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\headline={\hfill{\fivepoint  GMGJPS --- 19/Feb/93 }}
\def\H{{\cal H}}
\def\R{{\bf R}}
\def\C{{\bf C}}
\def\Z{{\bf Z}}
\def\mfr#1/#2{\hbox{$#1\over#2$}}
\def\upgamma{\raise1pt\hbox{$\gamma$}}
\def\gammaf{\upgamma_{\rm F}}
\def\uprho{\raise1pt\hbox{$\rho$}}
\def\qed{{$\, \vrule height 0.25cm width 0.25cm depth 0.01cm \,$}}
\def\Tr{{\rm Tr}}
\def\R{{\bf R}}
\def\C{{\bf C}}
\def\uZ{\underline{Z}}
\def\uR{\underline{\cal R}}
\def\CTF{\raise1pt\hbox{\rm c}_{\rm TF}}
\def\CD{\raise1pt\hbox{\rm c}_{\rm D}}
\def\iint{\int\!\!\int}
\def\orho{\overline{\uprho}}
\def\const{{\rm const}\,}
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\centerline{\bf A CORRELATION ESTIMATE WITH APPLICATIONS TO QUANTUM}
\centerline{\bf SYSTEMS WITH COULOMB INTERACTIONS}
\vskip 1cm
{\baselineskip=0.75\baselineskip
{\settabs 3 \columns
\+Gian Michele Graf{$^1$}{}&&Jan Philip Solovej{$^2$}\cr
\+Institut f\"ur theoretische Physik&&Department of Mathematics\cr
\+ETH H\"onggerberg&&Princeton University\cr
\+CH--8093 Z\"urich, Switzerland&& Princeton, N.J. 08544\cr\smallskip}
\vfootnote{$^1$}{\eightpoint Alfred P. Sloan fellow}
\vfootnote{$^2$} {\eightpoint Work partially supported by U.S.
National Science Foundation grant 92--03829}
\bigskip
\centerline{\it Dedicated to Elliott H. Lieb on his 60th birthday} 
\vskip 1cm
{\narrower{\it Abstract:\/}
We consider some two-body operators acting on a Fock
space with either fermionic or no statistics.  We prove that they are
bounded below by one-body operators which mimic exchange effects. This
allows us to compare two-body correlations of fermionic and bosonic
systems with those in Hartree-Fock, respectively Hartree theory.  Applications
of the fermionic estimate yield lower bounds for the ground state
energy of jellium at high densities and of molecules with large nuclear
charges.\smallskip}}
\vskip 2cm
\noindent{\bf I. INTRODUCTION}

One of the striking differences between classical and quantum mechanics
is the existence of non-trivial correlations in the ground state of a quantum
 mechanical
system. {F}rom a rigorous point of view very little is known about these
 correlations.
By neglecting these correlations one arrives at an approximating theory for
the ground state energy. In the case of a fermionic system this theory
is the Hartree-Fock approximation while in the case of a bosonic system
the theory is the Hartree approximation.
We shall define these approximations below.

The first step in understanding the ground state correlations is to investigate
the validity of these approximating theories.
Although such an investigation will not give any information about the actual
physical implications due to the correlations it will, however, give an estimate
 on their
effects.

In a recent very beautiful paper [1] Bach gives an estimate on the
ground state correlations of Coulomb systems which proves the exactness of the
Hartree-Fock approximation in the high density limit. The importance of this
result motivated us to simplify and generalize the method of Bach.
It is this simplification and generalization that we present here.

An important ingredient in the method is to prove that there is
condensation. By this we mean in the case of fermions that the ground state
is a fairly well-defined Fermi sea. It turns out that this is
not too difficult to prove in the high density limit of Coulomb systems.
In the case of bosons, condensation means Bose-Einstein condensation, i.e.,
that in the ground state almost all particles are in one state.

In the case of bosons it can actually be very difficult to prove condensation.
An example of this is the bosonic jellium model which we define below.
Here condensation has to the best of our knowledge never been rigorously
 established.
However, in this case a very impressive method (avoiding the issue of
 condensation)
for obtaining a correlation estimate of the right order was given by Conlon
 [4,5] and by
Conlon, Lieb and Yau [6].
Assuming condensation in the bosonic jellium model is also the essential feature
of the Bogoliubov approximation (where correlations are not neglected)
which was used by Foldy [10] to compute the ground state energy to leading order
in the high density limit. It is a very interesting open problem to give a
rigorous derivation of this result.

In the present paper we shall be concerned with Coulomb systems.
By this we mean that we consider $N$-body Hamiltonians of the form
$$
        H(N,V) =\sum_{i=1}^N(-\Delta_i-V(x_i))
        +\sum_{1\leq i<j\leq N}{1\over|x_i-x_j|}\quad,\eqno(1.1)
$$
acting on the many-body space $\H(N)=\bigotimes\limits^N\H$, where
the one-body space is $\H=L^2(\R^3;\C^m)$ and $x_i\in \R^3$, $i=1,\ldots,N$.
The space $\C^m$ refers to the spin variable. It corresponds to particles
having $m$ spin states.

The operator $H(N,V)$ acts also on the fermionic (antisymmetric) subspace
$\H_{\rm F}(N)=\bigwedge\limits^N\H$ and on the bosonic (symmetric) subspace.
In general for any operator $A$ on $\H$ we define operators $A_i$,
 $i=1,2,\ldots,N$
on $\H(N)$ by simply letting $A$ act on the $i^{\rm th}$ particle (i.e., on the
$i^{\rm th}$ factor in the tensor product).
The one-body operator $\sum_i^NA_i$ and the two-body operator
$\sum_{1\leq i<j\leq N}A_iA_j$ then act on $\H(N)$ as well as on the fermionic
 and
bosonic subspaces.

We shall only be interested in the ground state energy, i.e., the
lowest eigenvalue (or in general the bottom of the spectrum) of $H(N,V)$.
The bottom of the spectrum of $H(N,V)$ restricted to the bosonic
subspace is identical to the bottom of the spectrum of the operator
acting on the full space $\H(N)$. The unrestricted operator is therefore often
refered to as the bosonic case. We shall here mainly be interested in the
fermionic case.

The fermionic ground state energy is
$$
        E(N,V)=\inf_{\langle\ \cdot\ \rangle_{\rm F}}
               \langle H(N,V) \rangle_{\rm F}\quad,\eqno(1.2)
$$
where the infimum is over all fermionic states
$\langle\ \cdot\ \rangle_{\rm F}$.  By a fermionic state we mean a
positive linear functional on the bounded operators on $\H_{\rm F}(N)$
satisfying the normalization condition that its value on the identity
$I$ is $\langle I\rangle_{\rm F}=1$. Although, $H(N,V)$ is not bounded
we only consider potentials $V$ for which $H(N,V)$ is bounded from
below. The expectation $\langle H(N,V) \rangle_{\rm F}$ of $H(N,V)$ in a
fermionic state is then a meaningful quantity (possibly having the value
infinity). As explained above the bosonic ground state energy can be defined as
 in
(1.2) but with the infimum being over states on the full space $\H(N)$. The
bosonic energy is thus clearly smaller than the fermionic energy.

Corresponding to states $\langle\cdot\rangle_{\rm F}$ on $\H_{\rm F}(N)$
and $\langle\cdot\rangle$ on $\H(N)$ we define operators $\gammaf$ and $\gamma$
on the one-body space $\H$. These operators are called
{\it one-particle density matrices} for the states and are defined by the
 following
relations which should hold for all bounded operators $A$ on $\H$.
$$
        \Bigl\langle\sum_iA_i\Bigr\rangle={\rm Tr}[\gamma A] \quad\hbox{and}\quad
        \Bigl\langle\sum_iA_i\Bigr\rangle_{\rm F} ={\rm Tr}[\gammaf A]\quad.
$$
It is easy to see that $\Tr\gamma=N$ and $\Tr\gammaf=N$. Moreover, $\gammaf$
 satisfies
that as an operator $0\leq\gammaf\leq I$.

We shall restrict our attention to two different systems (when dividing into
bosonic and fermionic models we in fact get four systems). The first is a
 molecule in the Born-Oppenheimer approximation. Such a molecule with $N$ electrons
and with $K$ nuclei having prescribed positive charges $\uZ=(Z_1,\ldots,Z_K)$
and positions $\uR=({\cal R}_1,\ldots,{\cal R}_K)$ is described by a
Hamiltonian $H(N,\uZ,\uR):=H(N,V)$ as in (1.1) where
$$
        V(x)=\sum_{k=1}^K{Z_k\over|x-{\cal R}_k|}\quad.\eqno(1.3)
$$
We denote the ground state energy of the molecule by $E(N,\uZ,\uR)=E(N,V)$.
We are using the term Born-Oppenheimer here to indicate that we are neglecting
the nuclear motion completely. We shall not, as is often done in the Born-Oppenheimer 
approximation, study the nuclear motion  in the potential $\uR\mapsto E(N,\uZ,\uR)$.
We are interested in the limit as the total nuclear charge $Z=\sum_{k=1}^K Z_k$
tends to infinity. We shall not discuss the bosonic molecule here, but refer
 the reader
to [3] where the atomic case ($K=1$) is treated. (In [3] a somewhat weaker
 version
of inequality (2.8) given below is used. Our proof of (2.8) would slightly
 simplify
the presentation in [3].)

The second system is the neutral jellium model on a cube $\Lambda$ in $\R^3$
with volume $|\Lambda|$. This model is again described by a Hamiltonian
$H(N,\Lambda):=H(N,V)$ as in (1.1)
but this time with the potential $V$ coming from a uniform charge background
$$
        V(x)=\cases{{N\over|\Lambda|}\int_{\Lambda}dy|x-y|^{-1}& if
        $x\in\Lambda$\cr
        -\infty&if $x\not\in\Lambda$}\quad.\eqno(1.4)
$$
(The value $-\infty$ amounts to choosing the Dirichlet condition on the boundary
of the cube $\Lambda$.) The energy of jellium is denoted by
 $E(N,\Lambda)=E(N,V)$.
The existence of the thermodynamic limit of the ground state energy with the
self-energy of the background included, i.e.,
$$
        E_{\rm J}(\overline{\uprho})=
        \lim_{N,|\Lambda|\to\infty\atop N/|\Lambda|=\overline{\uprho}}
        |\Lambda|^{-1}
        \left(E(N,\Lambda)+\mfr1/2\orho^2\iint_{\Lambda\times\Lambda} dxdy
	|x-y|^{-1}
        \right)\quad,\eqno(1.5)
$$
was proved in [15] . We are interested in the high density limit
 ($\overline{\uprho}
\to\infty$) of $E_{\rm J}(\overline{\uprho})$.
Again we shall not discuss the bosonic case, but this time the
reason, as explained above, is that the method presented here does not work.
We have to refer to [4--6].

