0$. This
holds for all small enough $Y$, provided
$\alpha+5\beta<2$ which follows from the conditions below.
\item $\alpha>0$ in order that $\varepsilon\to 0$ for $Y\to 0$.
\item $3\beta-1>0$ in order that $Y^{-1}(a/\ell)^{3}\to 0$ for for $Y\to
0$.
\item $1-3\beta+\gamma>0$ in order that
$Y(\ell/a)^{3}(R^3-R_{0}^3)/\ell^3\to 0$ for for $Y\to 0$.
%END CHANGE
\item $1-\alpha-2\beta-\gamma>0$ to control the last factor in
(\ref{Kformula2}).
\end{itemize}
Taking
\begin{equation}\label{exponents}
\alpha=1/17,\quad \beta=6/17,\quad \gamma=3/17
\end{equation}
all these conditions are satisfied, and
%BEGIN CHANGE
\begin{equation}
\alpha= 3\beta-1=1-3\beta+\gamma=1-\alpha-2\beta-\gamma=1/17.
\end{equation}
It is also clear that
$2R/\ell\sim Y^{\gamma/3}=Y^{1/17}$, up to higher order terms.
%END CHANGE
This completes the proof of Theorems \ref{lbth} and \ref{lbthm2}, for
the case of potentials with finite range. By optimizing the
proportionality constants in (\ref{ans}) one can show that $C=8.9$ is
possible in Theorem \ref{lbth} \cite{S1999}. The extension to
potentials of infinite range but finite scattering length is obtained
by approximation by finite range potentials, controlling the change of
the scattering length as the cut-off is removed. See Appendix A in
\cite{LY2d} and Appendix B in \cite{LSY1999} for details. We remark
that a slower decrease of the potential than $1/r^3$ implies infinite
scattering length.
\hfill$\Box$\bigskip
The exponents (\ref{exponents}) mean in particular that
\begin{equation}a\ll R\ll \rho^{-1/3}\ll \ell \ll(\rho
a)^{-1/2},\end{equation}
whereas Dyson's method required $R\sim \rho^{-1/3}$ as already explained.
The condition $\rho^{-1/3}\ll \ell$ is required in order to have many
particles in each box and thus $n(n-1)\approx n^2$. The condition
$\ell \ll(\rho a)^{-1/2}$ is necessary for a spectral gap
gap $\gg e_{0}(\rho)$ in Temple's inequality. It is also clear that
this choice of $\ell$ would lead to a far too big
energy and no bound for $ e_{0}(\rho)$ if we had chosen Dirichlet instead of
Neumann boundary
conditions for the cells. But with the latter the method works!
%%%%%%%%%%
\bigskip
\section{The Dilute Bose Gas in 2D} \label{sect2d}
In contrast to the three-dimensional theory, the two-dimensional Bose
gas began to receive attention only relatively late. The first
derivation of the correct asymptotic formula was, to our knowledge,
done by Schick \cite{schick} for a gas of hard discs. He found
\begin{equation}
e(\rho) \approx 4\pi \mu \rho |\ln(\rho a^2) |^{-1}.
\label{2den}
\end{equation}
This was accomplished by an infinite summation of `perturbation series'
diagrams. Subsequently, a corrected modification of \cite{schick} was
given in \cite{hines}. Positive temperature extensions were given in
\cite{popov} and in \cite{fishho}. All this work involved an analysis in
momentum space, with the exception
of a method due to one of us that works directly in configuration space
\cite{Lieb63}. Ovchinnikov \cite{Ovch} derived \eqref{2den} by using,
basically, the method in \cite{Lieb63}. These derivations require
several unproven assumptions and are not rigorous.
In two dimensions the scattering length $a$ is defined using the zero
energy scattering equation (\ref{3dscatteq}) but instead of
$\psi(r)\approx 1-a/r$ we now impose the asymptotic condition
$\psi(r)\approx \ln(r/a)$.
This is explained in the appendix to \cite{LY2d}.
Note that in two dimensions the ground state energy
could not possibly be $e_0(\rho) \approx 4\pi \mu
\rho a$ as in three dimensions because that would be dimensionally wrong.
Since $e_0(\rho) $ should essentially be proportional to $\rho$,
there is apparently no room for an $a$ dependence --- which is
ridiculous! It turns out that this dependence comes about in the
$\ln(\rho a^2)$ factor.
One of the intriguing facts about \eqref{2den} is that the energy for $N$
particles is {\it not equal} to $N(N-1)/2$ times the energy for two
particles in the low density limit --- as is the case in
three dimensions. The latter quantity, $E_0(2,L)$, is, asymptotically
for large $L$, equal to $8\pi \mu L^{-2} \left[ \ln(L^2/a^2)
\right]^{-1}$. (This is seen in an analogous way as \eqref{partint}. The
three-dimensional boundary condition $\psi_0(|\x|=R)=1-a/R$ is
replaced by $\psi_0(|\x|=R)=\ln (R/a)$ and moreover it has to be taken
into account that with this normalization $\|\psi_0\|^2={\rm
(volume)}(\ln (R/a))^2$ (to leading order), instead of just the volume
in the three-dimensional case.) Thus, if the $N(N-1)/2$ rule were to
apply, \eqref{2den} would have to be replaced by the much smaller
quantity $4\pi \mu
\rho\left[ \ln(L^2/a^2) \right]^{-1}$. In other words, $L$, which tends
to $\infty$ in the thermodynamic limit, has to be replaced by the mean
particle separation, $\rho^{-1/2}$ in the logarithmic factor. Various
poetic formulations of this curious fact have been given, but the fact
remains that the non-linearity is something that does not occur in more
than two dimensions and its precise nature is hardly obvious,
physically. This anomaly is the main reason that the two-dimensional case is
not a trivial extension of the three-dimensional one.
Eq.\ \eqref{2den} was proved in \cite{LY2d} for nonnegative, finite
range two-body potentials by finding upper and lower bounds of the
correct form, using similar ideas as in the previous section for the
three-dimensional case. We discuss below the modifications that have to
be made in the present two-dimensional case. The restriction to
finite range can be relaxed as in three dimensions, but the
restriction to nonnegative $v$ cannot be removed in the current state
of our methodology. The upper bounds will have relative remainder
terms O($|\ln(\rho a^2)|^{-1}$) while the lower bound will have
remainder O($|\ln(\rho a^2)|^{-1/5}$). It is claimed in \cite{hines}
that the relative error for a hard core gas is negative and O$(\ln
|\ln(\rho a^2)||\ln(\rho a^2)|^{-1})$, which is consistent with our
bounds.
The upper bound is derived in complete analogy with the three
dimensional case. The function $f_0$ in the variational ansatz
\eqref{deff} is in
two dimensions also the zero energy scattering solution --- but for
2D, of course. The result is
\begin{equation}\label{upperbound3}
E_{0}(N,L)/N\leq \frac{2\pi\mu\rho}{\ln(b/a)-\pi\rho b^{2}}\left(1+{\rm
O}([\ln(b/a)]^{{-1}})\right).
\end{equation}
The minimum over $b$ of the leading term is obtained for
$b=(2\pi\rho)^{{-1/2}}$. Inserting this in \eqref{upperbound3} we thus
obtain
\begin{equation}\label{upperbound1}
E_{0}(N,L)/N\leq \frac{4\pi\mu\rho}{|\ln(\rho a^{2})|}\left(1+{\rm
O}(|\ln(\rho a^{2})|^{{-1}}\right).
