# On $F$ and $E$, in DFT

Paul E. Lammert<EMAIL_ADDRESS>Dept. of Physics, 104B Davey Lab
Pennsylvania State University
University Park, PA 16802-6300

###### Abstract.

Rigorous mathematical foundations of density functional theory are revisited,
with some use of infinitesimal (nonstandard) methods. A thorough treatment is
given of basic properties of internal energy and ground-state energy
functionals along with several improvements and clarifications of known
results. A simple metrizable topology is constructed on the space of densities
using a hierarchy of spatial partitions. This topology is very weak, but
supplemented by control of internal energy, it is, in a rough sense,
essentially as strong as $L^{1}$. Consequently, the internal energy functional
$F$ is lower semicontinuous with respect to it. With separation of positive
and negative parts of external potentials, very badly behaved, even infinite,
positive parts can be handled. Confining potentials are thereby incorporated
directly into the density functional framework.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivations
    2. 1.2 What is here, old and new
    3. 1.3 Particle types
  2. 2 Internal energy as function of state (${\mathcal{E}}$)
    1. 2.1 Kinetic energy
    2. 2.2 Interaction energy
      1. 2.2.1 Hardy inequality
    3. 2.3 ${\mathcal{E}}:{\mathcal{H}}\rightarrow\overline{{\mathbb{R}}}$ is lower semicontinuous
      1. 2.3.1 Lower semicontinuity and extended-real-valued functions
      2. 2.3.2 Lsc theorem
    4. 2.4 Extension to mixed states
      1. 2.4.1 Standard and nearstandard mixed states
      2. 2.4.2 Lower semicontinuity of ${\mathcal{E}}$
  3. 3 Density ($\rho$)
    1. 3.1 $\mathrm{dens}\,$ is continuous
    2. 3.2 Tameness and nearstandardness
      1. 3.2.1 Tameness
      2. 3.2.2 A fundamental spatial approximation
      3. 3.2.3 Tame implies nearstandard
  4. 4 Internal energy as function of density ($F^{\circ}$ and $F$)
    1. 4.1 Constrained energy minimization
    2. 4.2 Sobolev norms and kinetic energy bounds
      1. 4.2.1 Basic inequality
      2. 4.2.2 Two bounds
    3. 4.3 Effective domain
      1. 4.3.1 Transporting densities and wavefunctions
      2. 4.3.2 Wavefunction construction for $\rho\in{\mathcal{J}}_{\mathcal{N}}$
    4. 4.4 ${\mathcal{J}}_{\mathcal{N}}\subset L^{1}\cap L^{3}\cap W^{1,1}$
    5. 4.5 Internal energy is lower semicontinuous on $\mathsf{Dens}_{\mathcal{N}}$
      1. 4.5.1 Topologies on $\mathsf{Dens}_{\mathcal{N}}$
      2. 4.5.2 Pure-state internal energy
      3. 4.5.3 Mixed-state internal energy
  5. 5 Weak-${\mathfrak{P}}$ topology on $\mathsf{Dens}_{\mathcal{N}}$
    1. 5.1 Hierarchy of partitions
      1. 5.1.1 a metric for the weak-${\mathfrak{P}}$ topology
    2. 5.2 How weak is it?
    3. 5.3 Approximation by projection
      1. 5.3.1 Weak-${\mathfrak{P}}^{n}$ is weak-but-not-leaky
    4. 5.4 Weak-${\mathfrak{P}}$ with internal energy control is as good as $L^{1}$
      1. 5.4.1 Poincaré inequality[11, 12, Acosta+Duran03, 14]
      2. 5.4.2 A dichotomy
  6. 6 Ground-state energy, or ‘energy floor’ ($E$)
    1. 6.0.1 on the term “ground-state energy”
    2. 6.0.2 Habitability and stability
    3. 6.0.3 some notation
    4. 6.1 Bounded external potentials
      1. 6.1.1 Fenchel conjugate and biconjugate[15, 16, 17, 18]
      2. 6.1.2 application to $F$
    5. 6.2 $[0,+\infty]$-valued external potentials
      1. 6.2.1 sufficient condition for habitability of $\mathsf{Potl}^{\uparrow}$ external potentials
    6. 6.3 $F$-norm, and relatively $F$-small perturbations
    7. 6.4 $L^{3/2}+L^{\infty}$
    8. 6.5 Concavity and continuity
      1. 6.5.1 concavity
      2. 6.5.2 A locally lower-bounded convex function is locally Lipschitz
      3. 6.5.3 “Perturbative” continuity of $E$
  7. A Selective functional analysis refresher
    1. A.1 Lebesgue ($L^{p}$) and Sobolev ($H^{1}$, $W^{1,1}$) spaces
    2. A.2 the concept of ‘topology’
  8. B Infinitesimals
    1. B.1 seriously
    2. B.2 base datatypes
    3. B.3 limits and continuity
    4. B.4 metric, function, and topological spaces
    5. B.5 $*$-transform and Leibniz’ Principle
    6. B.6 overspill