We now introduce three approximations to the ground state energy of
the Hamiltonian (1.1). All statements below should be understood to hold
for the potentials in (1.3) and (1.4), but not necessarily for potentials in
general.
\medskip
\noindent{\bf Hartree-Fock Theory:}
In Hartree-Fock theory one restricts attention to states
 $\langle\cdot\rangle_{\rm F}$
that have {\it no correlations}. By this we mean that the expectation of any
 operator
$H$ is $\langle H\rangle_{\rm F}=\langle\Psi|H\Psi\rangle$
where $\Psi\in\H_{\rm F}(N)$ is of the Slater determinant type, i.e., of the
 form
$\Psi=(N!)^{-1/2}\psi_1\wedge\ldots\wedge\psi_N$, where 
$\psi_1,\ldots,\psi_N\in\H$
are orthonormal one-particle wave functions.
For such a state the one-particle density matrix is the projection operator $P$
in $\H$ projecting onto the space spanned by $\psi_1,\ldots,\psi_N$ (i.e., the
 Fermi sea).
It is easy to see that the one-particle density matrix $P$ determines the
uncorrelated state uniquely.
A straightforward computation shows that the expectation
$\langle\Psi|H(N,V)\Psi\rangle$ is equal to
$$
{\cal E}_{\rm HF}(P)
 =\Tr_\H[(-\Delta-V)P]+D(\uprho_P,\uprho_P)
        -\mfr1/2\sum_{\sigma,\sigma'}\iint dxdy
                {|P(x,\sigma;y,\sigma')|^2\over|x-y|}\quad,\eqno (1.6)
$$
where $P(x,\sigma;y,\sigma')$ is the integral kernel for $P$ with
$\sigma$ and $\sigma'$ denoting the spin variables. The density is
$ \uprho_P(x)=\sum_\sigma P(x,\sigma;x,\sigma)$ and we have introduced
the positive definite quadratic form
$$
        D(\uprho_1,\uprho_2)=\mfr1/2
        \iint dxdy \uprho_1(x)|x-y|^{-1}\uprho_2(y)\quad.\eqno(1.7)
$$
We call the functional ${\cal E}_{\rm HF}$ the Hartree-Fock functional.
If we allow the value infinity this functional is defined on all trace class
 operators
$P$ on $\H$. The term in (1.6) involving the quadratic form $D$ is called
the {\it direct Coulomb energy} while the last term in (1.6) is the
{\it exchange energy\/}.

The Hartree-Fock energy is
$$
        \eqalignno{E_{\rm HF}(N,V)=&\inf\Bigl\{{\cal E}_{\rm HF}(P)\mid P
        \hbox{ is an $N$ dimensional projection}\Bigr\}\cr
        =&\inf\Bigl\{{\cal E}_{\rm HF}(P)
        \mid 0\leq P\leq I,\ \Tr P=N\Bigr\}\quad .&(1.8)}
$$
That the variational problem can be formulated by the second line was realized
by Lieb in [14] (for a simple proof of this fact see also [1]) .
{F}rom the minimax principle it follows that $E(N,V)\leq E_{\rm HF}(N,V)$.
\medskip
{\it Remarks\/}:\enskip
(a) We shall measure the degree of non-condensation of a general state
$\langle\cdot\rangle_{\rm F}$, with one-particle density matrix $\gammaf$, into
the Fermi sea, given by the projection $P$,  by the positive quantity (see also [2])
$$
        N^{-1}\delta(\gammaf,P):=N^{-1}\Tr[\gammaf(I-P)]\quad,
$$
i.e., the relative number of particles outside the Fermi sea.
Thus if $N^{-1}\delta(\gammaf,P)\ll 1$ we say that the state is well condensated.
As in (1.8) we shall not limit $P$ to be a projection but in general allow $P$
to satisfy $0\leq P\leq I$ in the definition of $\delta(\gammaf,P)$.


(b) In case of bosons an uncorrelated state is given by a product wave function
$\Psi=\bigotimes\limits^N\psi$ where $\psi$ is a normalized function in $\H$.
The one-particle density matrix is $\gamma=NP$ where $P$ is the projection
onto $\psi$. Hartree theory is the restriction to these states.
Condensation is defined as in (a).
\medskip
\noindent{\bf Thomas-Fermi Theory:} The second approximating theory which is
 much
simpler than Hartree-Fock theory is the statistical description given
by the Thomas-Fermi functional
$${\cal E}_{\rm TF}(\uprho)
=\int dx\Bigl(\mfr3/5\CTF\uprho(x)^{5/3}-V(x)\uprho(x)\Bigr)
  +D(\uprho,\uprho)\quad,\eqno(1.9)
$$
where $\CTF=(6\pi^2/m)^{2/3}$. More precisely the statistical description
is given by the minimization problem
$$
  E_{\rm TF}(N,V)=
   \inf\Bigl\{
       {\cal E}_{\rm TF}(\uprho)\mid\uprho\geq 0,\,\int dx\ \uprho\leq N\
       \Bigr\}\quad.\eqno(1.10)
$$

It is known [13,16] that there is a unique minimizer $\uprho_{\rm TF}$. Together
with the potential $V$, this charge distribution gives raise to the Thomas-Fermi
potential
$$\phi_{\rm TF}=V-\uprho_{\rm TF}*{1\over|x|}\quad.\eqno(1.11)$$
\medskip
\noindent{\bf Dirac-Schwinger Theory:}  What we here call the Dirac-Schwinger
 theory
is an intermediate model between Hartree-Fock theory
and Thomas-Fermi theory defined in terms of the Hamiltonian
$$
  H_{\rm DS}(N,V)=
    \sum_{i=1}^N h_i-D(\uprho_{\rm TF},\uprho_{\rm TF})
    -\CD\int dx\uprho_{\rm TF}(x)^{4/3}\quad,\eqno(1.12)
$$
$$
   h=-\Delta-\phi_{\rm TF}(x)\quad,\eqno(1.13)
$$
acting on $\H_{\rm F}(N)$. Here $\CD=(2\pi)^{-3}m\CTF^2$ is the Dirac [7]
 constant.
In this model noninteracting electrons move in the Thomas-Fermi mean-field 
potential. The
term $D(\uprho_{\rm TF},\uprho_{\rm TF})$ is subtracted
because the first term
overcounts the Coulomb repulsion between the electrons.
The exchange energy is
taken into account simply by subtracting a constant. Correspondingly we have
$$
  E_{\rm DS}(N,V)=\inf_{\langle\ \cdot\ \rangle_{\rm F}}
               \langle H_{\rm DS}(N,V) \rangle_{\rm F}\quad.\eqno(1.14)
$$

The limits we are interested in (i.e., $Z\to\infty$ for the molecular problem
 and
$|\Lambda|\to\infty$ for jellium) can be considered as quasiclassical
limits for the Hamiltonian $H_{\rm DS}$.  Thomas-Fermi theory is the leading
 order
quasiclassical approximation to $H_{\rm DS}$ with the exchange term neglected.

In the molecular case the quasiclassical approximation of $H_{\rm DS}$ was
studied in [12,13,\break16,20]. It is known that the energies
$$E(N,\uZ,\uR),\ E_{\rm HF}(N,\uZ,\uR),\  E_{\rm DS}(N,\uZ,\uR)
\ \hbox{and}\ E_{\rm TF}(N,\uZ,\uR)$$
agree to the leading order which is $O(Z^{7/3})$ (for $N$ of order $Z$) as
 $Z\to\infty$.

In the atomic case ($K=1$) it was realized by Schwinger [18] that the
second correction\footnote{$^\dagger$}{\eightpoint The first correction is the 
Scott correction
due to the Coulomb singularities at the positions of the nuclei} to the
leading order quasiclassical approximation is of the same order
as the exchange energy originally computed by Dirac in [7]. The
order of the exchange energy is $O(Z^{5/3})$.
It would therefore not be adequate in this case to replace $H_{\rm DS}$
by what is called the Thomas-Fermi-Dirac functional which is
the Thomas-Fermi functional with the exchange integral (the last term in (1.12))
included.
The non-rigorous analysis of Schwinger was proved to be mathematically
correct in the atomic case in the monumental work of Fefferman and Seco
announced in [9]. We shall not here go into this very involved analysis.

Our main result in the molecular case is the following theorem first proved
in [1] and [2].

{\bf THEOREM 1.} {\it For some $\varepsilon>0$ we have
$$\eqalignno{
    E_{\rm HF}(N,\uZ,\uR)\geq E(N,\uZ,\uR)
   &\geq E_{\rm HF}(N,\uZ,\uR)-{\rm O}(Z^{5/3-\varepsilon})&(1.15)\cr
    E(N,\uZ,\uR)
   &\geq E_{\rm DS}(N,\uZ,\uR)-{\rm O}(Z^{5/3-\varepsilon})\quad,&(1.16)
}$$
as $Z\to\infty$ uniformly in $\uZ,\,\uR$ and $N\leq Z$.
}

In (1.16) we can only prove the lower bound. An upper bound presumably requires
an analysis similar to the one described in [9]. In [2] it was shown
how, assuming the results announced in [9], one can prove an upper bound.

In the case of fermionic jellium the semiclassical analysis becomes
exact in the thermodynamic limit and the main result in this case has
a more explicit form.