\end{equation}
To prove the lower bound the essential new step is to modify Dyson's lemma
for 2D. The 2D version of
Lemma \ref{dysonl} is:
\begin{lemma}\label{dyson2d}
Let $v(r)\geq0$ and $v(r)=0$ for $r>R_0$.
Let $U(r)\geq 0$ be any function satisfying
\begin{equation}\label{1dyson}
\int_0^\infty U(r)\ln(r/a)rdr \leq 1~~~~~{\rm and}~~~~~ U(r)=0 ~~~{\rm
for}~
r~~0$ such
that
\begin{eqnarray}\nonumber
\|f-\langle f\rangle_\Lambda\|_{L^2(\Lambda)}^2&\leq& \half C
\|\nabla f\|_{L^{6/5}(\Lambda)}^2\\ &\leq& C\left(\|\nabla
f\|_{L^{6/5}(\Omega)}^2+\|\nabla f\|_{L^{6/5}(\Omega^c)}^2\right)\
.
\end{eqnarray}
Applying H\"older's inequality $$ \|\nabla f\|_{L^{6/5}(\Omega)}
\leq \|\nabla f\|_{L^{2}(\Omega)}|\Omega|^{1/3} $$ (and the
analogue with $\Omega$ replaced by $\Omega^c$), we see that
(\ref{poinchom}) holds.
\end{proof}
Applying this result, we are now able to finish the proof of
Theorem \ref{hombecthm}. Denote by $\langle \Psi_0
\rangle_{\Lambda,\X}$ the average of $\Psi_0(\x,\X)$ over
$\x\in\Lambda$. Using Lemma \ref{lem2b}, with $\Omega=\Omega_\X$
and $f(\x)= \Psi_0(\x,\X)-\langle \Psi_0 \rangle_{\Lambda,\X}$, we
conclude that
\begin{eqnarray}\nonumber
&& \int d\X \int d\x \left[\Psi_0(\x,\X)-\langle
\Psi_0\rangle_{\Lambda,\X}\right]^2
\\ \nonumber && \leq C\int d\X\left[L^2\int_{\Omega_\X}
|\nabla_{\x} \Psi_0(\x,\X)|^2 d\x\right. \\ &&\left.
\qquad\quad\qquad + \left(\frac{4\pi}3\right)^{2/3}
N^{2/3-8/51} \int_\Lambda |\nabla_{\x} \Psi_0(\x,\X)|^2 d\x
\right], \label{21b}
\end{eqnarray}
where we used that $|\Omega_\X^c|\leq (4\pi/3) N^{1-4/17}$. The
first integral on the right side of (\ref{21b}) tends to zero as
$N\to\infty$ by (\ref{47}), and the second term vanishes in this limit
because
of (\ref{45}). Moreover,
\begin{eqnarray}\nonumber
&&\int\int d\X d\x \left[\Psi_0(\x,\X)-\langle
\Psi_0\rangle_{\Lambda,\X}\right]^2\\ &&=\int\int d\X d\x
|\Psi_0(\x,\X)|^2-\frac 1N \frac 1{L^3} \int\int \gamma(\x,\, \y)
d\x d\y\ .
\end{eqnarray}
We conclude that
\begin{equation}
\liminf_{N\to\infty} \frac 1N \frac 1{L^3} \int\int \gamma(\x,\,
\y) d\x d\y \geq \lim_{N\to\infty}\int \int d\X d\x
|\Psi_0(\x,\X)|^2=1\ ,
\end{equation}
and Theorem \ref{hombecthm} is proven.
\end{proof}
As stated, Theorem \ref{hombecthm} is concerned with a simultaneous
thermodynamic and $a\to 0$ limit, where, as $N\to \infty$, the box
length $L$ is proportional to $N^{1/3}$ and $a\sim N^{-2/3}$. By
scaling, the above result is equivalent to considering a Bose gas in a
{\it fixed box} of side length $L=1$, and keeping $Na$ fixed as
$N\to\infty$, i.e., $a\sim 1/N$. The ground state energy of the
system is then, asymptotically, $N\times 4\pi Na$, and Theorem
\ref{hombecthm} implies that the one-particle reduced density
matrix $\gamma$ of the ground state converges, after division by $N$,
to the projection
onto the constant function. An analogous result holds true for
inhomogeneous systems. This was recently shown in \cite{LS02} and
will be presented in Section \ref{becsect}.
\section{Gross-Pitaevskii Equation for Trapped Bosons} \label{sectgp}
In the recent experiments on Bose condensation (see, e.g.,
\cite{TRAP}), the particles are confined at very low temperatures
in a `trap' where the particle density is {\em inhomogeneous},
contrary to the case of a large `box', where the
density is essentially uniform.
We model the trap by a slowly varying confining
potential $ V$, with $V(\x)\to \infty $ as $|\x|\to \infty$.
The
Hamiltonian becomes
\begin{equation}\label{trapham}
H = \sum_{i=1}^{N}\left\{ -\mu \Delta_i +V(\x_i)\right\} +
\sum_{1 \leq i < j \leq N} v(|\x_i - \x_j|) \ .
\end{equation}
Shifting the energy scale if necessary
we can assume that $V$ is nonnegative.
The ground state energy, $\hbar\omega$, of
$- \mu \Delta + V(\x)$ is
a natural energy unit and the corresponding
length unit, $\sqrt{\hbar/(m\omega)}=\sqrt{2\mu/(\hbar\omega)}
\equiv L_{\rm osc}$, is a measure of the extension
of the trap.
In the sequel we shall be considering a limit
where $a/L_{\rm osc}$ tends to zero while $N\to\infty$. Experimentally
$a/L_{\rm osc}$ can be changed in two ways: One can either vary $L_{\rm
osc}$ or $a$. The first alternative is usually simpler in practice but
very recently a direct tuning of the scattering length itself has also
been shown to be feasible \cite{Cornish}. Mathematically, both
alternatives
are equivalent, of course. The first corresponds to writing
$V(\x)=L_{\rm osc}^{-2} V_1(\x/L_{\rm osc})$ and keeping $V_1$ and
$v$ fixed. The second corresponds to writing the interaction potential
as $v(|\x|)=a^{-2}v_1(|\x|/a)$ like
in \eqref{v1}, where $v_1$ has unit scattering length,
and keeping $V$ and $v_1$ fixed. This is equivalent to the first,
since for given $V_1$ and $v_1$ the ground state energy of (\ref{trapham}),
measured
in units of $\hbar\omega$, depends only on $N$ and $a/L_{\rm osc}$.
In the dilute limit when $a$ is much smaller than the mean particle
distance, the energy becomes independent of $v_1$.
We choose $L_{\rm osc}$ as a length unit. The energy unit is
$\hbar\omega=2\mu L_{\rm osc}^{-2}=2\mu$. Moreover, we find it
convenient to regard $V$ and $v_1$ as fixed. This justifies the notion
$E_0(N,a)$ for the quantum mechanical ground state energy.
The idea is now to use the information about the thermodynamic limiting
energy of the dilute Bose gas in a box to find the ground state
energy of (\ref{trapham}) in an appropriate limit. This has been done
in \cite{LSY1999, LSY2d} and in this section we
give an account of this work.
As we saw in Sections \ref{sect3d} and
\ref{sect2d} there is a difference in the $\rho$ dependence between
two and three dimensions, so we can expect a related difference now.
We discuss 3D first.
\subsection{Three Dimensions}
Associated with the quantum mechanical ground state energy problem
is the Gross-Pitaevskii (GP) energy functional \cite{G1961,G1963,P1961}
\begin{equation}\label{gpfunc3d}
\E^{\rm
GP}[\phi]=\int_{\R^3}\left(\mu|\nabla\phi|^2+V|\phi|^2+4\pi \mu
a|\phi|^4\right)d\x
\end{equation}
with the subsidiary condition \begin{equation}\label{norm}
\int_{\R^3}|\phi|^2=N.\end{equation}
As before, $a>0$ is the scattering length of $v$.