##  1\. Introduction

###  1.1. Motivations

In the density functional theory (DFT) literature, whenever the matter of
mathematically rigorous foundations arises, a 1983 paper by Elliott Lieb[1]
justly looms large. It propounds what could reasonably be called the “standard
framework”. Although (or perhaps because) the community seems generally to
regard it as a satisfactory foundation, the literature gives evidence that it
is not widely understood. Standard DFT textbooks[2, 3, 4, 5] and reviews[6]
spill very little ink on such matters since they have much else to cover. What
few expository treatments exist[7, 4] hew very close to Lieb, and are not easy
to obtain. Consequently, one more such exposition may have a place. However,
while these notes are a fairly complete treatment of the most fundamental
matters, it does not just add another set of footprints to the same patch of
ground. The standard framework locates densities in
$L^{1}({{\mathbb{R}}}^{3})\cap L^{3}({{\mathbb{R}}}^{3})$ and external one-
body potentials in the dual space
$L^{\infty}({{\mathbb{R}}}^{3})+L^{3/2}({{\mathbb{R}}}^{3})$, with the
internal energy function $F$ on the former and the ground-state energy $E$ on
the latter in a relation of Fenchel conjugacy. Rigid adherence to this
framework is both unnecessarily limiting and physically distorting. Consider
first the density side. In its primitive conception, density is a measure,
telling us the mass in every region. The natural topologies for measures are
weak topologies. This is reinforced by the fact (§4.5.3, apparently not
noticed before, that $F$ is lower semicontinuous with respect to topologies
weaker than weak-$L^{1}$. Keeping in mind that densities are non-negative and
normalized, not general elements of a vector space is important to seeing
that. On the potential side, the standard framework is even more constricting.
It is good physics, and may even be useful, to allow potentials to have vastly
more ill-behaved positive parts than negative parts. Why is $E(v)>-\infty$ for
every $v\in L^{\infty}+L^{3/2}$? The general theory of Fenchel conjugacy, by
itself, certainly does not make that expected. A very clear understanding
(Thm. 6.2) is reached by maintaining a clear view of what is required to keep
positive or negative unboundedness under control.

###  1.2. What is here, old and new

This section gives a brief tour of the contents of the paper, emphasizing the
new results and putting it in context with some initial casual motivation.

It all begins with the following problem of conventional Hilbert-space quantum
mechanics. A system of $\mathcal{N}$ indistinguishable particles (electrons,
if you wish) with kinetic energy (operator) $T$ and mutual interaction $W$ is
placed in an external one-body potential $v$, and we want to find the ground
state and its energy. That is, a minimizer of
$\langle\psi|T+W+\underline{v}|\psi\rangle$ is sought:

$E(v)=\inf_{\psi}\left\\{\langle\psi|T+W|\psi\rangle+\langle\psi|\underline{v}|\psi\rangle\right\\}.$

$\underline{v}$ here is the $\mathcal{N}$-body potential obtained by adding a
copy of $v$ for each particle in the system. But building $\underline{v}$ is a
step in the wrong direction. The external potential energy depends on the
state only through its one-particle density. So, instead of minimizing over
all states in one go, why not split it into two stages? First, for each
density $\rho$, find the state with lowest internal energy,

$F^{\circ}(\rho)=\inf_{\mathrm{dens}\,\psi=\rho}\langle\psi|T+W|\psi\rangle.$
(1.1)

Since all states with density $\rho$ have the same external energy, only the
best among them has any chance of being a ground state. Then, in stage two,
minimize over densities:

$E(v)=\inf_{\rho}\left\\{F^{\circ}(\rho)+\int v\rho\right\\}.$ (1.2)

This is the constrained-search approach of Levy[8] and Lieb[1] The original
problem involved particular $\mathcal{N}$, $W$ and $v$. This final formula
suggests an entirely different viewpoint. The system specified by
$\mathcal{N}$ and $W$ is fixed, and we study its response to many different
external potentials. All the relevant properties of the system are encoded in
$F^{\circ}(\rho)$.