{\bf THEOREM 2.} {\it For $\orho\geq\const m^2$ we have
$$
        E_{\rm J}(\overline{\uprho})=
	\mfr3/5\CTF\overline{\uprho}^{5/3}-
	\CD\overline{\uprho}^{4/3}
        -{\cal C}(\orho)\orho^{4/3}\quad,\eqno(1.17)
$$
where $0\leq{\cal C}(\orho)\leq
C_{\varepsilon} m^{2/15}\orho\phantom{)}^{-1/15+\varepsilon}$
for all $\varepsilon>0$ small enough.}
\medskip
{\it Remark\/}:\enskip A perturbative computation of Gell-Mann and Brueckner
[11] leads to the asymptotic behavior
${\cal C}(\orho)\sim\orho\phantom{)}^{-1/3}\log\orho$.
\medskip
The paper is organized as follows. In Sect.~II we give a purely algebraic
inequality which allows one to estimate correlations. It estimates a
two-body operator below in terms of one-body operators.
The estimate involves a one-body operator $P$ on $\H$.
The error term in the algebraic inequality is (almost) proportional to
the operator $I-P$.
In the applications $P$ will (almost) be the projection which minimizes
the Hartree-Fock functional and thus $I-P$ is the orthogonal projection
and the error term can be controlled by the degree of condensation.
In Sect.~III we show how the algebraic inequality can be applied to the Coulomb
 potential.
In Sect.~IV we study the jellium problem and in Sect.~V the more complicated
molecular problem.
\bigskip
\noindent{\bf II. THE ALGEBRAIC INEQUALITY}

We present here a purely algebraic inequality for operators
acting on tensor products of a Hilbert space.
Let $\H$ be a complex Hilbert space. For an integer $N>0$
we shall consider the $N$-fold tensor product $\H(N)=\bigotimes\limits^N\H$
and its antisymmetric subspace $\H_{\rm F}(N)=\bigwedge\limits^N\H$.
In terms of particle language $\H(N)$ is a full $N$ particle space and
$\H_{\rm F}(N)$ is its fermionic subspace.

\smallskip
{\bf THEOREM 3.} {\it Let $X$ and $P$ be bounded operators on $\H$ satisfying
$0\leq X$ and $0\leq P\leq I$ (where $I$ is the
identity operator on $\H$).  For all $\alpha\geq0$ we have the following
 operator
inequality on the $N$-particle space $\H(N)$
$$
        \sum_{1\leq i<j\leq N}X_iX_j\geq \sum_i
	\Bigl(\alpha X_i-\mfr1/2X_iP_iX_i\Bigr)
        -\mfr1/2\alpha^2 -\sum_iB_i\quad,\eqno(2.1)
$$
where the error term is given by the following operator on $\H$ depending on an
arbitrary parameter $\lambda\geq0$
$$
        \eqalignno{B=&\Bigl[\alpha+\mfr1/2\left(1+2\lambda+
	(4+\lambda+\lambda^{-1})
        (N-1)\right)\|PXP\|\Bigr](I-P)X(I-P)\cr
        &+\mfr1/2\lambda^{-1}(X^2+PX^2P)\quad.&(2.2)}
$$
However, if we restrict to the fermionic subspace $\H_{\rm F}$ inequality (2.1)
holds with the operator $B$ replaced by
$$
        \eqalignno{B_{\rm F}=&\Bigl[\alpha+\mfr1/2(5+3\lambda
        +\lambda^{-1}){\rm Tr}[PX]\Bigr] (I-P)X(I-P)\cr
        &+\mfr1/2\lambda^{-1}(X^2+PX^2P)\quad.&(2.3)}
$$
(Notice that in contrast to $B$ the operator $B_{\rm F}$ has no $N$ dependence.)
}

{\it Remark:\/} We want to emphasize that inequality (2.1) estimates a
two-body operator in terms of one-body operators.

{\it Proof:\/} For simplicity we shall denote $Q=I-P$. Then $0\leq Q\leq I$.
There are two main ingredients in the proof of (2.1).
The first is an operator identity which we write as
$ \sum_{1\leq i<j\leq N}X_iX_j=A_1+A_2+A_3$, where
$$\eqalignno{
        A_1=&\sum_{1\leq i<j\leq N}\Bigl(Q_iQ_j+Q_iP_j + P_iQ_j\Bigr)X_iX_j
        \Bigl(Q_iQ_j+Q_iP_j + P_iQ_j\Bigr)\quad,&(2.4)\cr
        \noalign{\hbox{and}}
        A_2=&\mfr1/2\sum_{i\ne j}\Bigl(P_iP_jX_iX_jP_iP_j+2P_iP_jX_iX_jP_iQ_j
        +2Q_iP_jX_iX_jP_iP_j\Bigr)\cr
        =&\mfr1/2 \biggl|\alpha-\sum_iP_iX_i(I+Q_i)\biggr|^2-\mfr1/2\alpha^2\cr
        &+\sum_i\Bigl\{\alpha X_i\ -\mfr1/2(I+Q_i)X_iP^2_iX_i(I+Q_i)
        -\alpha Q_iX_iQ_i\Bigr\}\cr
         &-2\sum_{i\ne j}Q_iX_iP_iP_jX_jQ_j\quad, &(2.5)\cr
        \noalign{\hbox{where we have used the convention $|T|^2=T^*T$ and}}
        A_3=&\sum_{i<j}\biggl(P_iP_jX_iX_jQ_iQ_j+Q_iQ_jX_iX_jP_iP_j\Bigr)
        =\mfr1/2\biggl|\lambda^{-1/2} \sum_iQ_iX_iP_i
        +\lambda^{1/2}\sum_iP_iX_iQ_i\biggr|^2\cr
        &-\mfr1/2\sum_i\Bigl\{\lambda Q_iX_iP_i^2X_iQ_i
        +\lambda^{-1}P_iX_iQ_i^2X_iP_i+
        P_iX_iQ_iP_iX_iQ_i+Q_iX_iP_iQ_iX_iP_i\Bigr\}\cr
        &-\mfr1/2(\lambda+\lambda^{-1}) \sum_{i\ne
 j}Q_iX_iP_iP_jX_jQ_j\quad.&(2.6)
}
$$
Since $X\geq0$, the expression $A_1\geq0$.
Furthermore, the first terms in the above expressions for
both $A_2$ and $A_3$ are also positive and can thus be neglected for the lower
 bound.

The second main ingredient in the proof is the estimate of the only remaining
 two-body
term, namely the last term in the expressions for $A_2$ and $A_3$.
In  estimating this term the positivity of the operator $X$ is again essential,
 as it
was in concluding that $A_1\geq0$.
It is also here that the difference between the full $N$-body space $\H(N)$ and
 the
antisymmetric fermionic subspace $\H_{\rm F}(N)$ plays an important role.
We proceed as follows
$$
        \eqalignno{\sum_{i\ne j}&Q_iX_iP_iP_jX_jQ_j=\sum_{i\ne j}P_iX_iP_iQ_jX_j
   Q_j
        -\mfr1/2\sum_{i\ne j} (Q_jP_i-Q_iP_j)X_iX_j(Q_jP_i-Q_iP_j)\cr
        \leq&\sum_{i\ne j}(Q_jX_jQ_j)^{1/2}(P_iX_iP_i)(Q_jX_jQ_j)^{1/2}
        \leq\cases{(N-1)\|PXP\|\sum_iQ_iX_iQ_i&on $\H(N)$\cr\cr
        \hbox{Tr}[PX]\sum_iQ_iX_iQ_i&on $\H_{\rm F}(N)$}\quad.}
$$
The last inequality follows since the operator $\sum_{i=1}^{N-1}P_iX_iP_i$ on
$\H(N-1)$ is bounded in norm by $(N-1)\|PXP\|$ and as an operator on $\H_{\rm
 F}(N-1)$
it is bounded in norm by Tr$[PX]$.

The proof of the theorem is now essentially complete, since we have found a
 lower bound
in terms of one-body operators. It only remains to write this lower bound in as
 simple
a form as possible. Since we are now only dealing with one-body operators
this last step involves only estimates on operators on the one-body space $\H$.

In estimating $A_2$ we single out the estimate
$$
        \eqalignno{(I+Q)XP^2X(I+Q)&=XPX-XPQX +QXP^2X+XP^2XQ+QXP^2XQ\cr
        &\leq XPX-XPQX+(1+\lambda)QXP^2XQ+\lambda^{-1}XP^2X\quad.}
$$
Combining this with the simple operator estimate $XPQX\geq PXPQXQ+QXPQXP$
we arrive at
$$
        \eqalignno{A_2+A_3\geq&\sum_i\Bigl(\alpha X_i-\mfr1/2X_iP_iX_i\Bigr)
        -\mfr1/2\alpha^2 -\sum_iB'_i\cr
        &-(2+\mfr1/2(\lambda+\lambda^{-1}))
        \cases{(N-1)\|PXP\|\sum_iQ_iX_iQ_i&on $\H(N)$\cr\cr
        \hbox{Tr}[PX]\sum_iQ_iX_iQ_i&on $\H_{\rm F}(N)$}\quad,&(2.7)}
$$
where
$$
        B'=\alpha
 QXQ+\mfr1/2(1+2\lambda)QXP^2XQ+\mfr1/2\lambda^{-1}(PXQ^2XP+XP^2X)\quad.
$$
Finally, using $\|X^{1/2}P^2X^{1/2}\|=\|PXP\|$ we get that
$QXP^2XQ\leq \|PXP\|QXQ\leq\hbox{Tr}[PX]QXQ$. Using this together with the
trivial estimates $P^2\leq P\leq I$ and $Q^2\leq Q\leq I$ we see that
(2.7) leads to the estimate (2.1) with (2.2) and (2.3) respectively.
\qed

