{\D}_{g,\lambda}  =  \lim_{V \to \infty}\frac{1}{V}\langle 
 N_\L \rangle_{H_{\L,g,\lambda}^{\D}} = \frac{\mu}{g\lambda}.
\end{equation}

To prove Bose condensation in the full model (\ref{H-int}), we subtract from
\eqref{U-Lambda} the van
der Waals-like behaving part of the interaction proportional to 
$N_\L^2/2V$, tune this with a factor, taking into account the optimal stability 
constants, and add it to the kinetic-energy term. The latter serves as our reference 
system \eqref{H-MF}, from which we establish a lower bound on the
condensate density $\rho^\Delta_{0,g}(\beta,\mu) =\lim_{V \to \infty}\langle 
N_0/V\rangle_{H_{\Lambda, g}^{\D}}$ in the fully interacting system (\ref{H-int}).  
In the lemma below, a lower bound on $\rho^\Delta_{0,g}(\beta,\mu)$ is given. 
\begin{lemma}\label{lemma-lb}
The condensate density $\rho^\Delta_{0,g}(\beta,\mu)$ in the thermodynamic limit
of grand-canonical Gibbs states of interacting systems (\ref{H-int}) with 
superstable two-body potentials $v$ (\ref{U-Lambda}), has the following lower
bound:
\begin{multline}\label{lb}
%\begin{split}
\rho^\Delta_{0,g}(\beta,\mu) \geq  \frac{\mu}{g\hat{v}(0)(1 -\epsilon)} +
\frac{g\hat{v}(0)}{2\Delta}\left(\rho^{P}(\beta,-\Delta)\right)^2\\ 
 -\frac{(\mu +\Delta)^2}{2\Delta g \hat{v}(0)} \frac{\epsilon}{(1-\epsilon)^2} 
 - \frac{\mu +\Delta}{\Delta(1-\epsilon)}\rho^{P}(\beta,-\Delta) \\ 
 - \frac{g v(0)}{2\Delta}\rho^{(\Delta = 0)}(\beta,\mu) - \rho^{P}_c(\beta).
%\end{split}
\end{multline}
Here $\rho_g^{(\Delta = 0)}(\beta,\mu)$ is the total
density of the interacting gas without gap \eqref{H-int}, and $\epsilon > 0$ is
an arbitrarily small constant. $\rho^{P}(\beta,-\Delta)$ refers to the total 
density of the PBG at inverse temperature $\beta$ and chemical potential 
$\mu = -\D$, $\rho_c^{P}(\beta)$ is the critical density of the PBG. The bound 
is valid for values $\mu > g\hat{v}(0)(1-\epsilon)\rho_c^{P}(\beta)$, and
dimensions $D = 3$.
\end{lemma}

\textit{Idea of the proof}.
Using the Bogoliubov convexity inequality \cite{zagrebnov:2001} one gets:
\begin{equation}\label{b-conv}
\frac{g}{V}\langle W_\L^\lambda \rangle_{H^{\D}_{\L,g}} \leq
p_\Lambda[H^{\D}_{\L,g,\lambda}] - p_\Lambda[H^{\Delta}_{\Lambda,g}]
\leq \frac{g}{V}\langle W_\Lambda^\lambda \rangle_{H^{\D}_{\L,g,\lambda}}.
\end{equation}
This gives upper and lower bounds on the difference of the thermodynamic 
pressure $p_\L[H_\L]$ of the mean-field reference Bose gas (\ref{H-MF}) and 
the full model (\ref{H-int}).  The operator $W_\L^\lambda$ is the difference 
between the interactions of the fully interacting and the mean-field Bose gases,
\ie $W_\Lambda^\lambda = U_\Lambda - \frac{\lambda}{2V}N_\L^2$.   
The expectation values in (\ref{b-conv}) can be estimated using, 
for the lower bound, the superstability properties of the interaction, and, 
for the upper bound, the properties of the equilibrium states of the 
mean-field reference Bose gas.
The lower bound in \eqref{b-conv} follows from  (\ref{sstab2}), and from
the tuning of the interaction parameter $\lambda$ for the mean-field reference 
Bose gas (\ref{H-MF}) to the constant $A$ in (\ref{sstab2}),
\begin{equation}\label{b1}
\frac{g}{V}\langle W_\Lambda^{A} \rangle_{H^{\D}_{\L,g}} \geq 
-\frac{gB}{V}\langle N_\Lambda \rangle_{H^{\D}_{\L,g}}.
\end{equation}  
Using the mode by mode gauge invariance of the Gibbs states of the mean-field
Bose gas we arrive at the following upper bound for the pressure difference
(\ref{b-conv}),
\begin{equation}\label{b2}
\frac{g}{V}\langle W_\L^{A} \rangle_{H^{\D}_{\L,g,A}}
\leq  \frac{g}{V^2} \langle C N_\L^2 
- \frac{\hat{v}(0)}{2}N_0^2 \rangle_{H^{\D}_{\L,g,A}},
\end{equation}
where $C = \hat{v}(0) -A/2$.
It follows from the properties of the mean-field Bose gas that the expectation
values in the rhs of (\ref{b2}) in the limit $(V\to\infty)$ are
given by $gC(\rho^{\D}_A(\beta,\mu))^2 - 
g\hat{v}(0)(\rho^{\D}_{0,A}(\beta,\mu))^2/2$.