This may remind the reader of statistical mechanics. In a prototypical
magnetic system, there are an enormous number of internal degrees of freedom,
but the only thing relevant for the interaction with an externally imposed
magnetic field is the net magnetization. The free energy as a function of
magnetization is analogous to $F^{\circ}(\rho)$, and the free energy as
function of external field, to $E(v)$. The two are related by a Legendre
transformation, similarly to (1.2. But thermodynamic conjugacy works both
ways. Do we also have

$F^{\circ}(\rho)=\sup_{v}\left\\{E(v)-\int v\rho\right\\}\,?$ (1.3)

Questions now come hard and fast; they are the subject of this paper. We have
lots of infima here. Are there, in fact, minimizers? Consider the functions
$f(x)=1/x$ and $g(x)=(\text{ if }\,x=0\,\,\text{ then }\,\,1\,\,\text{ else
}\,\,x^{2})$. Then, $\inf_{x}f(x)=\inf_{x}g(x)=0$, but neither function
actually takes the value $0$ anywhere. Take minimizing sequences
$f(x_{n})\stackrel{{\scriptstyle n\to\infty}}{{\to}}0$, and
$g(y_{n})\stackrel{{\scriptstyle n\to\infty}}{{\to}}0$. We would hope that at
least subsequences converge to minimizers. But it does not work. There is no
subsequence of $(x_{n})$ that converges at all. The sequence $(y_{n})$
converges to zero, but the function $g$ suddenly jumps up. In the case of $f$,
there is a failure of compactness, in the case of $g$, of lower-
semicontinuity. (If $g$ suddenly jumped down at zero, all would be well, so
only “half” of continuity is really needed.) In the context of function spaces
there are many distinct, legitimate and useful concepts of convergence. When
we say, “$F$ is lower semicontinuous”, we want to make the strongest possible
statement, so we seek the weakest topology with respect to which it is true.
Lower-semicontinuity is a pervasive theme in the following pages.

In §2, we study the internal energy as a function of pure and mixed states,
and its lower-semicontinuity. §3 turns attention to density, where the
important results involve continuity of the map from states to densities and
compactness of the set of low-energy states with given density. Pure state
($F^{\circ}$) and mixed-state ($F$) internal energy as a function of density
is studied in §4. Several results here are new. Lower-semicontinuity with
respect to topologies much weaker than the $L^{1}$ topology is demonstrated. A
general method of transporting densities and wavefunctions is used to make the
well-known demonstration of an upper bound on $F^{\circ}$ in terms of the
$H^{1}$ Sobolev norm of $\sqrt{\rho}$ less ad hoc and to facilitate the
calculation of a kinetic energy bound. We point out that not only do finite-
energy densities satisfy $\sqrt{\rho}\in H^{1}({{\mathbb{R}}^{3}})$ as is
well-known, but also $\rho\in W^{1,1}({\mathbb{R}}^{3})$. Section 5 presents
the construction of a convenient very weak yet metrizable topology on the
space of densities, based on a hierarchy of partitions of space into cubical
lattices. $F$ is lower semicontinuous with respect to this
weak-${\mathfrak{P}}$ topology. Despite its weakness, the
weak-${\mathfrak{P}}$ topology is, roughly speaking, essentially as strong as
$L^{1}$ under conditions of control on the internal energy. Finally, in §6,
attention is turned to the ground-state energy $E(v)$. Here we make a very
strong, and uncustomary, distinction between unbounded positive and unbounded
negative potentials. The physical motivation for this was touched upon in 1.1.
Rewards for separating them in the mathematical treatment are the possibility
of incorporating confining potentials directly into the density-functional
framework, as well as an improved understanding of the stability of Lieb’s
$L^{\infty}+L^{3/2}$ potentials, and continuity of $E(v)$. We will see that
Eq. (1.3) is not true, since the right-hand side is necessarily convex, but
$F^{\circ}$ is not. The defect is easily corrected by replacing $F^{\circ}$ by
the mixed-state internal energy $F$. Furthermore, the maximization need be
carried out only over potentials which are linear combinations of indicator
functions of cells in the partition hierarchy ${\mathfrak{P}}$. This is a new
result. Appendix A gives a refresher on basic functional analysis. In these
notes, we freely use simple methods of infinitesimal (or nonstandard)
analysis. Infinitesimal methods are a way of doing mathematics, and especially
of dealing with mathematical idealizations, that ought to be familiar to
physical scientists. Appendix B gives an account of what we need. It is
relegated to an appendix simply because it is not the real subject of these
notes, but only a tool. The reader may want to take an early look, since it
begins to be used already in 2.3.1.