As a simple consequence of this theorem we shall now give the estimates of Bach
 [1] (see also [3]).
Consider states $\langle\ \cdot\ \rangle$ and
$\langle\ \cdot\ \rangle_{\rm F}$ on the algebra of bounded
operators on $\H(N)$ and $\H_{\rm F}(N)$ respectively.
Let $\gamma$ and $\gammaf$  be the corresponding one-particle density matrices.
\smallskip
{\bf COROLLARY 4 (Bach's inequalities).} {\it If $X$ is a projection,
i.e., $X^2=X=X^*$ we have
$$\eqalignno{
        \left\langle\sum_{1\leq i<j\leq N}X_iX_j\right\rangle
        \geq&\mfr1/2\Bigl(\left({\rm Tr}[X\gamma]\right)^2
        -N^{-1}{\rm Tr}[\gamma X\gamma X]\Bigr)\cr
        &-\mfr1/2{\rm Tr}[X\gamma]\min\left\{1, \sqrt{\mfr57/2}
        \left[{\rm Tr}[X(\gamma-N^{-1}\gamma^2)]\right]
        ^{1/2}\right\}&(2.8)\cr
\noalign{\hbox{and}}
        \left\langle\sum_{1\leq i<j\leq N}X_iX_j\right\rangle_{\rm F}\geq&
        \mfr1/2\Bigl(\left({\rm Tr}[X\gammaf]\right)^2
        -{\rm Tr}[\gammaf X\gammaf X]\Bigr)\cr
        &-\mfr1/2{\rm Tr}[X\gammaf]\min\left\{1, \sqrt{\mfr77/2}
        \left[{\rm Tr}[X(\gammaf-\gammaf^2)
        ]\right]^{1/2} \right\}&(2.9)\cr
}
$$
}

{\it Proof:\/}
In both (2.8) and (2.9) the case of the min being equal to $1$ is proved
from (2.1) by  simply choosing $\alpha=\Tr[X\gamma]$ or $\alpha=\Tr[X\gammaf]$
and $P=I$ and $\lambda=\infty$. In this case the operators $B$ and $B_{\rm F}$
 are
both zero.

We next turn to the case when the min in (2.8) and (2.9) is smaller than $1$.
We consider first (2.8). We shall use (2.1) and (2.2) with
$\lambda^{-1}=\mfr1/4\sqrt{\mfr57/2}
        \left[{\rm Tr}[X(\gamma-N^{-1} \gamma^2)]\right] ^{1/2}$
and $\alpha=\hbox{Tr}[X\gamma]$ and $P=N^{-1}\gamma$. Since the min in (2.8) is
assumed to be smaller than $1$ we have $\lambda\geq4$.
To arrive at (2.8) we need the additional simple observation that
$N\|PXP\|\leq N\hbox{Tr}[PX] =\hbox{Tr}[\gamma X]$. We can then estimate the
error term in (2.2) in the following way using that $\lambda\geq4$.
$$
        \eqalign{\Tr[B\gamma]\leq&\mfr1/2(6+2\lambda+\lambda^{-1})
        \Tr[X\gamma]\Tr[X(\gamma-N^{-1}\gamma^2)]+\lambda^{-1}\Tr[X\gamma]\cr
        \leq&\mfr1/2\sqrt{\mfr57/2}\Tr[X\gamma]
        \left[{\rm Tr}[X(\gamma-N^{-1}\gamma^2)\right]^{1/2}
        \quad.}
$$

To get (2.9) we choose $\lambda^{-1}=\mfr1/4\sqrt{\mfr77/2}
\left[{\rm Tr}[X(\gammaf- \gammaf^2)
\right] ^{1/2}$ and $\alpha=\hbox{Tr}[X\gammaf]$ and $P=\gammaf$.
As before we have $\lambda\geq4$. Inserting this into (2.3) easily gives
(2.9).\qed

In the next section we will use the following inequality, which we state
for fermions only:
\smallskip
{\bf COROLLARY 5.} {\it Let $X$ and $P$ be bounded operators on $\H$ satisfying
$X=X^2=X^*$ and $0\leq P\leq I$, and let $\alpha\geq0$.
Then
$$\eqalignno{
       \left\langle\sum_{1\leq i<j\leq N}X_iX_j\right\rangle_{\rm F}\geq&
       \alpha{\rm Tr}[X\gammaf]-\mfr1/2\alpha^2-\mfr1/2{\rm Tr}[PXPX]&(2.10)\cr
      &-{\rm const}(\alpha+{\rm Tr}[XP]+{\rm Tr}[X\gammaf])
          \min\left\{
          1, \left[{\rm Tr}[X(I-P)\gammaf(I-P)]\right]^{1/2}
          \right\}\quad.&\cr
}
$$
}

{\it Proof:\/}
The case where the min is 1 is done as before. In the other case we first note
that for fermions we have
$$\eqalign{
  \sum_iX_iP_iX_i
 &\leq(1+\lambda^{-1})\sum_iP_iX_iP_iX_iP_i+(1+\lambda)\sum_iQ_iX_iP_iX_iQ_i\cr
 &\leq{\rm Tr}[PXPX]+\lambda^{-1}\sum_iP_iX_iP_i
   +(1+\lambda){\rm Tr}[XP]\sum_iQ_iX_iQ_i
}$$
since $\sum_iP_iX_iP_iX_iP_i\leq{\rm Tr}[PXPXP]={\rm Tr}[P^{1/2}XP^2XP^{1/2}]
\leq{\rm Tr}[P^{1/2}XPXP^{1/2}]={\rm Tr}[PXPX]$ and
$\|X^{1/2}PX^{1/2}\|=\|P^{1/2}XP^{1/2}\|\leq{\rm Tr}[XP]$. This shows that in
(2.1) $\sum_iX_iP_iX_i$ can be replaced by ${\rm Tr}[PXPX]$ at the expense of
changing the numerical constants in (2.3). We then let
$\lambda^{-1}={\rm Tr}[XQ\gammaf Q]^{1/2}$ and use $\lambda\geq1$:
$$\eqalign{
  {\rm Tr}[B_{\rm F}\gamma]
&\leq{\rm const}\lambda(\alpha+{\rm Tr}[XP]){\rm Tr}[QXQ\gammaf]+
   {\rm const}\lambda^{-1}({\rm Tr}[X\gammaf]+{\rm Tr}[PXP\gammaf])\cr
&\leq{\rm const}(\alpha+{\rm Tr}[XP]+{\rm Tr}[X\gammaf])   
{\rm Tr}[XQ\gammaf Q]^{1/2} }
$$
since ${\rm Tr}[PXP\gammaf]={\rm Tr}[(PXP)^{1/2}\gammaf(PXP)^{1/2}]
\leq{\rm Tr}[XP]$.
\qed
\bigskip

{\it Remark:\/}\enskip One may ask whether the exponent $1/2$ appearing
in (2.8) and (2.9) is optimal. The following example shows that,
indeed, it is. The example is for the fermionic case (2.9), but it is
easy to give a similar example in the case of (2.8). The idea behind the example
is to construct fermionic states depending on a parameter $\varepsilon$
such that $\Tr[X(\gammaf-\gammaf^2)]\to0$ as $\varepsilon\to0$ and such that 
the leading contribution to the error in Theorem~3 comes from $A_3$
estimated in (2.6).
 
{\it Example:\/}\enskip
Let $\psi_1,\ldots,\psi_{N+2}$ be an orthonormal family of functions.
Define normalized $N$-body functions by
$\Psi_1=(N!)^{-1/2}\psi_1\wedge\ldots\wedge\psi_N$ and
$\Psi_2=(N!)^{-1/2}\psi_3\wedge\ldots\wedge\psi_{N+2}$ and let
$ \Psi=(1-\varepsilon)^{1/2}\Psi_1 -\varepsilon^{1/2}\Psi_2 $. Then $\Psi$
is also normalized and if $P$ and $P'$ are the one-particle density
matrices (projections) for $\Psi_1$ and $\Psi_2$ respectively,
one easily sees that the one-particle density matrix corresponding to $\Psi$
is $\gammaf=(1-\varepsilon)P+\varepsilon P'$.
Hence $\gammaf-\gammaf^2=\varepsilon(1-\varepsilon)P''$
where $P''$ is the projection onto span$\{\psi_1,\psi_2,\psi_{N+1},\psi_{N+2}\}$.
Now let $X$ be the projection onto span$\{\psi_1+\psi_{N+1}, \psi_2+\psi_{N+2}\}$.
Then $\Tr[\gammaf X]=1$ and $\Tr[X(\gammaf-\gammaf^2)]=2\varepsilon(1-\varepsilon)$.
We can also compute $\langle\Psi_{1}|\sum_{i<j}X_iX_j\Psi_{1}\rangle
=\mfr1/2\Bigl(\Tr[XP]^2-\Tr[PXPX]\Bigr)$ and likewise for $\Psi_2$.
Moreover, we find that the cross-term is 
$\langle\Psi_1|\sum_{i<j}X_iX_j\Psi_2\rangle=\mfr1/4$.
Notice, that if we write $\sum_{i<j}X_iX_j=A_1+A_2+A_3$ as in the proof of Theorem~3
corresponding to the decompostion of the  one-body space by $P$ and $Q=I-P$ then 
the cross-term gets a contribution from $A_3$ only.
Hence as $\varepsilon\to0$
$$
        \langle\Psi|\sum_{1\leq i<j\leq N}X_iX_j\Psi\rangle
        -\mfr1/2\Bigl(\left({\rm Tr}[X\gammaf]\right)^2
        -{\rm Tr}[\gammaf X\gammaf X]\Bigr)=-\mfr1/2\varepsilon^{1/2}+O(\varepsilon).
$$
\bigskip
\noindent{\bf III. APPLICATIONS TO QUANTUM SYSTEMS}