The corresponding energy is
\begin{equation}\label{gpen3d}
E^{\rm GP}(N,a)=\inf_{\int|\phi|^2=N}\E^{\rm GP}[\phi]= \E^{\rm
GP}[\phi^{\rm GP}],\end{equation} with a unique, positive $\phi^{\rm
GP}$. The existence of the minimizer $\phi^{\rm GP}$ is proved by
standard techniques and it can be shown to be continuously
differentiable, see \cite{LSY1999}, Sect.~2 and Appendix A. The
minimizer depends on $N$ and $a$, of course, and when this is
important we denote it by $\phi^{\rm GP}_{N,a}$.
The variational equation satisfied by the minimizer is the
{\it GP equation}
\begin{equation}\label{gpeq}
-\mu\Delta\phi^{\rm GP}(\x)+ V(\x)\phi^{\rm GP}(\x)+8\pi\mu a \phi^{\rm
GP}(\x)^3 = \mu^{\rm GP} \phi^{\rm GP}(\x),
\end{equation}
where $\mu^{\rm GP}$ is the chemical potential, given by
\begin{equation}\label{mugp}
\mu^{\rm GP}=dE^{\rm GP}(N,a)/dN=E^{\rm GP}(N,a)/N+
(4\pi \mu a/N)\int |\phi^{\rm GP}(\x)|^4 d\x.
\end{equation}
The GP theory has the following scaling property:
\begin{equation}\label{scalen}
E^{\rm GP}(N,a)=N E^{\rm GP}(1,Na),
\end{equation}
and
\begin{equation}\label{scalphi}
\phi^{\rm GP}_{N,a}(\x)= N^{1/2} \phi^{\rm GP}_{1,Na}(\x).
\end{equation}
Hence we see that the relevant parameter in GP theory is the
combination $Na$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%
We now turn to the relation of $E^{\rm GP}$ and
$\phi^{\rm GP}$ to the quantum mechanical ground state.
If $v=0$, then the ground state of \eqref{trapham} is
$$\Psi_{0}(\x_{1},\dots,\x_{N})=\hbox{$\prod_{i=1}^{N}$}\phi_{0}(\x_{i})
$$
with $\phi_{0}$ the normalized ground state of $-\mu \Delta + V(\x)$.
In this case
clearly $\phi^{\rm GP}=\sqrt{N}\ \phi_{0}$, and then
$E^{\rm GP}=N\hbar\omega = E_0$. In the other extreme, if $V(\x)=0$ for
$\x$ inside a large box of volume $L^3$ and $V(\x)= \infty$ otherwise,
then $\phi^{\rm GP} \approx \sqrt{N/L^3}$ and we get $E^{\rm
GP}(N,a) = 4\pi \mu a N^2/L^3$, which is the previously considered energy
$E_0$ for the homogeneous gas in the low
density regime. (In this case, the gradient term in $\E^{\rm GP}$
plays no role.)
In general, we expect that for {\it dilute} gases in a suitable limit
\begin{equation}\label{approx}E_0
\approx E^{\rm GP}\quad{\rm and}\quad \rho^{\rm QM}(\x)\approx
\left|\phi^{\rm GP}(\x)\right|^2\equiv \rho^{\rm GP}(\x),\end{equation}
where the quantum mechanical particle density in the ground state is
defined by \begin{equation} \rho^{\rm
QM}(\x)=N\int|\Psi_{0}(\x,\x_{2},\dots,\x_{N})|^2d\x_{2}\cdots
d\x_{N}. \end{equation} {\it Dilute} means here that
\begin{equation}\bar\rho a^3\ll 1,\end{equation} where
\begin{equation}\label{rhobar}
\bar\rho=\frac 1N\int|\rho^{\rm GP}(\x)|^2 d\x
\end{equation}
is the {\it mean density}.
The limit in which \eqref{approx} can be expected to be true
should be chosen so that {\it all three} terms in
$\E^{\rm GP}$ make a contribution. The scaling relations \eqref{scalen} and \eqref{scalphi}
indicate that fixing
$Na$ as $N\to\infty$ is the right thing to do (and this is quite relevant since
experimentally $N$ can be quite large, $10^6$ and more, and
$Na$ can range from about 1 to $10^4$ \cite{DGPS}).
Fixing $Na$
(which we refer to as the GP
case) also means that
we really are dealing with a dilute limit, because the mean density
$\bar \rho$ is then of the order $N$ (since $\bar\rho_{N,a}=N\bar\rho_{1,Na}$) and hence
\begin{equation}
a^3\bar\rho\sim N^{-2}.
\end{equation}
The precise statement of \eqref{approx} is:
\begin{theorem}[\textbf{GP limit of the QM ground state energy and
density}]\label{thmgp3} If $N\to\infty$ with $Na$ fixed, then
\begin{equation}\label{econv}
\lim_{N\to\infty}\frac{{E_0(N,a)}}{ {E^{\rm GP}(N,a)}}=1,
\end{equation}
and
\begin{equation}\label{dconv}
\lim_{N\to\infty}\frac{1}{ N}\rho^{\rm QM}_{N,a}(\x)= \left
|{\phi^{\rm GP}_{1,Na}}(\x)\right|^2
\end{equation}
in the weak $L^1$-sense.
\end{theorem}
To describe situations where $Na$ is very large, it is appropriate
to consider a limit where, as $N\to\infty$, $a\gg
N^{-1}$, i.e. $Na\to\infty$, but still $\bar\rho a^3\to 0$.
In
this case, the gradient term in the GP functional becomes
negligible compared to the other terms and the so-called {\it
Thomas-Fermi (TF) functional}
\begin{equation}\label{gtf}
\E^{\rm TF}[\rho]=\int_{\R^3}\left(V\rho+4\pi \mu a\rho^2\right)d\x
\end{equation}
arises. (Note that this functional has nothing to do with the
fermionic theory invented by Thomas and Fermi in 1927, except
for a certain formal analogy.) It is
defined for nonnegative functions $\rho$ on $\R^3$. Its ground
state energy $E^{\rm TF}$ and density $\rho^{\rm TF}$ are defined
analogously to the GP case. (The TF functional is especially
relevant for the two-dimensional Bose gas. There $a$ has to
decrease exponentially with $N$ in the GP limit, so the TF limit
is more adequate; see Subsection \ref{sub2d} below).
Our second main result of this section is that minimization of
(\ref{gtf}) reproduces correctly the ground state energy and
density of the many-body Hamiltonian in the limit when
$N\to\infty$, $a^3\bar \rho\to 0$, but $Na\to \infty$ (which we
refer to as the TF case), provided the external potential is
reasonably well behaved. We will assume that $V$ is asymptotically
equal to some function $W$ that is homogeneous of some order $s>0$, i.e.,
$W(\lambda\x)=\lambda^s W(\x)$ for all $\lambda>0$,
and locally H\"older continuous (see \cite{LSY2d} for a precise
definition). This condition can be relaxed, but it seems adequate
for most practical applications and simplifies things
considerably.
\begin{theorem}[\textbf{TF limit of the QM ground state energy
and density}]\label{thm2} Assume that $V$ satisfies
the conditions
stated above. If $\g\equiv Na\to\infty$ as $N\to\infty$, but still
$a^3\bar\rho\to 0$, then
\begin{equation}\label{econftf}
\lim_{N\to\infty}\frac{E_0(N,a)} {E^{\rm TF}(N,a)}=1,
\end{equation}
and
\begin{equation}\label{dconvtf}
\lim_{N\to\infty}\frac{\g^{3/(s+3)}}{N}\rho^{\rm
QM}_{N,a}(\g^{1/(s+3)}\x)= \tilde\rho^{\rm TF}_{1,1}(\x)
\end{equation}
in the weak $L^1$-sense, where $\tilde\rho^{\rm TF}_{1,1}$ is the
minimizer of the TF functional under the condition $\int\rho=1$,
$a=1$, and with $V$ replaced by $W$.