The pressure $p_\L[H^\D_\L]$ is an increasing convex function of
$\D \geq 0$. Since the condensate density $\rho^\Delta_{0,g}(\beta,\mu)$ is 
the derivative 
of the pressure with respect to $\D$, by convexity we 
find a lower bound for the condensate density $\rho^\Delta_{0,g}(\beta,\mu)$, 
given by
\begin{equation}\label{c1}
\frac{1}{V}\langle N_0 \rangle_{H_{\L, g}^{\D}} 
\geq  \frac{p_\L[H^\D_{\L,g}] - 
p_\L[H^{(\D =0)}_{\L,g}]}{\D}.
\end{equation}
Analogously, by virtue of the same convexity property, 
the difference of the pressures between the mean-field Bose gas 
with gap and without gap, is bounded from below by the condensate density 
for the mean-field gas without gap, \ie
\begin{equation}\label{c2}
\frac{p_\Lambda[H^{\D}_{\L,g,A}] - p_\L[H^{(\D =0)}_{\L,g,A}]}{\D} \geq 
\frac{1}{V}\langle N_0 \rangle_{H_{\L,g,A}^{(\D=0)}} .
\end{equation}
Adding up these two inequalities and using the Bogoliubov convexity inequality
(\ref{b-conv}) once at $\D >0$ and once at $\D = 0$, together with the
bounds (\ref{b1}) and (\ref{b2}), yields the 
following lower bound for the condensate density $\rho^\D_{0,g}(\beta,\mu)$ 
in the thermodynamic limit ($V \to \infty$), \ie
\begin{multline}\label{lb-1}
\rho^\D_{0,g}(\beta,\mu)  \geq  \rho^{(\D=0)}_{0,g,A}(\beta,\mu)
+  g\frac{\hat{v}(0)}{2\Delta}(\rho^{\Delta}_{0,g,A}(\beta,\mu))^2\\
 -\frac{g}{\D}\left( B \rho^{(\Delta=0)}_{g}(\beta,\mu) +
C(\rho^{\D}_{g,A}(\beta,\mu))^2 \right).
\end{multline}
The lower bound (\ref{lb}) is now found using the explicit expressions for total 
density and the condensate 
density of the mean-field Bose gas with gap, (\ref{r-mf-D}) and (\ref{r0-mf-D}),
and the well-known expression for the condensate density in the gapless
mean-field model, \ie if $\mu > g \lambda\rho^{P}_c(\beta)$, 
\begin{equation*}
\rho^{(\D =0)}_{0,g,\lambda} = \lim_{V \to \infty}\frac{1}{V}\langle N_0
  \rangle_{H_{\L,g,\lambda}^{(\D =0)}} = \frac{\mu}{g\lambda}
  \-\rho_c^{P}(\beta),
\end{equation*}
where $\rho^{P}_c(\beta)$ is the critical density for the perfect Bose Gas at
inverse temperature $\beta$.
As a last step, we need the optimal superstability constants for continuous 
$L^1$ potentials of positive type \cite{lewis:1984}, \ie we take
$A = (1-\epsilon)\hat{v}(0)$, and $B = v(0)/2$,
and get the expression for the lower bound (\ref{lb}) in the lemma.
\hfill \textit{QED}