The results of the preceding section have applications to Coulomb systems
thanks to the Fefferman-de la Llave representation [8] of the Coulomb
potential:
$$
   {1\over|x-y|}={1\over\pi}\int dz\int_0^\infty{dr\over r^5}
              \chi_{B(r,z)}(x)\chi_{B(r,z)}(y)\quad,\eqno(3.1)
$$
where $\chi_{B(r,z)}$ is the characteristic function of the ball
$B(r,z)=\{x\in\R^3\mid |x-z|\leq r\}$.\par
For any density matrix $0\leq P\leq I$ with ${\rm Tr}P<+\infty$ acting on
$\H={\rm L}^2(\R^3,\C^m)$ let $\uprho_P\in{\rm L}^1(\R^3)$ be its density. It
satisfies ${\rm Tr}[Pf]=\int dx\uprho_P(x)f(x)$ for all
$f\in{\rm L}^\infty(\R^3)$. As before, we consider an $N$-particle fermionic
state $\langle\ \cdot\ \rangle_{\rm F}$ with corresponding one-particle
density matrix $\gamma$. The following is a generalization of Lemma 5 in [1]:
\smallskip
{\bf LEMMA 6.} {\it Fix $0<\varepsilon\leq 1/6$. For any density matrix $P$ and
   any
density $\uprho_0(x)\geq0$ we have
$$\eqalignno{
 \left\langle\sum_{1\leq i<j\leq N}{1\over|x_i-x_j|}\right\rangle_{\rm F}
     \geq& 2D(\uprho_0,\uprho_\gamma)-D(\uprho_0,\uprho_0)
        -\mfr1/2\sum_{\sigma,\sigma'}\iint dxdy
                {|P(x,\sigma;y,\sigma')|^2\over|x-y|}&\cr
     &-{\rm const}\|\rho\|_{5/3}^{5/6}\|\rho\|_1^{1/6+\varepsilon}
            \delta(\gamma,P)^{1/3-\varepsilon}&(3.2)
}$$
where $\rho=\uprho_0+\uprho_P+\uprho_\gamma$.
}

{\it Remark:\/}
The best choice for $\uprho_0$ is $\uprho_0=\uprho_\gamma$, since
$2D(\uprho_0,\uprho_\gamma)-D(\uprho_0,\uprho_0)=
-D(\uprho_\gamma,\uprho_\gamma)
-D(\uprho_0-\uprho_\gamma,\uprho_0-\uprho_\gamma)$.

{\it Proof:\/}
Let $X_{r,z}$ denote the  operator on $\H$ acting by multiplication with
$\chi_{B(r,z)}$; let $\alpha_{r,z}=\int_{B(r,z)}dx\uprho_0(x)$. Then
(2.10) yields
$$\eqalign{
       \left\langle\sum_{1\leq i<j\leq N}(X_{r,z})_i(X_{r,z})_j
       \right\rangle_{\rm F}\geq&
       \alpha_{r,z}{\rm Tr}[X_{r,z}\gamma]-\mfr1/2\alpha_{r,z}^2-
       \mfr1/2{\rm Tr}[PX_{r,z}PX_{r,z}]\cr
      &-{\rm const}\ E(r,z)
}
$$
where
$E(r,z)=(\int_{B(r,z)}dx\rho(x))
\min\{1,(\int_{B(r,z)}dx\uprho_{Q\gamma Q}(x))^{1/2}\}$. The claim follows
from this and from (3.1), except for the error term which we now estimate. Let
$f^*$ be the Hardy-Littlewood maximal function of $f\in{\rm L}^1(\R^3)$. Then
$$\eqalignno{
      E(r,z)&\leq(4\pi/3)r^3\rho^*(z)
      \min\{1,(4\pi/3)^{1/2}r^{3/2}\uprho_{Q\gamma Q}^*(z)^{1/2}\}&\cr
\noalign{\hbox{and}}
     \int_0^\infty{dr\over r^5}E(r,z)
      &\leq{\rm const}\rho^*(z)
       \Bigl(\uprho_{Q\gamma Q}^*(z)^{1/2}\int_0^R dr r^{-1/2}
             +\int_R^\infty dr r^{-2}\Bigr)&\cr
      &\leq{\rm const}\rho^*(z)\uprho_{Q\gamma Q}^*(z)^{1/3}&\cr
\noalign{\hbox{for $R=\uprho_{Q\gamma Q}^*(z)^{-1/3}$. Thus, for fixed
$0<\varepsilon\leq 1/5$,}}
     \int dz\int_0^\infty{dr\over r^5}E(r,z)
     &\leq{\rm const}\|\rho^*(\uprho_{Q\gamma Q}^*)^{1/3}\|_1
      \leq{\rm const}\|\rho^*\|_{3/(2+\varepsilon)}
                     \|\uprho_{Q\gamma Q}^*\|_{1/(1-\varepsilon)}^{1/3}&\cr
     &\leq{\rm const}\|\rho\|_{3/(2+\varepsilon)}
                     \|\uprho_{Q\gamma Q}\|_{1/(1-\varepsilon)}^{1/3}&\cr
     &\leq{\rm const}\|\rho\|_{5/3}^{(5-5\varepsilon)/6}
                     \|\rho\|_1^{(1+5\varepsilon)/6}
                     \|\uprho_{Q\gamma Q}\|_{5/3}^{5\varepsilon/6}
                     \|\uprho_{Q\gamma Q}\|_1^{(2-5\varepsilon)/6}&(3.3)\cr
}$$
by using the H\"older inequality, the Hardy-Littlewood maximal inequality
[19] and
$\|f\|_p\leq
 \|f\|_{5/3}^{{5\over 2}-{5\over 2p}}\|f\|_1^{{5\over 2p}-{3\over 2}}$ for
$1\leq p\leq 5/3$. {F}rom
$Q\gamma Q\leq (I-P)\gamma(I-P)+(I+P)\gamma(I+P)=2(\gamma+P\gamma P)\leq
 2(\gamma+P)$ we obtain $\uprho_{Q\gamma Q}\leq 2\rho$. On the other hand,
$\|\uprho_{Q\gamma Q}\|_1=\Tr[Q\gamma Q]\leq\delta(\gamma,P)$. The error
estimate in (3.2) follows by applying these last two inequalities to the last
two factors in (3.3).\qed

\bigskip
\noindent{\bf IV. THE JELLIUM MODEL}

We shall here prove Theorem~2. The proof is divided into a proof of an
upper bound and a lower bound on the energy $E_{\rm J}(\orho)$ .
We first turn to the upper bound, i.e., to
the proof that ${\cal C}(\orho)\geq0$.

{\it Proof of upper bound in (1.17)\/}: Since the
thermodynamic limit is independent of the shape of the domain $\Lambda$
as proved in [15] we may for convenience assume that $\Lambda$ is a ball
 centered
at the origin. For the lower bound we shall return to $\Lambda$ being a cube.

We shall use that $E(N,\Lambda)\leq E_{\rm HF}(N,\Lambda)$ and appeal
to the Lieb variational principle (the second line in (1.8)).
We define an operator $P$ on $\H$ by the integral kernel
$$
        P(x,\sigma;y,\sigma')=\delta_{\sigma,\sigma'}(2\pi)^{-3}\int_{|p|\leq F}
   dp g(x)
        e^{ip(x-y)}g(y)\quad,\eqno(4.1)
$$
where $g$ is a function we shall choose below to be spherically symmetric,
supported in the ball $\Lambda$ and satisfying $0\leq g\leq 1$.
The parameter $F$ is to be chosen such that $\Tr P=N$.
By Parseval's identity $0\leq P\leq I$ and the density corresponding to $P$ is
$$
        \uprho_P(x)=(m/6\pi^2)F^3g(x)^2.
$$
Thus $F=\left(6\pi^2N/(m\int g^2)\right)^{1/3}=\CTF^{1/2}(N/\int g^2)^{1/3}$.
It is easy to compute the integral kernel of $P$ explicitly indeed
$$
        P(x,\sigma;y,\sigma')=\delta_{\sigma,\sigma'}F^3g(x)G_0(|x-y|F)g(y
 )
        \quad, \eqno(4.2)
$$
where $G_0(t)=(2\pi^2)^{-1}t^{-3}(\sin t-t\cos t)$.

We shall let $\orho$ denote both the value $N/|\Lambda|$ and the background
 density which
is equal to the constant $N/|\Lambda|$ in $\Lambda$ and is zero outside
 $\Lambda$.
{F}rom (1.6) we now see that the right side  of (1.5) can be estimated as follows
$$
        \eqalignno{|\Lambda|^{-1}\Bigl(E(N,\Lambda)+D(\orho,\orho)\Bigr) \leq
        |\Lambda|^{-1}\Bigl(&\Tr[-\Delta P] +D(\uprho_P-\orho,\uprho_P-\orho)\cr
        &-\mfr1/2\sum_{\sigma,\sigma'}\iint dxdy {|P(x,\sigma;y,\sigma')|^2\over
   |x-y|}
        \Bigr).\quad&(4.3)}
$$
A simple computation gives
$\Tr[-\Delta P]= \mfr3/5\CTF\int\uprho_P(x)^{5/3}+(m/6\pi^2)F^3\|\nabla g\|_2^2$
   .

Let $R=(3|\Lambda|/4\pi)^{1/3}$ be the radius of $\Lambda$ we can choose $g=g_R$
such that  (a) $g_R(x)=1$ if $|x|\leq R(1-R^{-\alpha})$ and
(b) $\|\nabla g_R\|^2_2\leq \const R^{1+\alpha}$ for any $\alpha$ satisfying
 $1<\alpha<2$.
It is then clear that in the thermodynamic limit $|\Lambda|\to\infty$ with
$N/|\Lambda|=\orho$ fixed we have
$$
        |\Lambda|^{-1}\Tr[-\Delta P]\to\mfr3/5\CTF\orho^{5/3}\quad. \eqno(4.4)
$$