###  1.3. Particle types

We deal throughout with a system of $\mathcal{N}$ indistinguishable particles.
A condensed matter physicist or chemist generally thinks about DFT only as a
theory of electrons. Our default assumption is indeed that the particles are
fermions, but in principle, the theory could be applied to many different
kinds of particles. Statistics only appear in the explicit construction of a
wavefunction in 4.3.2, so for bosons a better bound on $F^{\circ}$ would be
achievable. Spin is nothing but an annoyance here; it is always summed over
and makes no qualitative difference. Finally, although the mutual interaction
of most interest is the Coulomb interaction, the crucial ingredient is
relative form-boundedness by kinetic energy. We do use positivity of the
interaction in the form $F\geq 0$, but that is a matter of convenience and
could be relaxed.

##  2\. Internal energy as function of state (${\mathcal{E}}$)

###  2.1. Kinetic energy

Throughout, the Hilbert space of $\mathcal{N}$-particle wavefunctions of
appropriate symmetry is denoted by ${\mathcal{H}}$. In units such that
$\hbar^{2}/2m$ has value 1, the kinetic energy of the wavefunction
$\psi(\underline{x},\underline{\sigma})$ is

${\mathcal{E}}_{0}(\psi):{\kern-5.0pt}=\langle\psi|T|\psi\rangle=\int|\nabla\psi|^{2}=\|\nabla\psi\|_{2}^{2}$
(2.1)

######  Remark 2.1 (integration and configuration variable conventions).

$\underline{x}$ denotes the full set of $\mathcal{N}$-particle configuration
space coordinates $(x_{1},x_{2},\ldots,x_{\mathcal{N}})$, and similarly
$\underline{\sigma}$, the collection of spin coordinates. The integral in
(2.1) illustrates some conventions that we will use. Integrals are over all
available variables unless a restriction is indicated by a subscript on the
integral sign. For example, ‘$\int_{x_{1}=x}\cdots$’ means that the position
coordinate of particle 1 is held fixed at $x$. The gradient symbol is subject
to a different convention. The gradient in (2.1) is
$3\mathcal{N}$-dimensional; a subscript would indicate that differentiation is
with respect to only the indicated coordinates. Integration over spin
variables means summation.

In case $\psi$ is in the domain of the operator $T$, so that $T\psi\in
L^{2}({\mathbb{R}})^{3}$ the quadratic form (2.1) is equal to the inner
product of $\psi$ with $T\psi$, which would be written
$\langle\psi|T\psi\rangle$. The quadratic form, understood as a function
$L^{2}({\mathbb{R}}^{3})\rightarrow\overline{{\mathbb{R}}}$ into the topped
reals (see 2.3.1)
$\overline{{\mathbb{R}}}:{\kern-5.0pt}={{\mathbb{R}}}\cup\\{+\infty\\}$ has
the advantage that it is meaningful for every $\psi$, and there are many
wavefunctions satisfying ${\mathcal{E}}_{0}(\psi)<+\infty$ for which $T\psi$
does not exist. In general, Hamiltonians appear in DFT in the guise of
expectation values. Quadratic form definitions are therefore appropriate.

###  2.2. Interaction energy

Now we add the interaction energy of the particles.

${\mathcal{E}}_{\lambda}(\psi)={\mathcal{E}}_{0}(\psi)+\langle\psi|W|\psi\rangle.$
(2.2)

Usually, the interaction $W$ is a Coulomb interaction,
$\sum_{i<j}|x_{i}-x_{j}|^{-1}$, but we consider generally what properties we
want to require.

Call $a>0$ an ${\mathcal{E}}_{0}$ form-bound for $W$ if there is some $b\geq
0$ such that

$\forall\psi\in L^{2}({{\mathbb{R}}}^{3}),\quad|\langle\psi|W|\psi\rangle|\leq
a{\mathcal{E}}_{0}(\psi)+b\|\psi\|^{2}.$ (2.3)

Then, if $a_{0}$ is the infimum of all ${\mathcal{E}}_{0}$ form-bounds for
$W$, says that $W$ is ${\mathcal{E}}_{0}$-form-bounded with relative bound
$a_{0}$. The relative form bound of the Coulomb interaction is zero, as shown
in 2.2.1. For any $a>0$, there is some constant $b$ such that $W_{Coul}$ is
dominated by ${\mathcal{E}}_{0}+b$. This fact is crucial to the stability of
non-relativistic one-electron atoms of arbitrary $Z$. The motif of dominating
an attractive energy with a repulsive one and its significance for stability
will recur. In the following, we assume that the interaction $W$ is non-
negative and ${\mathcal{E}}_{0}$ bounded with relative form bound zero (“Kato
tiny relative to the kinetic energy”). Then, for some constants $c,d>0$,

$d\|\nabla\psi\|_{2}\leq E(\psi)\leq c\|\psi\|_{H^{1}}.$ (2.4)

####  2.2.1. Hardy inequality

The proof that the Coulomb interaction is Kato tiny relative to kinetic energy
uses the following inequality.