\end{theorem}
%\noindent{\it Remark.} The theorems are independent of actual form
%of the interaction potential $v_1$ in (\ref{v1}), they depend only
%on the scattering length $a$. This means that in the limit we
%consider only the scattering length effects the ground state
%properties, and not the details of the potential. Note also that
%the particular limit we consider is {\it not} a mean field limit,
%since the interaction potential is very hard in this limit; in
%fact the term $4\pi a|\phi|^4$ is mostly kinetic energy (see also
%Theorem \ref{compthm} below).
%\medskip
In the following, we will present the essentials of the proofs
Theorems \ref{thmgp3} and \ref{thm2}.
We will derive appropriate upper and lower bounds on the ground
state energy $E_0$.
The proof of the lower bound in Theorem \ref{thmgp3}
presented here is a modified version of (and partly simpler than)
the original proof in \cite{LSY1999}.
The convergence of the densities follows from
the convergence of the energies in the usual way by variation with
respect to the external potential. For simplicity, we set $\mu\equiv 1$ in
the following.
\begin{proof}[Proof of Theorems \ref{thmgp3} and \ref{thm2}] {\it Part 1:
Upper bound to the QM energy.} To derive an upper bound on $E_0$ we
use a generalization of a trial wave function of Dyson \cite{dyson},
who used this function to give an upper bound on the ground state
energy of the homogeneous hard core Bose gas (c.f.\ Section
\ref{upsec}). It is of the form
\begin{equation}\label{ansatz}
\Psi(\x_{1},\dots,\x_{N})
=\prod_{i=1}^N\phi^{\rm
GP}(\x_{i})F(\x_{1},\dots,\x_{N}),
\end{equation}
where $F$ is constructed in the following way:
\begin{equation}F(\x_1,\dots,\x_N)=\prod_{i=1}^N
f(t_i(\x_1,\dots,\x_i)),\end{equation} where $t_i =
\min\{|\x_i-\x_j|, 1\leq j\leq i-1\}$ is the distance of $\x_{i}$
to its {\it nearest neighbor} among the points
$\x_1,\dots,\x_{i-1}$, and $f$ is a function of $r\geq 0$. As in
\eqref{deff} we
choose it to be
\begin{equation}
f(r)=\left\{\begin{array}{cl} f_{0}(r)/f_0(b) \quad
&\mbox{for}\quad r~~**0$, that will eventually tend to $\infty$, and
restrict ourselves to boxes inside a cube $\Lambda_M$ of side length
$M$. Since $v\geq 0$ the contribution to
\eqref{5.26} of boxes outside this cube is easily estimated from below by
$-8\pi Na \sup_{\x\notin \Lambda_M}\rho^{\rm GP}(\x)$, which, divided
by $N$, is arbitrarily small for $M$ large, since $Na$ is fixed and
$\pgp/N^{1/2}=\pgp_{1,Na}$ decreases faster than exponentially at
infinity (\cite{LSY1999}, Lemma A.5).
%%%%%%%%%%%%%%
For the boxes inside the cube $\Lambda_M$ we want to use Lemma
\ref{dysonl} and therefore we must approximate $\rho^{\rm GP}$ by
constants in each box. Let $\rmax$ and $\rmin$, respectively, denote
the maximal and minimal values of $\rho^{\rm GP}$ in box $\al$. Define
\begin{equation}
\Psi_\alpha(\x_1,\dots, \x_{n_\al})=F_\alpha(\x_1,\dots, \x_{n_\al})
\prod_{k=1}^{n_\al}\phi^{\rm GP}(\x_k),
\end{equation}
and
\begin{equation}
\Psi^{(i)}_\alpha(\x_1,\dots, \x_{n_\al})=F_\alpha(\x_1,\dots, \x_{n_\al})
\prod_{\substack{k=1 \\ k\neq
i}}^{n_\al}\phi^{\rm GP}(\x_k).
\end{equation}
We have, for all $1\leq i\leq n_\al$,
\begin{equation}\label{5.29}
\begin{split}
&\int\prod_{k=1}^{n_\alpha}\rho^{\rm GP}(\x_k)\left(|\nabla_i
F_\alpha|^2\right.+\half\sum_{j\neq i}\left.
v(|\x_i-\x_j|)|F_\alpha|^2\right)
\\&\geq
\rmin\int \left(|\nabla_i\Psi^{(i)}_\alpha|^2\right. +\half\sum_{j\neq i}\left.
v(|\x_i-\x_j|)|\Psi^{(i)}_\alpha|^2\right).
\end{split}
\end{equation}
We now use
Lemma \ref{dysonl} to get, for all $0\leq \eps\leq 1$,
\begin{equation}\label{5.30}
(\ref{5.29})\geq \rmin\int \left(\varepsilon |\nabla_i\Psi^{(i)}_\alpha|^2
+a(1-\eps)U(t_i)|\Psi^{(i)}_\alpha|^2\right)
\end{equation}
where $t_i$ is the distance to the nearest neighbor of $\x_i$, c.f.,
\eqref{2.29}, and $U$ the potential \eqref{softened}.
Since $\Psi_\alpha=\pgp(\x_i)\Psi^{(i)}_\al$ we can estimate
\begin{equation}\label{tpsial}
|\nabla_i\Psi_\alpha|^2 \leq 2\rmax |\nabla_i\Psi^{(i)}_\alpha|^2 +
2|\Psi^{(i)}_\alpha|^2 N C_M
\end{equation}
with
\begin{equation}
C_M=\frac1N\sup_{\x\in\Lambda_M}|\nabla\pgp(\x)|^2=\sup_{\x\in\Lambda_M}
|\nabla\pgp_{1,Na}(\x)|^2. \end{equation} Since $Na$ is fixed, $C_M$
is independent of $N$. Inserting \eqref{tpsial} into
\eqref{5.30}, summing over $i$ and using $\rho^{\rm GP}(\x_i)\leq \rmax$ in
the
last term of \eqref{ener3} (in the box $\al$), we get
\begin{equation}\label{qalfal}
Q_\al(F_\alpha)\geq \frac{\rmin}{\rmax}E^{U}_\eps(n_\al,L)-8\pi a\rmax n_\al -
\eps C_M n_\al,
\end{equation}
where $L$ is the side length of the box and $E^{U}_\eps(n_\al,L)$ is
the ground state energy of
\begin{equation}\label{eueps}
\sum_{i=1}^{n_\al}(-\half\eps \Delta_i+(1-\eps)aU(t_i))
\end{equation}
in the box (c.f.\ \eqref{halfway}). We want to minimize \eqref{qalfal}
with respect to $n_\al$ and drop the subsidiary condition
$\sum_\al{n_\al}=N$ in \eqref{5.26}. This can only lower the minimum.
For the time being we also ignore the last term in
\eqref{qalfal}. (The total contribution of this term for all boxes is
bounded by $\eps C_M N$ and will be shown to be negligible compared to
the other terms.)