{F}rom the choice of $g_R$ it follows that
$|\Lambda|(1-R^{-\alpha})^3\leq \int g_R^2\leq |\Lambda|$.
We then have  $\uprho_P(x)=(N/\int g_R^2)-\widetilde{\uprho}(x)$ where
 $\widetilde{\uprho}$
is a positive density supported in $R(1-R^{-\alpha})\leq|x|\leq R$ and with
$\int \widetilde{\uprho}\leq  N[(1-R^{-\alpha})^{-3}-1]$.
Since $\iint_{\Lambda\times\Lambda}dx dy|x-y|^{-1}=\const R^5$ and
$N/|\Lambda|-N/(\int g_R^2)\leq 0$ we obtain
$$      \eqalign{D(\uprho_P-\orho,\uprho_P-\orho)=&\mfr1/2
        ({N\over|\Lambda|}-{N\over\int g_R^2})^2\iint_{\Lambda\times\Lambda}dx
 dy|x-y|^{-1}\cr
        &+({N\over|\Lambda|}-{N\over\int g_R^2})
        \int dx\widetilde{\uprho}(x)\int_{\Lambda}dy |x-y|^{-1}
        +D(\widetilde{\uprho},\widetilde{\uprho})\cr
        &\leq \const \left({N\over|\Lambda|}\right)^2
        [(1-R^{-\alpha})^{-3}-1]^2 R^5\left(1+(1-R^{-\alpha})^{-1}\right)\quad,}
$$
where in the last inequality we have used
 $D(\widetilde{\uprho},\widetilde{\uprho})\leq
(\int \widetilde{\uprho})^2 R^{-1}(1-R^{-\alpha})^{-1}$.
Since $\alpha>1$ we find in the thermodynamic limit that
$$
        \lim_{|\Lambda|\to\infty}|\Lambda|^{-1}
        D(\uprho_P-\orho,\uprho_P-\orho)\leq \const\orho^2\lim_{R\to\infty}
        R^{2-2\alpha}=0\quad.\eqno(4.5)
$$
Finally, using (4.2) and the change of variables $s={x+y\over 2},\,r=F(x-y)$
(whose Jacobian is $(FR^{-1})^3$) we compute
$$\displaylines{
  \mfr1/2|\Lambda|^{-1}\sum_{\sigma,\sigma'}\iint dxdy
  {|P(x,\sigma;y,\sigma')|^2\over|x-y|}\hfill\cr
  \hfill
  ={m\over2}(|\Lambda|^{-1}R^3)F^4\iint drds g_R(Rs+F^{-1}{r\over2})^2
   {G_0(|r|)^2\over|r|} g_R(Rs-F^{-1}{r\over2})^2\quad.
}$$
In the termodynamic limit $F\to\CTF^{1/2}\orho^{1/3}$ and
$g_R(Rs\pm F^{-1}{r\over2})^2\to\chi_{B(1,0)}(s)$ pointwise. By dominated
convergence the above expression tends to
$$
        {m\over 2}\CTF^2\orho^{4/3}
        \int dr {G_0(|r|)^2\over |r|}=
        2\pi m\CTF^2\orho^{4/3}
        \int_0^\infty\ dt\  t\ G_0(t)^2= \CD\orho^{4/3}\quad,\eqno(4.6)
$$
where we used the value $(2\pi)^{-4}$ of the last integral.
Combining (4.6) with (4.3--5) proves the upper bound.

{\it Proof of lower bound in (1.17)\/}:
We now assume that $\Lambda$ is a cube of length
$L=|\Lambda|^{1/3}$ and we consider states $\langle\cdot\rangle_{\rm F}$
 such that
in the thermodynamic limit
$$
        |\Lambda|^{-1}\Bigl(\Bigl\langle H(N,\Lambda)\Bigr\rangle_{\rm F}
        +D(\orho,\orho)\Bigr) \to E_{\rm J}(\orho)\quad. \eqno(4.7)
$$

We shall use Lemma~6 with $\uprho_0=\orho$ and $P$ being the projection with
integral kernel
$$
        P(x,\sigma;y,\sigma')=\cases{
        |\Lambda|^{-1}\delta_{\sigma,\sigma'}\sum
        \limits_{p\in 2\pi L^{-1}\Z^3,\atop |p|\leq p_{\rm TF}}
        e^{ip(x-y)}&, if $x,y\in\Lambda$\cr
        0&, otherwise} \eqno(4.8)
$$
where $p_{\rm TF}=\CTF^{1/2}\orho^{1/3}$.
Notice that $P$ is a spectral projection of the Laplacian with {\it periodic}
 boundary
conditions on $\Lambda$ and that $\uprho_P(x)$ converges to $\orho$ in the
thermodynamic limit.
Since $\langle V\rangle_{\rm F}= 2D(\orho,\uprho_{\gamma})$ we get from Lemma~6
 that
$$
        \eqalignno{\Bigl\langle
 H(N,\Lambda)\Bigr\rangle_{\rm F}+D(\orho,\orho)\geq&
        \Bigl\langle \sum_i-\Delta_i\Bigr\rangle_{\rm F}
        -\mfr1/2\sum_{\sigma,\sigma'}\iint dxdy 
	{|P(x,\sigma;y,\sigma')|^2\over|x-y|}
        \cr
        &-\const\|\rho\|_1^{1/6+\varepsilon'}\|\rho\|_{5/3}^{5/6}
       \delta(\gamma,P)^{1/3-\varepsilon'}\quad.
        &(4.9)}
$$
Recall that $\uprho=\uprho_{\gamma}+\uprho_P+\orho$.

We first point out that the same computation which led to (4.6) gives
$$
        \lim_{|\Lambda|\to\infty}\mfr1/2|\Lambda|^{-1}\sum_{\sigma,\sigma'}
        \iint dxdy
 {|P(x,\sigma;y,\sigma')|^2\over|x-y|}=\CD\orho^{4/3}\quad.\eqno(4.10)
$$

Secondly, we use (4.7) (and the upper bound on $E_{\rm J}$ proved above),
 (4.9)
and (4.10) together with the Lieb-Thirring estimate [17]
$\Bigl\langle \sum_i-\Delta_i\Bigr\rangle_{\rm F} \geq \const m^{-2/3}
\int \uprho_{\gamma}^{5/3}$
and the trivial estimate $\delta(\gamma,P)\leq \Tr\gamma=N$ to obtain that
for $\orho\geq \const m^2$
$$
        \limsup_{|\Lambda|\to\infty}|\Lambda|^{-1}\int\rho_{\gamma}^{5/3}
        \leq\const\orho^{5/3}\quad.\eqno(4.11)
$$

Since the form domain of the Dirichlet Laplacian is included in the form domain
of the Laplacian with periodic boundary conditions we may write
$$
        \Bigl\langle\sum_i-\Delta\Bigr\rangle_{\rm F}=\Tr[(-\Delta)\gamma]
        = \sum\limits_{p\in 2\pi L^{-1}\Z^3}|p|^2M(p)\quad,\eqno(4.12)
$$
where $M(p)=|\Lambda|^{-1}\sum_{\sigma}\iint
 dxdye^{ip(y-x)}\gamma(x,\sigma;y,\sigma)$.
In order to control
$\delta(\gamma,P)=\langle \sum_{i=1}^N(I\break-P_i)\rangle_{\rm F}$
we view $\langle\ \cdot\ \rangle_{\rm F}$ as an approximate ground state of
$\sum_{i=1}^N\Delta_i $ which we probe by means of the perturbation
$\sum_{i=1}^N(I-P)_i $, a method known as the Feynman-Hellmann technique.
More precisely we want to estimate
$$
  \Bigl\langle\sum_i-\Delta\Bigr\rangle_{\rm F}-\alpha\delta(\gamma,P)
   =\sum\limits_{p\in 2\pi L^{-1}\Z^3}(p^2-p_{\rm TF}^2
    +\alpha\theta(p_{\rm TF}^2-p^2))M(p)+(p_{\rm TF}^2-\alpha)N\quad,$$
where we have used that $\sum_pM(p)=\Tr\gamma=N$ and $\delta(\gamma,P)=
\Tr[\gamma(I-P)]=\sum_{|p|>p_{\rm TF}}M(p)$.
Here $\theta$ is the step function
$\theta(t)=1$ if $t\geq 0$ and $\theta(t)=0$ if $t<0$. Note that
$0\leq M(p)\leq m$.
 The sum is thus bounded below by
$$       \inf\Bigl\{\sum\limits_{p\in 2\pi L^{-1}\Z^3}
    (p^2-p_{\rm TF}^2+\alpha\theta(p_{\rm TF}^2-p^2))M(p)\mid
     0\leq M(p)\leq m\Bigr\}\quad,
$$
whose minimizer is
$M(p)=m\theta(p_{\rm TF}^2-p^2-\alpha\theta(p_{\rm TF}^2-p^2))
     =m\theta(p_{\rm TF}^2-\alpha_+-p^2)$ with $t_+=\max(t,0)$. For
$\alpha\geq 0$ we then have
$$\eqalign{
  \Bigl\langle\sum_i-\Delta\Bigr\rangle_{\rm F}
  -\mfr3/5\CTF|\Lambda|\orho^{5/3}&-\alpha\delta(\gamma,P)\cr
  &\geq m\sum\limits_{p^2\leq p_{\rm TF}^2-\alpha}(p^2-p_{\rm TF}^2+\alpha)
    -\CTF^{-3/2}|\Lambda|(\mfr3/5p_{\rm TF}^5
    -(p_{\rm TF}^2-\alpha)p_{\rm TF}^3)\cr
  &= -\CTF^{-3/2}|\Lambda|\Bigl(\mfr2/5(p_{\rm TF}^2-\alpha)_+^{5/2}
    -\mfr2/5p_{\rm TF}^5+p_{\rm TF}^3\alpha\Bigr)
   +{\rm o}(|\Lambda|)\cr
  &= -\CTF^{-3/2}|\Lambda|\mfr1/2\cdot\mfr3/2
   (p_{\rm TF}^2-\alpha')_+^{1/2}\alpha^2
   +{\rm o}(|\Lambda|)\cr
  &\geq -{\rm const}\,m^{2/3}|\Lambda|\orho^{1/3}\alpha^2
   +{\rm o}(|\Lambda|)
}$$
for some $0\leq\alpha'\leq\alpha$ (by Taylor's formula).
Optimization over $\alpha\geq 0$ then yields
$$
        |\Lambda|^{-1}\Bigl\langle\sum_i-\Delta\Bigr\rangle_{\rm F}\geq
        \mfr3/5\CTF\orho^{5/3}+\const m^{-2/3}\orho^{-1/3}
        (|\Lambda|^{-1}\delta(\gamma,P))^2+{\rm o}(1)\eqno (4.13)
$$
Inserting (4.10), (4.11) and (4.13) into (4.9) gives for large $\orho$
the following estimate:
$$\eqalign{
      |\Lambda|^{-1}\Bigl(\Bigl\langle H(N,\Lambda)\Bigr\rangle
        +D(\orho,\orho)\Bigr)
 \geq&\mfr3/5\CTF\orho^{5/3}-\CD\orho^{4/3}
        +\const m^{-2/3}\orho^{-1/3}(|\Lambda|^{-1}\delta(\gamma,P))^2\cr
       &-\const \orho^{1+\varepsilon'} (|\Lambda|^{-1}\delta(\gamma,P))^{1/3-
	\varepsilon'}
        +{\rm o}(1)\cr
 \geq&\mfr3/5\CTF\orho^{5/3}-\CD\orho^{4/3}
      -\const m^{2/15}\orho^{4/3-1/15+\varepsilon}
      +{\rm o}(1)\quad.
}$$
The last inequality which proves the lower bound was found by minimizing over
the quantity $|\Lambda|^{-1}\delta(\gamma,P)$.
\qed
\bigskip
\noindent{\bf V. BORN-OPPENHEIMER MOLECULES}