######  Lemma 2.1 (Hardy’s inequality).

For $\psi\in H^{1}({\mathbb{R}}^{3})$,

$\int_{{\mathbb{R}}^{3}}\frac{|\psi|^{2}}{4r^{2}}\,dx\leq\int_{{\mathbb{R}}^{3}}|\nabla\psi|^{2}\,dx$

###### Proof.

(sketch) We demonstrate the inequality for real-valued $\psi\in
C^{\infty}_{c}({\mathbb{R}}^{3})$. Real and imaginary parts separate, and an
approximation argument extends it to $H^{1}({\mathbb{R}}^{3})$.

Define $\phi=r^{1/2}\psi$, and note that $\phi(0)=0$. Now,

$\displaystyle(\nabla\psi)^{2}$ $\displaystyle=\left(-\frac{\nabla
r}{r^{3/2}}\phi+r^{-1/2}\nabla\phi\right)^{2}\geq\frac{1}{4r^{3}}\phi^{2}-2\left(\frac{\phi}{2r^{3/2}}r^{-1/2}\right)\left(\frac{\partial\phi}{\partial
r}\right)$
$\displaystyle=\frac{1}{4r^{2}}\psi^{2}-\frac{1}{2r^{2}}\frac{\partial\phi^{2}}{\partial
r},$

where in the inequality, we’ve thrown out a positive term. The second term in
the final expression integrates to zero, since $\phi(x)=0$ at $x=0$ and for
large $x$. ∎

A couple of tricks are needed to apply the Hardy inequality to our situation.
First, note that $1/r$ differs from $f(r|\epsilon)=(\text{ if
}\,r<\epsilon\,\text{ then }\,1/r\,\text{ else }\,0)$ by a bounded function
and $f(r|\epsilon)\leq\epsilon/r^{2}$. Second, the expectation of
$|x_{i}-x_{j}|^{2}$ can be handled by change of variables. Write the required
integral as
$\int|\psi|^{2}/|x_{1}-x_{2}|^{2}\,d(x_{1}-x_{2})\,dx_{2}\,dx_{3}\cdots
dx_{\mathcal{N}}$, and notice that
$|\nabla_{x_{1}-x_{2}}\psi|^{2}\leq|\nabla\psi|^{2}$.

###  2.3. ${\mathcal{E}}:{\mathcal{H}}\rightarrow\overline{{\mathbb{R}}}$ is
lower semicontinuous

####  2.3.1. Lower semicontinuity and extended-real-valued functions

We will often be interested in functions taking values in the topped reals

$\overline{{\mathbb{R}}}:{\kern-5.0pt}={\mathbb{R}}\cup\\{+\infty\\}.$ (2.5)

A neighborhood of $+\infty$ contains an interval of the form $(a,\infty]$ for
some $a\in{\mathbb{R}}$. Infinitesimally, this means that “$x\simeq+\infty$”
is to be understood literally. ${{}^{*}}{\overline{{\mathbb{R}}}}$ consists of
${{}^{*}}{{\mathbb{R}}}$ together with the new point $+\infty$. If
$x\in{{}^{*}}{{\mathbb{R}}}$ is illimited, then ${}^{\circ}{x}=+\infty$.

Continuity of a function $f:X\rightarrow{{\mathbb{R}}}$ at a point $x\in X$ is
defined this way: Given a tolerance $\epsilon$, there is a neighborhood $U$ of
$x$ such that $|f(y)-f(x)|<\epsilon$ for every $y\in U$. If we split the
condition into $f(y)>f(x)-\epsilon$ and $f(y)<f(x)+\epsilon$, then the first
characterizes lower semicontinuity at $x$, and the latter, upper
semicontinuity. In case $f$ is a map into $\overline{{\mathbb{R}}}$, we extend
this definition for a point $x$ with $f(x)=+\infty$. $f$ is lower
semicontinuous at $x$ if for any $M\in{\mathbb{R}}$, there is a neighborhood
$U$ of $x$ such that $f(y)>M$ whenever $y\in U$. In an infinitesimal idiom,
$f:X\rightarrow{\mathbb{R}}$ is lower semicontinuous (abbreviated ‘lsc’) at
$x$ if $f(y)\gtrsim f(x)$ for all $y$ in the halo of $x$. Equivalently,
${}^{\circ}{f(y)}\geq f(x)$. If $f$ is lsc everywhere, then we call it simply
“lower semicontinuous”. This is the property we are usually interested in.