Now we get our main result:
\begin{theorem}\label{theorem}
Consider a three dimensional system of interacting Bose particles (\ref{H-int}), with
a superstable two-body interaction.
Take any $\eta > 0$, fix a temperature and a chemical potential $(\beta,\mu)$
such that $\mu > g\hat{v}(0)(\rho^{P}_c(\beta) + 2 \eta)$,
then there exists  a minimal gap $\Delta_{min}$ such that for all 
$\Delta \geq \Delta_{min}$,
\begin{multline*}
\left|\frac{g\hat{v}(0)}{2\Delta}\left(\rho^{P}(\beta,-\Delta)\right)^2  
- \frac{g v(0)}{2\Delta}\rho_g^{(\Delta = 0)}(\beta,\mu) 
\right. \\ \left.
- 2\frac{\mu +
\Delta}{\Delta}\rho^{P}(\beta,-\Delta)\right| < \eta
\end{multline*} 
For these values of the gap $\Delta$ we have $
\rho^\Delta_{0,g}(\beta,\mu) > 0$, and hence proved condensation.
\end{theorem}

\textit{Proof.} It is a consequence of Lemma~\ref{lemma-lb} and 
the fact that $\epsilon >0$ can be chosen arbitrarily small. 
\hfill \textit{QED}


\section{discussion}
First of all let us remark that our proofs hold 
without any change in all dimensions $D \geq 3$. For $D = 1$ or $2$ a similar 
lower bound on the condensate density can be derived, on the basis of modified 
convexity arguments (\ref{c1})--(\ref{c2}), \ie one has to consider pressure 
differences of the form $p_\Lambda[H^{\D}_{\L}] - p_\L[H^{\D_0}_{\L}]$, with 
$0 < \D_0 < \D$, instead of with $\D_0 =0$ in (\ref{c1})--(\ref{c2}). This yields 
the substitution of $\rho^{\D_0}_{0,g,A}(\beta,\mu)$ and
$\rho_g^{\D_0}(\beta,\mu)$ for $\rho^{(\D=0)}_{0,g,A}(\beta,\mu)$ and
$\rho_g^{(\D=0)}(\beta,\mu)$ in (\ref{lb-1}). Hence, also in one and two 
dimensional interacting Bose gases with gap \eqref{H-int}, Bose 
condensation is proved, in contrast to the Bogoliubov--Hohenberg 
theorem \cite{bratteli:1996} which yields the absence of BEC for 
$1D$ or $2D$ translation invariant continuous Bose systems without gap.   


Remark that our bound (\ref{lb}) contains a term $\rho_g^{(\D=0)}(\beta,\mu)$, 
the total density of the fully interacting model (\ref{H-int}) without gap 
$(\D =0)$, which is not explicitly known. This term can be 
eliminated using an alternative reference system, yielding a new bound. 
This alternative reference system is an adaptation of the original reference 
systems (\ref{H-MF}) with an extra interaction term, linear in the total 
number operator $N_\L$ (cf.\ \eqref{sstab2}), \ie yielding a shift in the 
chemical potential. The derivation of this lower bound on the Bose condensate 
density, is then as above. It is explicitly given by
\begin{multline}\label{lb-a}
\rho^\Delta_{0,g}(\beta,\mu) \geq  \frac{2\mu + gv(0)}{g\hat{v}(0)(1 -\epsilon)} +
\frac{g\hat{v}(0)}{2\Delta}\left(\rho^{P}(\beta,-\Delta)\right)^2\\ 
 -\frac{(2\mu +2\D + gv(0))^2}{2\D g \hat{v}(0)} \frac{\epsilon}{(1-\epsilon)^2} 
  - \rho^{P}_c(\beta) \\
 - \frac{2\mu + 2\D + gv(0)}{\D \hat{v}(0)(1-\epsilon)}\left(\frac{v(0)}{2}+ 
 \rho^{P}(\beta,-\Delta)\right), 
\end{multline}
for $\mu \geq g\hat{v}(0)1-\epsilon)\rho^{P}_c(\beta)$.
The properties of this bound are completely analogous to the properties of
Lemma~\ref{lemma-lb}. However, as all quantities in this bound are
explicitly known, it is a useful one to compute numerical estimations on the 
condensate density. 