The proof of (1.15,16) is a local version of that given in the previous section.
We begin by reviewing the classical picture of the quantum ground state as it
emerges e.g. from [13]. The minimizer $\uprho_{\rm TF}$ of (1.10) satisfies
$\int dx\,\uprho_{\rm TF}(x)=\min(N,Z)=N$ and the Thomas-Fermi equation
$$
   \CTF\uprho_{\rm TF}^{2/3}=(\phi_{\rm TF}+\mu)_+\eqno (5.1)
$$
for some $\mu\leq0$ called the chemical potential. Let $M(p,q)$ be the
ensemble over phase space $(\R^6,\,(2\pi)^{-3}dpdq)$ obtained by placing $N$
electrons in such a way as to minimize the expectation of the Hamiltonian
function $h(p,q)=p^2-\phi_{\rm TF}(q)$ under the constraint of putting at most
$m$ particles per unit cell. That is, $M(p,q)=m\theta(\mu-h(p,q))$. Indeed,
$(2\pi)^{-3}\int dp\ M(p,q)=\uprho_{\rm TF}(q)$ and
$$
  E_{\rm C}=(2\pi)^{-3}m\iint_{h(p,q)\leq\mu}dpdq\ h(p,q)
  =E_{\rm TF}+D(\uprho_{\rm TF},\uprho_{\rm TF})\quad,
$$
as it is seen by carrying out the $p$-integration and by using (5.1).

This picture is translated into quantum mechanics by means of coherent states
$f_{pq}(x)$ $=g(x-q)e^{ipx}$ with $g$ even and $\|g\|_2=1$. The
corresponding projections are
$\Pi_{pq}=f_{pq}\langle f_{pq}\mid\ \cdot\ \rangle$.
A semiclassical approximation for the one-particle density matrix $\gamma$ of
the quantum ground state should then be given by the weak integral
$$
  P_{\rm SC}=(2\pi)^{-3}\iint_{h(p,q)\leq\mu}dpdq\ \Pi_{pq}\quad.
$$
We note that $0\leq P_{\rm SC}\leq I$ and that $\Tr P_{\rm SC}=N$. Its
integral kernel is $P(x,y)\delta_{\sigma,\sigma'}$ with
$$
   P(x,y)
 =(2\pi)^{-3}\iint_{h(p,q)\leq\mu}dpdq\ g(x-q)\bar g(y-q)e^{ip(x-y)}
 =\int dq\ G(q,x-y)g(x-q)\bar g(y-q)\quad,
$$
where $G(q,r)=p_{\rm TF}(q)^3 G_0(p_{\rm TF}(q)|r|),\,
p_{\rm TF}=\CTF^{1/2}\uprho_{\rm TF}^{1/3}$ and $G_0$ is as in (4.2).
By setting $f(s,r)=g(s+\mfr r/2)\bar g(s-\mfr r/2)$ we can rewrite this as
$P(x,y)=G*_1f(\mfr x+y/2,x-y)$, where $*_1$ stands for convolution in the first
variable only. In particular, $P_{\rm SC}$ has density
$\uprho_{\rm SC}=\uprho_{\rm TF}*|g|^2$. We now turn to the exchange energy:
Since the Jacobian of the change of variables $s=(x+y)/2,\,r=x-y$ is 1, it is
estimated as
$$
  {m\over 2}\iint dxdy {|P(x,y)|^2\over|x-y|}
     ={m\over 2}\iint {dr\over r}ds|G*_1f(s,r)|^2
     \leq{m\over 2}\iint {dr\over r}ds|G(s,r)|^2
     =\CD\int dx\uprho_{\rm TF}^{4/3}\eqno(5.2)
$$
by using Young's inequality. The last equality follows as in (4.6).

The constants appearing below are uniform in the variables of the theorem,
namely in $\uZ,\,\uR$ and $N\leq Z$. They may however depend on $m,\,K$ and
$\varepsilon$.

Also of interest will be the operator
$$\eqalignno{
  h_{\rm SC}&=(2\pi)^{-3}\iint dpdq\ h(p,q)\Pi_{pq}&\cr
            &=-\Delta-\phi_{\rm TF}*|g|^2+\|\nabla g\|_2^2
             =h+(\phi_{\rm TF}-\phi_{\rm TF}*|g|^2)+\|\nabla g\|_2^2&(5.3)
}$$
with $h$ as in (1.13). We will need that
$\|\phi_{\rm TF}-\phi_{\rm TF}*|g|^2\|_{5/2}
 \leq\const Z\||x|^{1/2}|g|^2\|_1^{2/5}$. Indeed,
$$\eqalignno{
  \bigl| |x|^{-1}-|x|^{-1}*|g|^2\bigr|
  &\leq\int dy\bigl| |x|^{-1}-|x-y|^{-1}\bigr||g(y)|^2&\cr
  &\leq\left(\int dy\bigl| |x|^{-1}-|x-y|^{-1}\bigr|^{5/2}|g(y)|^2\right)^{2/5}
       \left(\int dy|g(y)|^2\right)^{3/5}&\cr
\noalign{\hbox{so that}}
  \| |x|^{-1}-|x|^{-1}*|g|^2\|_{5/2}^{5/2}
 &\leq\int dy\left(\int dx\bigl||x|^{-1}-|x-y|^{-1}\bigr|^{5/2}\right)|g(y)|^2
 \quad.&
}$$
Clearly, $|x|^{-1}-|x-y|^{-1}\in{\rm L}^{5/2}(\R^3)$ for fixed $y$. By scaling,
$\int dx\bigl| |x|^{-1}-|x-y|^{-1}\bigr|^{5/2}={\rm const}|y|^{1/2}$. Hence
$\|\phi_{\rm TF}-\phi_{\rm TF}*|g|^2\|_{5/2}\leq
 {\rm const}(\sum_{k=1}^K Z_k+\|\uprho_{\rm TF}\|_1)
  \||x|^{1/2}|g(x)|^2\|_1^{2/5}$.

As a final preliminary we give a bound for the ${\rm L}^{5/3}$ norms of the
densities met so far: $\|\rho\|_{5/3}^{5/3}\leq\const Z^{7/3}$
either for $\rho=\uprho_{\rm TF}$ (and hence for $\rho=\uprho_{\rm SC}$) or,
using the notation of Sect.~III, for $\rho=\uprho_\gamma$ provided
$\langle H(N,\uZ,\uR) \rangle_{\rm F}$ $\leq 0$. To see this we write
$|x|^{-1}=V_1+V_2$ with
$V_1=\min(|x|^{-1},\lambda),\,V_2=(|x|^{-1}-\lambda)_+$ for any $\lambda>0$.
Then $\|V_1\|_\infty=\lambda,\,\|V_2\|_{5/2}=\const\lambda^{-1/5}$.
Hence
$\int dx\rho|x|^{-1}
  \leq\lambda\|\rho\|_1+\const\lambda^{-1/5}\|\rho\|_{5/3}$,
which is
$\leq{\rm const}\|\rho\|_1^{1/6}\|\rho\|_{5/3}^{5/6}$ after optimizing
$\lambda$. For $\rho=\uprho_{\rm TF}$ this implies
$$
 0\geq E_{\rm TF}
 \geq{\rm const}\|\rho\|_{5/3}^{5/3}-\const ZN^{1/6}\|\rho\|_{5/3}^{5/6}
 \geq{\rm const}\|\rho\|_{5/3}^{5/6}(\|\rho\|_{5/3}^{5/6}-\const Z^{7/6})
 \quad,
$$
where we dropped the repulsion between electrons. The Lieb-Thirring inequality
[17] yields a lower bound for $\langle H(N,\uZ,\uR) \rangle_{\rm F}$ of the
same form but with $\rho=\uprho_\gamma$.

{\it Proof of (1.16):\/}
We take $\uprho_0=\uprho_{\rm TF},\,P=P_{\rm SC}$ in (3.2) and use the bounds
for the ${\rm L}^{5/3}$ norms of the occurring densities, as well as (5.2). The
result is
$$
  \langle H(N,\uZ,\uR) \rangle_{\rm F}\geq
  \langle H_{\rm DS}(N,\uZ,\uR) \rangle_{\rm F}
  -\const Z^{7/6}Z^{1/6+\varepsilon}
   \delta(\gamma,P_{\rm SC})^{1/3-\varepsilon}
$$
provided $\langle H(N,\uZ,\uR) \rangle_{\rm F}$ $\leq 0$. Since
$E(N,\uZ,\uR)\leq E(1,\uZ,\uR)<0$ for $Z>0$, we see that all is needed is an
estimate of the form $\delta(\gamma,P_{\rm SC})={\rm O}(Z^{1-\varepsilon})$
for states with $\langle H(N,\uZ,\uR) \rangle_{\rm F}$ arbitrarily close to
$E(N,\uZ,\uR)$. Just by using the trivial estimates
$\delta(\gamma,P_{\rm SC})\leq\Tr\ \gamma=N\leq Z$ and
$\|\uprho_{\rm TF}\|_{4/3}^{4/3}
 \leq\|\uprho_{\rm TF}\|_1^{1/2}\|\uprho_{\rm TF}\|_{5/3}^{5/6}
 \leq\const Z^{5/3}$ we have
$$
  \langle H(N,\uZ,\uR) \rangle_{\rm F}\geq
  \Tr[h\gamma]-D(\uprho_{\rm TF},\uprho_{\rm TF})-\const Z^{5/3}\quad,
$$
a fact which can also be derived [13] without using the lemma. Now (5.3)
implies
$$
  \langle H(N,\uZ,\uR) \rangle_{\rm F}\geq
  \Tr[h_{\rm SC}\gamma]-D(\uprho_{\rm TF},\uprho_{\rm TF})
  -{\rm const}(Z^{12/5}\||x|^{1/2}|g|^2\|_1^{2/5}
               +Z\|\nabla g\|_2^2+Z^{5/3})\eqno(5.4)
$$
since
$\Tr[(\phi_{\rm TF}-\phi_{\rm TF}*|g|^2)\gamma]
 \leq\|\phi_{\rm TF}-\phi_{\rm TF}*|g|^2\|_{5/2}\|\uprho_\gamma\|_{5/3}$.
We set $g(x)=\lambda^{-3/2}g_0(x/\lambda)$ for some smooth, even $g_0$ with
$\|g_0\|_2=1$, so that $\|g\|_2=1$ as required. Then the above error is
${\rm O}(Z^{12/5}\lambda^{1/5}+Z\lambda^{-2}+Z^{5/3})$, i.e.
${\rm O}(Z^{7/3-2/33})$ once we take the optimal value $\lambda=Z^{-7/11}$.