######  Definition 2.1.

A ${\mathbb{R}}$-valued function $f$ on a topological space $X$ is lower
semicontinuous if $f^{-1}(-\infty,a]$ is closed for every $a\in{\mathbb{R}}$.

We usually work with the infinitesimal characterization, but there is also a
useful geometrical intuition attached to the stated definition.

######  Definition 2.2.

If $f:X\rightarrow\overline{{\mathbb{R}}}$, then its effective domain is

$\mathrm{dom}\,f:{\kern-5.0pt}=\\{x\in X:f(x)<\infty\\}.$ (2.6)

The epigraph of $f$ is the set on or above its graph in $X\times{\mathbb{R}}$:

$\mathrm{epi}\,f:{\kern-5.0pt}=\left\\{(x,a)\in
X\times{\mathbb{R}}\,:\,f(x)\leq a\right\\}.$ (2.7)

This implies that if $f(x)=\infty$, then there are no points in
$\mathrm{epi}\,f$ of the form $(x,a)$. Implicitly, $X\times{\mathbb{R}}$ is
equipped with the product topology, so that $(y,b)\simeq(x,a)$ if and only if
both $y\simeq x$ and $b\simeq a$.

We have introduced the notion of epigraph here because it provides another way
to look at lower semicontinuity. $f:X\rightarrow\overline{{\mathbb{R}}}$ is
lower semicontinuous if and only if its epigraph is closed. This perspective
also gives a nice proof of the fact that the pointwise supremum of any set of
lower semicontinuous functions is also lsc. Let $g=\sup_{\alpha\in
I}f_{\alpha}$, with all the $f_{\alpha}$’s lsc. Then,
$\mathrm{epi}\,g=\cap_{\alpha}\mathrm{epi}\,f_{\alpha}$. But, the intersection
of any collection of closed sets is closed, hence $g$ is lsc.

####  2.3.2. Lsc theorem

######  Theorem 2.1.

${\mathcal{E}}:{{\mathcal{H}}}\rightarrow\overline{{\mathbb{R}}}$ is lower
semicontinuous.

###### Proof.

Since ${\mathcal{E}}\sim{\mathcal{E}}_{0}$, it suffices to prove this for
${\mathcal{E}}_{0}$. And that is very easily done by Fourier transformation:

${\mathcal{E}}_{0}(\psi)=\|\nabla\psi\|_{2}^{2}=\int
q^{2}|\hat{\psi}(q)|^{2}\,dq=\lim_{Q\to\infty}p_{Q}(\psi),$

where

$p_{Q}(\psi):{\kern-5.0pt}=\int_{|q|\leq Q}q^{2}|\hat{\psi}(q)|^{2}\,dq.$

But, $p_{Q}:{\mathcal{H}}\rightarrow{{\mathbb{R}}}$ is clearly continuous, and
$p_{Q}(\psi)$ is monotonically non-decreasing in $Q$. As supremum of
continuous functions, then, ${\mathcal{E}}_{0}$ is lsc. ∎

###  2.4. Extension to mixed states

####  2.4.1. Standard and nearstandard mixed states

To the vector ${\psi}\in{\mathcal{H}}$ is associated the pure state
$|\psi\rangle\langle\psi|\in\mathsf{State}^{\circ}$, which is insensitive to
the phase of $\psi$. A mixed state is a probabalistic mixture of pure states,
and has the normal form

$\gamma=\sum_{i\in{\mathcal{I}}}|\psi_{i}\rangle\langle\psi_{i}|,\quad
j\not=k\Rightarrow\langle\psi_{j}|\psi_{k}\rangle=0,\quad\|\psi_{1}\|\geq\|\psi_{2}\|\geq\cdots>0,\quad\sum_{i\in{\mathcal{I}}}\|\psi_{i}\|^{2}<\infty.$
(2.8)

The index set ${\mathcal{I}}$ here is either ${\mathbb{N}}$, or $[1,\ldots,n]$
for some $n\in{\mathbb{N}}$. Note, the normal form is not uniquely specified
in case of degenerate eigenvalues. On a few occasions, we have use for an
alternate normal form

$\gamma=\sum_{i\in{\mathcal{I}}}c_{i}|\hat{\psi_{i}}\rangle\langle\hat{\psi_{i}}|.$
(2.9)

The two are related by $c_{i}=\|\psi_{i}\|^{2}$, so that the
$\hat{\psi_{i}}$’s are unit vectors.