Since the lower bound for the energy of Theorem
\ref{lbthm2} was obtained precisely from a lower bound to the operator
\eqref{eueps}, we can use the statement and proof of Theorem \ref{lbthm2}.
{F}rom this we see that
\begin{equation}\label{basicx}
E^{U}_\eps(n_\al,L)\geq (1-\varepsilon)\frac{4\pi
an_\al^2}{L^3}(1-CY_\al^{1/17})
\end{equation}
with $Y_\al=a^3n_\al/L^3$, provided $Y_\al$ is small enough,
$\eps\geq Y_\al^{1/17}$ and
$n_\al\geq {\rm (const.)} Y_\al^{-1/17}$. The condition on $\eps$ is
certainly fulfilled if we choose $\eps=Y^{1/17}$ with
$Y=a^3N/L^3$. We now want to show that the $n_\alpha$ minimizing
the right side of \eqref{qalfal} is large enough for \eqref{basicx} to apply.
If the minimum of the right side of \eqref{qalfal}
(without the last term) is taken for some $\bar n_\al$, we have
\begin{equation}\label{minnal}
\frac{\rmin}{\rmax}
\left(E^{U}_\eps(\bar n_\al+1,L)-E^{U}_\eps(\bar n_\al,L)\right)\geq
8\pi a\rmax.
\end{equation}
On the other hand, we claim that
\begin{lemma} For any $n$
\begin{equation}\label{chempot}
E^{U}_\eps( n+1,L)-E^{U}_\eps(n,L)\leq 8\pi a\frac{
n}{L^3}.
\end{equation}
\end{lemma}
\begin{proof}
Denote the operator \eqref{eueps} by $\tilde H_n$, with $n_\alpha=n$, and
let $\tilde
\Psi_n$
be its ground state. Let $t_i'$ be the distance to the nearest neighbor
of $\x_i$ among the $n+1$ points $\x_1,\dots,\x_{n+1}$ (without $\x_i$) and $t_i$ the
corresponding distance excluding $\x_{n+1}$. Obviously, for $1\leq i\leq n$,
\begin{equation}
U(t_i')\leq U(t_i)+U(|\x_i-\x_{n+1}|)
\end{equation}
and
\begin{equation}
U(t_{n+1}')\leq \sum_{i=1}^nU(|\x_i-\x_{n+1}|).
\end{equation}
Therefore
\begin{equation}
\tilde H_{n+1}\leq \tilde H_{n}-\half\eps\Delta_{n+1}+2a\sum_{i=1}^nU(|\x_i-\x_{n+1}|).
\end{equation}
Using $\tilde\Psi_n/L^{3/2}$ as trial function for $\tilde H_{n+1}$ we arrive at
\eqref{chempot}.
\end{proof}
Eq.\ \eqref{chempot} together with \eqref{minnal} shows that
$\bar n_\al$ is at least $\sim \rmax L^3$.
We shall choose
$L\sim N^{-1/10}$, so
the conditions needed for (\ref{basicx}) are fulfilled for $N$ large enough,
since $\rmax\sim N$ and hence
$\bar n_\al\sim N^{7/10}$ and
$Y_\al\sim N^{-2}$.
In order to obtain a lower bound on $Q_\al$ we therefore have to minimize
\begin{equation}\label{qalpha}
4\pi
a\left(\frac{\rmin}{\rmax}\frac{n_\al^2}{L^3}\left(1-CY^{1/17}\right)
-2n_\al\rmax\right).
\end{equation}
We can drop the
requirement that $n_\al$ has to
be an integer. The minimum of (\ref{qalpha}) is obtained for
\begin{equation}
n_\al= \frac{\rmax^2}{\rmin}\frac{L^3}{(1-CY^{1/17})}.
\end{equation}
By Eq.\ (\ref{ener2}) this gives the following lower bound,
including now the last term in \eqref{qalfal} as well as the
contributions from the
boxes outside $\Lambda_M$,
\begin{equation}\label{almostthere}
\begin{split}
&E_0(N,a)-E^{\rm GP}(N,a)\geq \\
&4\pi a\int|\rho^{\rm GP}|^2-4\pi a\sum_{\al\subset\Lambda_M}
\rmin^2
L^3\left(\frac{\rmax^3}{\rmin^3}\frac{1}{(1-CY^{1/17})}\right)\\ &-Y^{1/17}NC_M-4\pi
aN\sup_{\x\notin\Lambda_M}\rho^{\rm GP}(\x).
\end{split}
\end{equation}
Now $\rho^{\rm GP}$ is differentiable and strictly
positive. Since all the boxes are in the fixed cube $\Lambda_M$ there are
constants
$C'<\infty$, $C''>0$,
such that
\begin{equation}
\rmax-\rmin\leq NC'L,\quad \rmin\geq NC''.
\end{equation}
Since $L\sim N^{-1/10}$ and $Y\sim N^{-17/10}$ we therefore have, for large $N$,
\begin{equation}
\frac{\rmax^3}{\rmin^3}\frac{1}{(1-CY^{1/17})}\leq
1+{\rm (const.)}N^{-1/10}
\end{equation}
Also,
\begin{equation}
4\pi a\sum_{\al\subset\Lambda_M} \rmin^2 L^3\leq 4\pi a\int
|\rho^{\rm GP}|^2\leq E^{\rm
GP}(N,a).
\end{equation}
Hence, noting that $ E^{\rm GP}(N,a)=N E^{\rm GP}(1,Na)\sim N$ since $Na$ is fixed,
\begin{equation}\label{there}
\frac{E_0(N,a)}{E^{\rm GP}(N,a)}\geq 1-{\rm (const.)}(1+C_M)N^{-1/10}-{\rm (const.)}
\sup_{\x\notin \Lambda_M}|\pgp_{1,Na}|^2,
\end{equation}
where the constants depend on $Na$. We can now take $N\to\infty$ and
then $M\to\infty$.
\bigskip
\noindent {\it Part 3: Lower bound to the QM energy, TF case.}
In the above proof of the lower bound in the GP case we did not attempt to
keep track of the dependence of the constants on $Na$. In the TF case
$Na\to\infty$, so one would need to take a closer look at this
dependence if one wanted to carry the proof directly over to this
case. But we don't have to do so, because there is a simpler direct
proof. Using the explicit form of the TF minimizer, namely
\begin{equation}\label{tfminim}
\rho^{\rm TF}_{N,a}(\x)=\frac 1{8\pi a}[\mu^{\rm TF}-V(\x)]_+,
\end{equation}
where $[t]_+\equiv\max\{t,0\}$ and $\mu^{\rm TF}$ is chosen so
that the normalization condition $\int \rho^{\rm TF}_{N,a}=N$
holds, we can use
\begin{equation}\label{vbound}
V(\x)\geq \mu^{\rm TF}-8\pi a \rho^{\rm
TF}(\x)
\end{equation}
to get a replacement as in (\ref{repl}), but without
changing the measure. Moreover, $\rho^{\rm TF}$ has compact
support, so, applying again the box method described above, the
boxes far out do not contribute to the energy. However, $\mu^{\rm
TF}$ (which depends only on the combination $Na$) tends to
infinity as $Na\to\infty$. We need to control the
asymptotic behavior of $\mtf$, and this leads to
the restrictions on $V$ described in the paragraph preceding
Theorem \ref{thm2}. For simplicity, we shall here only consider the case when $V$
itself is homogeneous, i.e., $V(\lambda\x)=\lambda^sV(\x)$ for all $\lambda>0$ with
some $s>0$.