\begin{figure}[h]
\includegraphics[width=0.3\textwidth]{graph2.1}
%graph2.1 is a MetaPost generated EPS figure with internal LaTeX. 
\caption{\label{fig1} $(\mu,\D)$-graph of the lower bound.}
\end{figure}
An example of such a computation is represented in Fig.~\ref{fig1}.
In this $(\mu,\D)$ graph the values of the lower bound are given, for a
choice of the parameters $\beta = g = v(0) = \hat{v}(0) = \frac{\hbar^2}{2m} = 1$
and $\epsilon = 0.01 $. The lowest line $(a)$ is the line for which the lower 
bound (\ref{lb-a}) is zero, \ie above this curve we have BEC, the higher lines 
$(b)$,  $(c)$, and $(d)$, are the contours where the numerical values of the 
lower bound are $15,30$ resp.\ $45$. Clearly, the density of the 
condensate increases with increasing $\mu$ (\ie increasing total density) 
and increasing gap $\D$.
For a given chemical potential $\mu$, there is a minimal value for the 
gap $\D$ (cf.~Theorem~\ref{theorem}) below which we cannot prove condensation.
Notice that, if one fixes $\D$ larger than this minimal value, our lower bound
yields BEC between a $\mu_{min}$ and a $\mu_{max}$ of the chemical potential.
The $\mu_{max}$ increases with decreasing $\epsilon$. 
Considering the high $\mu$ 
regime and taking $\epsilon$ of order $\mu^{-2}$, one estimates the absolute 
lowest bound on $\D$ in order to have a positive lower bound on de condensate 
density (\ref{lb-a}), \ie condensation occurs when 
\begin{equation*}
\D > g\left(\frac{v(0)}{2} + \hat{v}(0)\rho^{P}(\beta,-\D)\right),
\end{equation*}
meaning that for non-zero interaction (\ref{U-Lambda}) $(g > 0)$, there is a
non-zero lower bound on the gap-width. This minorant is proportional to the
coupling constant $g \geq 0$. 

Finally we want to stress that our results are for homogeneous systems. On the
other hand we have to mention here the interesting recent result about the
condensation for trapped Bose gases \cite{lieb:2002}, \ie for inhomogeneous
systems where also a rigorous proof is given of Bose condensation in a realistic
setting.

\begin{acknowledgments}
 J. L. gratefully acknowledges financial support from K.U.Leuven 
 grant FLOF-10408, and V. A. Z. acknowledges ITF K.U.Leuven for hospitality.
\end{acknowledgments}

%\bibliographystyle{apsrev}
%\bibliography{/loc_home/lauwers/artikels/biblio}

\begin{thebibliography}{20}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
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  \def\bibnamefont#1{#1}\fi
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  \def\citenamefont#1{#1}\fi
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\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

\bibitem[{\citenamefont{Lebowitz and Penrose}(1966)}]{lebowitz:1966}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Lebowitz}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{O.}~\bibnamefont{Penrose}},
  \bibinfo{journal}{J. Math. Phys.} \textbf{\bibinfo{volume}{7}},
  \bibinfo{pages}{98 } (\bibinfo{year}{1966}).

\bibitem[{\citenamefont{Lieb}(1966)}]{lieb:1966}
\bibinfo{author}{\bibfnamefont{E.~H.} \bibnamefont{Lieb}}, \bibinfo{journal}{J.
  Math. Phys.} \textbf{\bibinfo{volume}{7}}, \bibinfo{pages}{1016 }
  (\bibinfo{year}{1966}).