We can also think of the set of mixed states $\mathsf{State}$ as a convex cone
in the Banach space of trace-class operators equipped with the trace norm,

$\|A\|_{\mathrm{Tr}}=\sum\langle\phi_{i}||A|\phi_{i}\rangle,\,\,\\{\phi_{i}\\}\,\text{
an ONB}.$ (2.10)

For a positive trace-class operator, the trace norm is just the trace, and for
a mixed state this is the generalization of norm-squared for a state vector.

For ${\psi}\in{{}^{*}}{{\mathcal{H}}}$, ${}^{\circ}{\psi}$ is the standard
vector infinitely close (“near”) to it, if such exists, else it is undefined.
The map $\psi\mapsto|\psi\rangle\langle\psi|$ from ${\mathcal{H}}$ to
$\mathsf{State}^{\circ}$ ought to be continuous, so
$\mathrm{st}\,|\psi\rangle\langle\psi|=|{}^{\circ}{\psi}\rangle\langle{}^{\circ}{\psi}|$.
Continuing in this vein, to deduce the normal form of $\mathrm{st}\,\gamma$
without explicitly working with the trace norm, note that both addition in
$\mathsf{State}$ and $\mathrm{Tr}$ should be continuous. Combined with the
standard part operation in $\mathsf{State}^{\circ}$, we conclude that

$\mathrm{st}\,\gamma=\,\,\text{ if
}{}^{\circ}{(\sum_{{\mathcal{I}}}\|\psi_{i}\|^{2})}=\sum_{{}^{\circ}{\mathcal{I}}}\|{}^{\circ}{\psi_{i}}\|^{2}\,\text{
then
}\,\sum_{i\in{}^{\circ}{\mathcal{I}}}|{}^{\circ}{\psi_{i}}\rangle\langle{}^{\circ}{\psi_{i}}|\,\,\text{else
}\mathsf{undefined}.$ (2.11)

Thus, $\gamma$ is remote if any of the $\psi_{i}$’s is so, or if its norm is
illimited, or if an appreciable weight is carried by infinitesimal
$\psi_{i}$’s, or if the norm of $\gamma$ is illimited (the if-clause fails
because the first standard part is undefined).

######  Definition 2.3.

The set of mixed states based on ${\mathcal{H}}$ is denoted $\mathsf{State}$,
and equipped with the trace norm. The subset of pure states is denoted by
$\mathsf{State}^{\circ}$, and the normalized states by $\mathsf{State}_{1}$.
‘State’ with no modifier means a member of $\mathsf{State}$, and the
distinction between a pure state and a corresponding state vector is often
elided.

####  2.4.2. Lower semicontinuity of ${\mathcal{E}}$

With (2.11) and the nonstandard characterization of lower-semicontinuity, we
can now extend Thm. 2.1 to $\mathsf{State}$. However, it will not be used in
the following.

######  Corollary 2.1.

${\mathcal{E}}:{\mathsf{State}}\rightarrow\overline{{\mathbb{R}}}$ is lower
semicontinuous.

###### Proof.

Take $\gamma\in{\mathsf{State}}$ a standard state and and suppose
$\gamma^{\prime}=\sum_{{\mathcal{I}}}|\psi_{i}^{\prime}\rangle\langle\psi_{i}^{\prime}|$
is near $\gamma$. We need to show that
${\mathcal{E}}(\gamma^{\prime})\gtrsim{\mathcal{E}}(\gamma)$. Since
${\mathcal{E}}$ is non-negative and lsc on $\mathsf{State}^{\circ}$,

${}^{\circ}{{\mathcal{E}}(\gamma^{\prime})}\geq\mathrm{st}\,\sum_{i=1}^{n}{\mathcal{E}}(\psi_{i}^{\prime})=\sum_{i=1}^{n}{}^{\circ}{{\mathcal{E}}(\psi_{i}^{\prime})}\geq\sum_{i=1}^{m}{\mathcal{E}}(\psi_{i}).$

If ${\mathcal{I}}$ is finite, just take $n=|{\mathcal{I}}|$, otherwise, the
limit $n\to\infty$ yields
${}^{\circ}{{\mathcal{E}}(\gamma^{\prime})}\geq{\mathcal{E}}(\gamma)$. ∎

##  3\. Density ($\rho$)

The one-particle density associated with an $\mathcal{N}$-particle
wavefunction $\psi$ is

$(\mathrm{dens}\,\psi)(x)=\mathcal{N}\int_{x_{1}=x}|\psi|^{2}\,.$ (3.1)

The right-hand side here is a linear function of the rank-one operator
$|\psi\rangle\langle\psi|$ and thus has an immediate linear extension to mixed
states:

$\mathrm{dens}\,\left(\sum c_{i}|\psi_{i}\rangle\langle\psi_{i}|\right)=\sum
c_{i}\mathrm{dens}\,\psi_{i}.$ (3.2)

We will overload the notation ‘$\mathrm{dens}\,$’ by writing
$\mathrm{dens}\,\psi$ for $\mathrm{dens}\,(|\psi\rangle\langle\psi|)$.