In the same way as in \eqref{mugp} we have, with $g=Na$,
\begin{equation}\label{mutf}
\mu^{\rm TF}(g)=dE^{\rm TF}(N,a)/dN=E^{\rm TF}(1,g)+
4\pi g\int |\rho^{\rm TF}_{1,g}(\x)|^2 d\x.
\end{equation}
The TF energy, chemical potential and minimizer
satisfy the scaling relations
\begin{equation}
E^{\rm TF}(1,g)=g^{s/(s+3)}E^{\rm TF}(1,1),
\end{equation}
\begin{equation}
\mu^{\rm TF}(g)=g^{s/(s+3)} \mu^{\rm TF}(1) ,
\end{equation}
and
\begin{equation}
g^{3/(s+3)}\rho^{\rm TF}_{1,g}(g^{1/(s+3)}\x)= \rho^{\rm TF}_{1,g}(\x) .
\end{equation}
We also introduce the scaled interaction potential, $\widehat v$, by
\begin{equation}
\widehat v(\x) =g^{2/(s+3)}v(g^{1/(s+3)}\x)
\end{equation}
with scattering length
\begin{equation}
\widehat a=g^{-1/(s+3)}a.
\end{equation}
Using \eqref{vbound}, \eqref{mutf} and
the scaling relations we obtain
\begin{equation}
E_0(N,a)\geq E^{\rm TF}(N,a)+4\pi N g^{s/(s+3)}\int |\rho^{\rm TF}_{1,1}|^2 +
g^{-2/(s+3)}Q
\end{equation}
with
\begin{equation}
Q=\inf_{\int|\Psi|^2=1}\sum_{i}\int\left(|\nabla_i\Psi|^2\right.+\half \sum_{j\neq i}
\left.\widehat v(\x_i-\x_j)|\Psi|^2-8\pi
N\widehat a \rtf_{1,1}(\x_i)|\Psi|^2\right).
\end{equation}
We can now proceed exactly as in Part 2 to arrive at the the analogy of
Eq.\ \eqref{almostthere}, which in the present case becomes
\begin{equation}\label{almosttherex}
\begin{split}
&E_0(N,a)-E^{\rm TF}(N,a)\geq \\
&4\pi N g^{s/(s+3)}\int |\rho^{\rm TF}_{1,1}|^2-4\pi N\widehat a\sum_{\al}
\rmax^2
L^3(1-C\widehat Y^{1/17})^{-1}.
\end{split}
\end{equation}
Here $\rmax$ is the maximum of $\rho^{\rm TF}_{1,1}$ in the box
$\alpha$, and $\widehat Y=\widehat a^3 N/L^3$. This holds as long as $L$ does not
decrease too fast with $N$. In particular, if $L$ is simply fixed, this
holds for all large enough $N$. Note that
\begin{equation}
\bar\rho=N\bar\rho_{1,g}\sim N g^{-3/(s+3)} \bar\rho_{1,1},
\end{equation}
so that $\widehat a^3 N\sim a^3 \bar
\rho$ goes to zero as $N\to\infty$ by assumption. Hence, if we first let
$N\to\infty$ (which implies $\widehat Y\to 0$) and then take $L$ to zero, we
of arrive at the desired
result
\begin{equation}\label{lowertf}
\liminf_{N\to\infty}\frac{E_0(N,a)}{E^{\rm TF}(N,a)}\geq 1
\end{equation}
in the limit $N\to\infty$, $a^3\bar\rho\to 0$. Here
we used the fact that (because $V$, and hence $\rtf$, is continuous by
assumption) the Riemann sum $\sum_\al\rmax^2 L^3$ converges to
$\int|\rtf_{1,1}|^2$ as $L\to 0$. Together with the upper bound (\ref{ubd})
and
the fact that $E^{\rm GP}(N,a)/E^{\rm TF}(N,a)=E^{\rm GP}(1,Na)/E^{\rm
TF}(1,Na)\to 1$ as $Na\to\infty$, which holds under our regularity
assumption on $V$ (c.f.\ Lemma 2.3 in \cite{LSY2d}), this proves
(\ref{econv}) and (\ref{econftf}).
\bigskip
\noindent {\it Part 4: Convergence of the densities.} The
convergence of the energies implies the convergence of the
densities in the usual way by variation of the external potential.
We show here the TF case, the GP case goes analogously. Set again
$\g=Na$. Making the replacement
\begin{equation}
V(\x)\longrightarrow V(\x)+\delta\g^{s/(s+3)}Z(\g^{-1/(s+3)}\x)
\end{equation}
for some positive $Z\in C_0^\infty$ and redoing the upper and
lower bounds we see that (\ref{econftf}) holds with $W$ replaced
by $W+\delta Z$. Differentiating with respect to $\delta$ at
$\delta=0$ yields
\begin{equation}
\lim_{N\to\infty}\frac{\g^{3/(s+3)}}N\rho^{\rm
QM}_{N,a}(\g^{1/(s+3)}\x) =\tilde\rho^{\rm TF}_{1,1}(\x)
\end{equation}
in the sense of distributions. Since the functions all have
$L^1$-norm 1, we can conclude that there is even weak
$L^1$-convergence.
\end{proof}
\subsection{Two Dimensions}\label{sub2d}
%%%%%%%%%%%%%%%%
In contrast to the three-dimensional case the energy per particle for
a dilute gas in two dimensions is {\it nonlinear} in $\rho$. In view
of Schick's formula \eqref{2den} for the energy of the homogeneous gas
it would appear natural to take the interaction into account in two
dimensional GP theory by a term
\begin{equation}
4\pi\int_{\R^2} |\ln(|\phi(\x)|^2 a^2)|^{-1}|\phi(\x)|^4{
d}\x,\end{equation}
and such a term has, indeed, been suggested in
\cite{Shev} and \cite{KoSt2000}. However, since the nonlinearity
appears only in a logarithm, this term is unnecessarily complicated
as far as leading order computations are concerned. For dilute gases
it turns out to be sufficient, to leading order, to use an interaction
term of the same form as in the three-dimensional case, i.e, define the
GP functional as (for simplicity we put $\mu=1$ in this section)
\begin{equation}\label{2dgpfunc}
\E^{\rm
GP}[\phi]=\int_{\R^2}\left(|\nabla\phi|^2+V|\phi|^2+4\pi
\alpha|\phi|^4\right)d\x,
\end{equation}
where instead of $a$ the coupling constant is now
\begin{equation}\label{alpha}\alpha=|\ln(\bar\rho_N
a^2)|^{-1}\end{equation}
with $\bar\rho_N$ the {\em mean density}
for the GP functional
at coupling constant
$1$ and particle number $N$. This is defined analogously to \eqref{rhobar}
as
\begin{equation}
\bar\rho_N=\frac1N\int|\phi^{\rm GP}_{N,1}|^4d\x
\end{equation}
where $\phi^{\rm GP}_{N,1}$ is the minimizer of \eqref{2dgpfunc} with
$\alpha=1$ and subsidiary condition $\int|\phi|^2=N$.
Note that $\alpha$ in \eqref{alpha} depends on
$N$ through the mean density.
%%%%%%%%%%%%%%%%%%%
Let us denote the GP energy
for a given $N$ and coupling constant
$\alpha$ by $E^{\rm GP}(N,\alpha)$ and the corresponding minimizer by
$\phi^{\rm GP}_{N,\alpha}$.
As in three dimensions the scaling relations
\begin{equation}E^{\rm GP}(N,\alpha)=NE^{\rm GP}(1,N\alpha)\end{equation}
and
\begin{equation}N^{-1/2}\phi^{\rm GP}_{N,\alpha}=\phi^{\rm GP}_{1,N\alpha},
\end{equation}
hold, and the relevant parameter is
\begin{equation}g\equiv N\alpha.\end{equation}
In three dimensions, where $\alpha=a$,
it is natural to consider the limit $N\to\infty$ with $g=Na$= const.