We will now estimate
$\delta(\gamma,P_{\rm SC})=\langle \sum_{i=1}^N(I-P_{\rm SC})_i\rangle_{\rm F}$
by using the Feynman-Hellmann technique, as in the previous section.
Specifically,
$$\displaylines{
    \Tr[h_{\rm SC}\gamma]-E_{\rm C}-\alpha\delta(\gamma,P_{\rm SC})
  =\Tr[(h_{\rm SC}-\alpha(I-P_{\rm SC}))\gamma]-E_{\rm C}\hfill\cr
  \hfill
  =(2\pi)^{-3}\left(
   \iint dpdq(h-\mu+\alpha\theta(\mu-h))\Tr[\Pi_{pq}\gamma]
    -m\iint_{h\leq\mu} dpdq(h-\mu+\alpha)\right)\quad,
}$$
since $\Tr\ \gamma=N=(2\pi)^{-3}m\iint_{h\leq\mu} dpdq$. Note that
$0\leq\Tr[\Pi_{pq}\gamma]\leq\Tr\ \Pi_{pq}\leq m$. The first integral is
therefore bounded below by
$$
\inf\Bigl\{\iint dpdq[h(p,q)-\mu+\alpha\theta(\mu-h(p,q))]M(p,q)
           \mid 0\leq M(p,q)\leq m\Bigr\}\quad.
$$
The obvious minimizer is
$M(p,q)=m\theta(\mu-h(p,q)-\alpha\theta(\mu-h(p,q)))
       =m\theta(\mu-\alpha_+-h(p,q))$. For $\alpha\geq 0$ we thus have
$$\eqalign{
    \Tr[h_{\rm SC}\gamma]-E_{\rm C}-\alpha\delta(\gamma,&P_{\rm SC})
  \geq(2\pi)^{-3}m\left(
   \iint_{h\leq\mu-\alpha}-\iint_{h\leq\mu} \right)dpdq(h-\mu+\alpha)\cr
 &=-(2\pi)^{-3}m\iint_{-\alpha\leq h-\mu\leq0} dpdq(h-\mu+\alpha)\cr
 &=-(2\pi)^{-3}m\int_0^\alpha d\alpha'\iint_{-\alpha'\leq h-\mu\leq0} dpdq\cr
 &=-(2\pi)^{-3}m\int_0^\alpha d\alpha'{4\pi\over 3}\int dq
   [(\phi_{\rm TF}+\mu)_+^{3/2}-(\phi_{\rm TF}+\mu-\alpha')_+^{3/2}]\cr
 &=-(2\pi)^{-2}m\int_0^\alpha d\alpha'\int_0^{\alpha'}  d\alpha''\int dq
   (\phi_{\rm TF}+\mu-\alpha'')_+^{1/2}
}$$
since $h-\mu+\alpha=\int_0^\alpha d\alpha'\theta(h-\mu+\alpha')$ if
$-\alpha\leq h-\mu\leq0$. The innermost integral is bounded by a constant
times $\alpha''^{-1/4}$, as will be shown in the lemma below. This yields
$$
    \Tr[h_{\rm SC}\gamma]-E_{\rm C}
  \geq\alpha\delta(\gamma,P_{\rm SC})-\const\alpha^{7/4}\quad,
$$
which is $\geq\const\delta(\gamma,P_{\rm SC})^{7/3}$ after optimizing
$\alpha\geq0$. Inserting this into (5.4) we obtain
$$
  \langle H(N,\uZ,\uR) \rangle_{\rm F}\geq
  E_{\rm TF}+{\rm const}\ \delta(\gamma,P_{\rm SC})^{7/3}
                    -\const Z^{{7/3}-{2/33}}\quad.
$$
This is an extension of the lower bound contained in the result of [13,16,20]
which we mentioned in the introduction, namely
$E=E_{\rm TF}+{\rm O}(Z^{7/3-2/33})$. By the upper bound therein there are
states with
$\langle H(N,\uZ,\uR) \rangle_{\rm F}\leq E_{\rm TF}
   +\const Z^{{7/3}-{2/33}}$. For such states we conclude that
$$
   \delta(\gamma,P_{\rm SC})\leq\const Z^{1-{2/77}}\quad,\eqno(5.5)
$$
which is what we needed.\qed

{\it Proof of (1.15):\/}
By taking $\uprho_0=\uprho_\gamma,\,P=\gamma$ in (3.2) we obtain
$$
  \langle H(N,\uZ,\uR) \rangle_{\rm F}\geq
  {\cal E}_{\rm HF}(\gamma)-\const Z^{4/3+\varepsilon}
  \delta(\gamma,\gamma)^{1/3-\varepsilon}\quad.
$$
We note that
$\delta(\gamma,P_{\rm SC})=N-\Tr[\gamma P_{\rm SC}]=\delta(P_{\rm SC},\gamma)$
and that
$$\eqalign{
  \delta(\gamma,\gamma)
  &=\Tr[P_{\rm SC}^{1/2}\gamma(I-\gamma)P_{\rm SC}^{1/2}]+
    \Tr[(I-P_{\rm SC})^{1/2}\gamma(I-\gamma)(I-P_{\rm SC})^{1/2}]\cr
  &\leq\Tr[P_{\rm SC}^{1/2}(I-\gamma)P_{\rm SC}^{1/2}]+
    \Tr[(I-P_{\rm SC})^{1/2}\gamma(I-P_{\rm SC})^{1/2}]
   =\delta(P_{\rm SC},\gamma)+\delta(\gamma,P_{\rm SC})\;.
}$$
We then conclude as before using (5.5).\qed

Left to be proved is the following estimate which is similar to one in [1].
\smallskip
{\bf LEMMA 7.} {\it For $\alpha>0$,

$$
 \int dx(\phi_{\rm TF}(x)+\mu-\alpha)_+^{1/2}
 \leq\const\alpha^{-1/4}\quad.
$$
}

{\it Proof:\/}
[13] Let $\phi_k(x)=\CTF^3(3/\pi)^2|x-{\cal R}_k|^{-4},\,k=1,\ldots, K$ and
$\phi(x)=\sum_{k=1}^K\phi_k(x)$. For $x\neq{\cal R}_k,\,k=1,\ldots, K$ we then
have $(4\pi)^{-1}\Delta\phi_k(x)=\CTF^{-3/2}\phi_k(x)^{3/2}$ and
$(4\pi)^{-1}\Delta\phi(x)$ $\leq\CTF^{-3/2}\phi(x)^{3/2}$, due to
$\sum_{k=1}^K\phi_k^{3/2}\leq(\sum_{k=1}^K\phi_k)^{3/2}$. We claim that
$$
  \phi_{\rm TF}+\mu\leq\phi\quad.\eqno(5.6)
$$
To show this, let $\psi=\phi_{\rm TF}+\mu-\phi$ and
$B=\{x\in\R^3\mid \psi(x)>0\}$. Then (1.11) implies ${\cal R}_k\notin\bar B$
and, on the open set $B$,
$ -(4\pi)^{-1}\Delta\psi=(4\pi)^{-1}\Delta\phi-\uprho_{\rm TF}
  \leq\CTF^{-3/2}[\phi^{3/2}-(\phi_{\rm TF}+\mu)_+^{3/2}]\leq 0$
because $\phi_{\rm TF}+\mu>\phi\geq 0$. That is, $\psi$ is subharmonic
on $B$. Since $\psi\big|_{\partial B}=0$ and
$\lim_{|x|\to\infty}\psi=\mu\leq0$, we have $\psi\leq 0$ on $B$. Hence
$B=\emptyset$ and (5.6) holds true. We can now estimate
$$\eqalign{
    \int dx(\phi_{\rm TF}+\mu-\alpha)_+^{1/2}
  &\leq\int dx(\phi-\alpha)_+^{1/2}
   =\int dx\bigl[\sum_{k=1}^K(\phi_k-\alpha K^{-1})\bigr]_+^{1/2}\cr
  &\leq\int dx\bigl[\sum_{k=1}^K(\phi_k-\alpha K^{-1})_+\bigr]^{1/2}
   \leq\int dx\sum_{k=1}^K(\phi_k-\alpha K^{-1})_+^{1/2}\cr
  &=K\int dx(\CTF^3(3/\pi)^2|x|^{-4}-\alpha K^{-1})_+^{1/2}
   =\const K^{5/4}\alpha^{-1/4}
}$$
by scaling the last integral.\qed
\bigskip
{\it Acknowledgements:\/} We thank B.~Simon for hospitality at Caltech, where
part of this work was done. We would also like to thank V.~Bach for many helpful 
comments. G.M.G. gratefully acknowledges support from
the Sloan Foundation.
\bigskip
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