\bibitem[{\citenamefont{Buffet et~al.}(1983)\citenamefont{Buffet, {de Smedt},
  and Pul{\`e}}}]{buffet:1983b}
\bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Buffet}},
  \bibinfo{author}{\bibfnamefont{Ph.}~\bibnamefont{{de Smedt}}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{J.~V.}
  \bibnamefont{Pul{\'e}}}, \bibinfo{journal}{J. Phys. A: Math. Gen.}
  \textbf{\bibinfo{volume}{16}}, \bibinfo{pages}{4309 } (\bibinfo{year}{1983}).

\bibitem[{\citenamefont{Dorlas et~al.}(1993)\citenamefont{Dorlas, Lewis, and
  Pul{\'e}}}]{dorlas:1993}
\bibinfo{author}{\bibfnamefont{T.~C.} \bibnamefont{Dorlas}},
  \bibinfo{author}{\bibfnamefont{J.~T.} \bibnamefont{Lewis}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.~V.} \bibnamefont{Pul{\'e}}},
  \bibinfo{journal}{Commun. Math. Phys.} \textbf{\bibinfo{volume}{156}},
  \bibinfo{pages}{37 } (\bibinfo{year}{1993}).

\bibitem[{\citenamefont{Zagrebnov and Bru}(2001)}]{zagrebnov:2001}
\bibinfo{author}{\bibfnamefont{V.~A.} \bibnamefont{Zagrebnov}}
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{J.-B.} \bibnamefont{Bru}},
  \bibinfo{journal}{Phys. Rep.} \textbf{\bibinfo{volume}{350}},
  \bibinfo{pages}{291 } (\bibinfo{year}{2001}).

\bibitem[{\citenamefont{Robinson}(1976)}]{robinson:1976}
\bibinfo{author}{\bibfnamefont{D.~W.} \bibnamefont{Robinson}},
  \bibinfo{journal}{Commun. Math. Phys.} \textbf{\bibinfo{volume}{50}},
  \bibinfo{pages}{53 } (\bibinfo{year}{1976}).

\bibitem[{\citenamefont{Landau and Wilde}(1979)}]{landau:1979}
\bibinfo{author}{\bibfnamefont{L.~J.} \bibnamefont{Landau}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{I.~F.} \bibnamefont{Wilde}},
  \bibinfo{journal}{Commun. Math. Phys.} \textbf{\bibinfo{volume}{70}},
  \bibinfo{pages}{43 } (\bibinfo{year}{1979}).

\bibitem[{\citenamefont{Ruelle}(1969)}]{ruelle:1969}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Ruelle}},
  \emph{\bibinfo{title}{Statistical Mechanics, Rigorous Results}}
  (\bibinfo{publisher}{W. A. Benjamin, Reading}, \bibinfo{year}{1969}).

\bibitem[{\citenamefont{Lewis et~al.}(1984)\citenamefont{Lewis, Pul{\`e}, and
  {de Smedt}}}]{lewis:1984}
\bibinfo{author}{\bibfnamefont{J.~T.} \bibnamefont{Lewis}},
  \bibinfo{author}{\bibfnamefont{J.~V.} \bibnamefont{Pul{\`e}}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{{de
  Smedt}}}, \bibinfo{journal}{J. Stat. Phys.} \textbf{\bibinfo{volume}{35}},
  \bibinfo{pages}{381 } (\bibinfo{year}{1984}).

\bibitem[{\citenamefont{Huang}(1967)}]{huang:1967}
\bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Huang}},
  \emph{\bibinfo{title}{Statistical Mechanics}} (\bibinfo{publisher}{Wiley,
  London}, \bibinfo{year}{1967}).

\bibitem[{\citenamefont{Davies}(1972)}]{davies:1972}
\bibinfo{author}{\bibfnamefont{E.~B.} \bibnamefont{Davies}},
  \bibinfo{journal}{Commun. Math. Phys.} \textbf{\bibinfo{volume}{28}},
  \bibinfo{pages}{69 } (\bibinfo{year}{1972}).