The analogue of Theorem \ref{thmgp3} in two dimensions is
\begin{theorem}[{\bf Two-dimensional GP limit
theorem}]
\label{2dlimit}
If, for $N\to\infty$,\linebreak $a^2\brtf_N\to 0$ with
$g=N/|\ln(a^2\brtf_N)|$ fixed, then
\begin{equation}\label{econv2}
\lim_{N\to\infty}\frac{E_{0}(N,a)}{\Egp(N,1/|\ln(a^2\brtf_N)|)}=
1
\end{equation}
and
\begin{equation}\label{dconv2}
\lim_{N\to\infty}\frac{1}{ N}\rho^{\rm QM}_{N,a}(\x)= \left
|{\phi^{\rm GP}_{1,g}}(\x)\right|^2
\end{equation}
in the weak $L^1$-sense.
\end{theorem}
This result, however, is of rather limited use in practice. The reason is
that in two dimensions the scattering length has to
decrease exponentially with $N$ if $g$ is fixed.
The parameter $g$ is
typically {\it very large} in two dimensions
so it is more appropriate to consider the
limit $N\to\infty$ and $g\to\infty$ (but still $\bar\rho_N a^2\to
0$).
For potentials $V$ that are {\it homogeneous} functions of $\x$, i.e.,
\begin{equation}\label{homog}V(\lambda \x)=\lambda^sV(\x)\end{equation}
for some $s>0$, this limit can be described by the a
`Thomas-Fermi' energy functional like \eqref{gtf} with coupling constant unity:
\begin{equation}\label{tffunct}
\E^{\rm TF}[\rho]=\int_{\R^2}\left(V(\x)\rho(\x)+4\pi \rho(\x)^2\right)
{ d}\x.
\end{equation}
This is just the GP functional without the gradient term and $\alpha=1$.
Here $\rho$ is a nonnegative function on $\R^2$ and the normalization
condition is
\begin{equation}\label{norm2}\int\rho(\x)d\x=1.\end{equation}
The minimizer of \eqref{tffunct} can be given explicitly. It is
\begin{equation}\label{tfminim2}\rho^{\rm
TF}_{1,1}(\x)=(8\pi)^{-1}[\mu^{\rm TF}-V(\x)]_+\end{equation}
where the chemical potential
$\mu^{\rm TF}$ is determined by the normalization condition \eqref{norm2}
and $[t]_{+}=t$
for $t\geq 0$ and zero otherwise.
We denote the corresponding energy by $E^{\rm TF}(1,1)$.
By scaling one obtains
\begin{equation}\lim_{g\to\infty}
E^{\rm GP}(1,g)/g^{s/(s+2)}=E^{\rm TF}(1,1),\end{equation}
\begin{equation}\label{gptotf}\lim_{g\to\infty}g^{2/(s+2)}
\rho^{\rm
GP}_{1,g}(g^{1/(s+2)}\x)=\rho^{\rm TF}_{1,1}(\x),\end{equation}
with the latter limit in the strong $L^2$ sense.
Our main result about two-dimensional
Bose gases in external potentials satisfying \eqref{homog}
is that analogous limits also hold for the many-particle quantum mechanical
ground state at
low densities:
\begin{theorem}[{\bf Two-dimensional TF limit theorem}]\label{thm22}
In two dimensions, if
$a^2\bar\rho_N\to 0$, but $g=N/|\ln(\bar\rho_N a^2)|\to \infty$ as
$N\to\infty$
then
\begin{equation}\lim_{N\to \infty}\frac{E_0(N,a)}{g^{s/s+2}}=
E^{\rm TF}(1,1)\end{equation}
and, in the weak $L^1$ sense,
\begin{equation}\label{conv}\lim_{N\to\infty}\frac{g^{2/(s+2)}}N
\rho^{\rm
QM}_{N,a}(g^{1/(s+2)}\x)=\rho^{\rm TF}_{1,1}(\x).\end{equation}
\end{theorem}
\noindent {\it Remarks:} 1. As in Theorem \ref{thm2}, it is sufficient that $V$ is
asymptotically equal to some homogeneous potential, $W$. In this case,
$E^{\rm TF}(1,1)$ and $\rho^{\rm TF}_{1,1}$ in Theorem \ref{thm22}
should be replaced by the corresponding quantities for $W$.
2. From Eq.\ \eqref{gptotf} it follows that
\begin{equation}\bar\rho_N\sim N^{s/(s+2)}\end{equation} for large $N$.
Hence the low density criterion
$a^2\bar\rho\ll 1$, means that
$a/L_{\rm osc}\ll N^{-s/2(s+2)}$.
%%%%%%%%%%%%%%%%%
We shall now comment briefly on the proofs of Theorems
\ref{2dlimit} and \ref{thm22},
mainly pointing out the differences from the 3D case considered previously.
The upper bounds for the energy are obtained exactly in a same way as
in three dimensions. For the lower bound in Theorem \ref{2dlimit} the
point to notice is that the expression
\eqref{qalpha}, that has to be minimized over $n_\al$, is in 2D
replaced by
\begin{equation}\label{qalpha2}
4\pi
\left(\frac{\rmin}{\rmax}\frac{n_\al^2}{L^2}\frac1{|\ln(a^2n_\alpha/L^2)|}
\left(1-\frac C{|\ln(a^2N/L^2)|^{1/5}}\right)
-\frac{2n_\al\rmax}{|\ln(a^2\bar\rho_N)|}\right),
\end{equation}
since Eq.\ \eqref{basicx} has to be
replaced by the analogous inequality for 2D (c.f.\ \eqref{lower}).
To minimize \eqref{qalpha2} we use the following lemma:
\begin{lemma}\label{xb}
For $0 0$ we have
\begin{equation}
\frac{x^2}{b^2}\frac{|\ln b|}{|\ln x|}
-2\frac xb\geq\frac{|\ln b|}{b^2}ed x^{2+d}-\frac{2x}{b}
\geq c(d)(b^ded\,|\ln b|)^{-1/(1+d)}
\end{equation}
with
\begin{equation}
c(d)=2^{(2+d)/(1+d)}\left(\frac 1{(2+d)^{(2+d)/(1+d)}}-\frac 1
{(2+d)^{1/(1+d)}}\right)\geq -1-\frac 14d^2.
\end{equation}
Choosing $d=1/|\ln b|$ gives the desired result.
\end{proof}
Applying this lemma with $x=a^2n_\al/L^2$, $b=a^2\rmax$ and
\begin{equation}k=\frac{\rmax}{\rmin}\,
\left(1-\frac C{|\ln(a^2N/L^2)|^{1/5}}\right)^{-1}\frac{|\ln(a^2\rmax)|}
{|\ln(a^2\bar\rho_N)|}
\end{equation}
we get the bound
\begin{equation}
\eqref{qalpha2}\geq -4\pi\frac{\rmax^2L^2}{|\ln(a^2\bar\rho_N)|}
\left(1+\frac1{4|\ln(a^2\rmax)|^2}\right) k.
\end{equation}
In the limit considered, $k$ and the factor in parenthesis both tend to 1 and
the Riemann sum over the boxes $\alpha$ converges to the integral as $L\to 0$.
The TF case, Thm.\ \ref{thm22}, is treated in the same way as in three
dimensions, with modifications analogous to those just discussed when passing
from 3D to 2D in GP theory.