\bibitem[{\citenamefont{Fannes and Verbeure}(1980)}]{fannes:1980b}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Fannes}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Verbeure}},
  \bibinfo{journal}{J. Math. Phys.} \textbf{\bibinfo{volume}{21}},
  \bibinfo{pages}{1809 } (\bibinfo{year}{1980}).

\bibitem[{\citenamefont{Buffet and Pul{\`e}}(1983)}]{buffet:1983}
\bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Buffet}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.~V.} \bibnamefont{Pul{\`e}}},
  \bibinfo{journal}{J. Math. Phys.} \textbf{\bibinfo{volume}{24}},
  \bibinfo{pages}{1608 } (\bibinfo{year}{1983}).

\bibitem[{\citenamefont{{Van Den Berg} et~al.}(1984)\citenamefont{{Van Den
  Berg}, Lewis, and {de Smedt}}}]{berg:1984}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{{van den Berg}}},
  \bibinfo{author}{\bibfnamefont{J.~T.} \bibnamefont{Lewis}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{{de Smedt}}},
  \bibinfo{journal}{J. Stat. Phys.} \textbf{\bibinfo{volume}{37}},
  \bibinfo{pages}{697 } (\bibinfo{year}{1984}).

\bibitem[{\citenamefont{Papoyan and Zagrebnov}(1986)}]{papoyan:1986}
\bibinfo{author}{\bibfnamefont{Vl.~V.} \bibnamefont{Papoyan}} 
\bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{V.~A.} \bibnamefont{Zagrebnov}},
  \bibinfo{journal}{Theor. Math. Phys.} \textbf{\bibinfo{volume}{69}},
  \bibinfo{pages}{1240 } (\bibinfo{year}{1986}).

\bibitem[{\citenamefont{Lewis et~al.}(1988)\citenamefont{Lewis, Pul{\'e}, and
  Zagrebnov}}]{lewis:1988}
\bibinfo{author}{\bibfnamefont{J.~T.} \bibnamefont{Lewis}},
  \bibinfo{author}{\bibfnamefont{J.~V.} \bibnamefont{Pul{\'e}}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{V.~A.}
  \bibnamefont{Zagrebnov}}, \bibinfo{journal}{Helv. Phys. Acta}
  \textbf{\bibinfo{volume}{61}}, \bibinfo{pages}{1063 } (\bibinfo{year}{1988}).

\bibitem[{\citenamefont{Cannon}(1973)}]{cannon:1973}
\bibinfo{author}{\bibfnamefont{J.~T.} \bibnamefont{Cannon}},
  \bibinfo{journal}{Commun. Math. Phys.} \textbf{\bibinfo{volume}{29}},
  \bibinfo{pages}{89 } (\bibinfo{year}{1973}).

\bibitem[{\citenamefont{Lewis and Pul{\`e}}(1974)}]{lewis:1974}
\bibinfo{author}{\bibfnamefont{J.~T.} \bibnamefont{Lewis}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.~V.} \bibnamefont{Pul{\`e}}},
  \bibinfo{journal}{Commun. Math. Phys.} \textbf{\bibinfo{volume}{36}},
  \bibinfo{pages}{1 } (\bibinfo{year}{1974}).

\bibitem[{\citenamefont{Bratteli and Robinson}(1996)}]{bratteli:1996}
\bibinfo{author}{\bibfnamefont{O.}~\bibnamefont{Bratteli}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{D.~W.} \bibnamefont{Robinson}},
  \emph{\bibinfo{title}{Operator Algebras and Quantum Statistical Mechanics 2}}
  (\bibinfo{publisher}{Springer-Verlag, Berlin}, \bibinfo{year}{1996}).

\bibitem[{\citenamefont{Lieb and Seiringer}(2002)}]{lieb:2002}
\bibinfo{author}{\bibfnamefont{E.~H.} \bibnamefont{Lieb}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Seiringer}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{88}},
  \bibinfo{pages}{170409} (\bibinfo{year}{2002}).

\end{thebibliography}


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