\bigskip
\section{BEC for Dilute Trapped Gases}\label{becsect}
It is gratifying to see the experimental realization, in traps, of
the long-predicted Bose-Einstein condensation (BEC) of gases. From
the theoretical point of view, however, a rigorous demonstration
of this phenomenon -- starting from the many-body Hamiltonian of
interacting particles -- has not yet been achieved. Following
\cite{LS02}, we will provide in this section such a rigorous
justification for the ground state of 2D or 3D bosons in a trap with
repulsive pair potentials, and in the well-defined limit in which the
Gross-Pitaevskii (GP) formula is applicable. It is the first proof of
BEC for interacting particles in a continuum (as distinct from
lattice) model and in a physically realistic situation. The
Gross-Pitaevskii limit under discussion here is, of course, a
physically simpler limit than the usual thermodynamic limit in which
the average density is held fixed as the particle number goes to
infinity. In the GP limit one also lets the range of the potential go
to zero as $N$ goes to infinity, but in such a way that the overall
effect is non-trivial. That is, the combined effect of the infinite
particle limit and the zero range limit is such as to leave a
measurable residue --- the GP function.
It was shown in the previous section (see also Theorem
\ref{compthm} below) that, for each fixed $Na$, the minimization
of the GP functional correctly reproduces the large $N$
asymptotics of the ground state energy and density of $H$ -- but
no assertion about BEC in this limit was made. We will now extend
this result by showing that in the Gross-Pitaevskii limit there
is indeed 100\% Bose condensation in the ground state. This is a
generalization of the homogeneous case considered in Theorem
\ref{hombecthm}. In the following, we concentrate on the 3D case,
but analogous considerations apply also to the 2D case.
For use later, we define the projector
\begin{equation}
P^{\rm GP}= |\varphi^{\rm GP}\rangle\langle \varphi^{\rm GP}|\ .
\end{equation}
Here (and everywhere else in this section) we denote $\varphi^{\rm
GP}=\phi^{\rm GP}_{1,Na}$ for simplicity, where $\phi^{\rm
GP}_{1,Na}$ is the minimizer of the GP functional (\ref{gpfunc3d})
with parameter $Na$ and normalization condition $\int|\phi|^2=1$
(compare with (\ref{scalphi})). Moreover, we set $\mu\equiv 1$.
In the following, $\Psi_0$ denotes the (nonnegative and normalized)
ground state of the Hamiltonian (\ref{trapham}). BEC refers to the
reduced one-particle density matrix $ \gamma(\x,\x')$ of $\Psi_0$,
defined in (\ref{defgamma}).
Complete (or 100\%) BEC is defined to be the property that
$\mbox{$\frac{1}{N}$}\gamma(\x,\x')$ not only has an eigenvalue of
order one, as in the general case of an incomplete BEC, but in the
limit it has only one nonzero eigenvalue (namely 1). Thus,
$\mbox{$\frac{1}{N}$}\gamma(\x,\x')$ becomes a simple product
$\varphi(\x)^*\varphi(\x')$ as $N\to \infty$, in which case $\varphi$ is called
the {\it condensate wave function}. In the GP limit, i.e.,
$N\to\infty$ with $N a$ fixed, we can show that this is the case, and
the condensate wave function is, in fact, the GP minimizer $\varphi^{\rm GP}$.
\begin{theorem}[\textbf{Bose-Einstein condensation}]\label{becthm}
For each fixed $Na$ $$ \lim_{N\to\infty} \frac 1 N \gamma(\x, \x')
= \varphi^{\rm GP}(\x)\varphi^{\rm GP}(\x')\ . $$ in trace norm, i.e., $\Tr \left|\frac 1
N \gamma - P^{\rm GP} \right| \to 0$.
\end{theorem}
We remark that Theorem \ref{becthm} implies that there is also 100\%
condensation for all $n$-particle reduced density matrices
\begin{eqnarray}\nonumber
&&\gamma^{(n)}(\x_1,\dots,\x_n;\x_1',\dots,\x_n')\\&&=n!\binom{N}{n}\int
\Psi_0(\x_1,\dots,\x_N)\Psi_0(\x_1',\dots,\x_n',\x_{n+1},
\dots\x_N)d\x_{n+1}\cdots d\x_N\nonumber \\
\end{eqnarray}
of
$\Psi_0$, i.e., they converge, after division by the normalization factor,
to the one-dimensional projector onto
the $n$-fold tensor product of $\varphi^{\rm GP}$. In other words, for
$n$ fixed particles the probability of finding them all in the same state
$\varphi^{\rm GP}$ tends to 1 in the
limit considered. To see this,
let $a^*, a$ denote the boson creation and annihilation operators
for the state $\varphi^{\rm GP}$, and observe that
\begin{equation}
1\geq \lim_{N\to\infty} N^{-n}\langle \Psi_0 | (a^*)^n a^n|\Psi_0\rangle =
\lim_{N\to\infty} N^{-n} \langle \Psi_0 | (a^*a)^n|\Psi_0\rangle \ ,
\end{equation}
since the terms coming from the commutators $[a, a^*]=1$ are of
lower order as $N\to \infty$ and vanish in the limit. From
convexity it follows that
\begin{equation}
N^{-n} \langle \Psi_0 | (a^*a)^n|\Psi_0\rangle \geq N^{-n} \langle
\Psi_0 | a^*a|\Psi_0\rangle ^n \,
\end{equation}
which converges to $1$ as $N\to\infty$, proving our claim.
Another corollary, important for the interpretation of
experiments, concerns the momentum distribution of the ground
state.
\begin{corollary}[\textbf{Convergence of momentum distribution}] Let
$$\widehat\rho (\k)=\int \int\gamma(\x, \x') \exp [i \k\cdot (\x
-\x')]
d\x d\x'$$
denote the one-particle momentum density of $\Psi_0$. Then, for
fixed $Na$, $$ \lim_{N\to\infty} \frac 1N
\widehat\rho(\k)=|\widehat\varphi^{\rm GP}(\k)|^2 $$ strongly in
$L^1(\R^3)$. Here, $\widehat\varphi^{\rm GP}$ denotes the Fourier
transform of $\varphi^{\rm GP}$.
\end{corollary}
\begin{proof} If ${\mathcal F}$ denotes the (unitary) operator `Fourier
transform' and if $h$ is an arbitrary $L^\infty$-function,
then
\begin{eqnarray}\nonumber
\left|\frac 1N\int \widehat\rho h-\int |\widehat\varphi^{\rm
GP}|^2 h\right|&=&\left|\Tr[{\mathcal F}^{-1}
(\gamma/N-P^{\rm GP}){\mathcal F}h]\right|\\ \nonumber
&\leq& \|h\|_\infty \Tr |\gamma/N-P^{\rm GP}|,
\end{eqnarray}
from which we conclude that $$\|\widehat\rho/N-|\widehat\varphi^{\rm
GP}|^2 \|_1\leq \Tr|\gamma/N-P^{\rm GP}|\ .$$
\end{proof}
Before proving Theorem \ref{becthm}, let us state some prior
results on which we shall build. Then we shall formulate two
lemmas, which will allow us to prove Theorem \ref{becthm}.
The following theorem is an extension of Theorem \ref{thmgp3}.
\begin{theorem}[\textbf{Asymptotics of the energy components}]\label{compthm}
If $\psi_0$ denotes \linebreak the solution to the zero-energy scattering equation for
$v$ (under the boundary condition
$\lim_{|\x|\to\infty}\psi_0(\x)=1$) and
$s=\int|\nabla\psi_0|^2/(4\pi a)$, then $0**