###  3.1. $\mathrm{dens}\,$ is continuous

Integrating the previous display over $x$, one finds

$\|\mathrm{dens}\,\gamma\|_{1}=\mathcal{N}\|{\gamma}\|_{tr}.$ (3.3)

Densities are, of course, non-negative integrable functions. We use the
notations

######  Definition 3.1.

$\displaystyle\mathsf{Dens}$ $\displaystyle=\left\\{\rho\in
L^{1}({\mathbb{R}}^{3}):\rho\geq 0\right\\},$
$\displaystyle\mathsf{Dens}_{\mathcal{N}}$
$\displaystyle=\left\\{\rho\in\mathsf{Dens}:\int\rho\,dx=\mathcal{N}\right\\}.$
(3.4)

######  Theorem 3.1.

$\mathrm{dens}\,:\mathsf{State}\rightarrow\mathsf{Dens}$ is continuous.

###### Proof.

(3.3) shows that $\mathrm{dens}\,$ is actually a bounded linear map. ∎

###  3.2. Tameness and nearstandardness

####  3.2.1. Tameness

A density in $\mathsf{Dens}_{\mathcal{N}}$ always integrates to $\mathcal{N}$.
Thus, given a density $\rho$, there is always some function (many in fact)
$R(\epsilon):(0,N]\rightarrow{{\mathbb{R}}}^{+}$ such that $\int_{|x|\geq
R(\epsilon)}\rho\,dx<\epsilon$ for all $\epsilon$. When a set $S$ of densities
can all be controlled in this way by the same falloff function, $S$ would be
said to be tight. In a nonstandard context, we borrow this term to apply to a
single ∗density in ${{}^{*}}{\mathsf{Dens}_{\mathcal{N}}}$. By Leibniz’
Principle, $\rho\in{{}^{*}}{\mathsf{Dens}_{\mathcal{N}}}$ has a $*$-falloff
function. We say it is tight if $R(\epsilon)$ is limited for non-infinitesimal
$\epsilon$. Here is an alternative phrasing.

######  Definition 3.2.

A $*$-density $\rho$ is tight if $\int_{|x|\geq R}\rho\,dx\simeq 0$ for every
illimited $R$. A $*$-state $\gamma$ is tame if it has limited internal energy
and norm, and $\mathrm{dens}\,\gamma$ is tight. A ∗-state which is not tame is
wild.

####  3.2.2. A fundamental spatial approximation

Our approximation scheme has two ingredients, a spatial truncation method and
a family of finite-dimensional subspaces of ${\mathcal{H}}$. Start with the
former. Let $0\leq\eta({x})\leq 1$ be a continuously differentiable cutoff
function equal to one in the cubical box $\mathchoice{\vbox{\hrule
height=0.8pt\hbox{\vrule width=0.8pt,height=6.0pt\kern 6.0pt\vrule
width=0.8pt} \hrule height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}(1/2)=[-1/2,1/2]^{3}$ and supported in $\mathchoice{\vbox{\hrule
height=0.8pt\hbox{\vrule width=0.8pt,height=6.0pt\kern 6.0pt\vrule
width=0.8pt} \hrule height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}({1})=[-1,1]^{3}$, for example, a product of three 1D cutoff
functions, and for $R>0$, let $(\eta_{R})(x)=\eta(x/R)$. Then, define the
$\mathcal{N}$-particle cutoff operator by

$(\Lambda_{R}\psi)(\underline{x},\underline{\sigma})=\left(\prod_{i=1}^{\mathcal{N}}\eta_{R}({x}_{i})\right)\psi(\underline{x},\underline{\sigma}).$
(3.5)

Thus, the state $\Lambda_{R}\psi$ is supported in the $\mathcal{N}$-particle
box $\mathchoice{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}^{\mathcal{N}}(R)$, which is to say that the density is zero
outside $\mathchoice{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}(R)$. $\Lambda_{R}\psi$ agrees with $\psi$ inside
$\mathchoice{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}^{\mathcal{N}}(\frac{R}{2})$ and is an attenuated version
outside, so we can get an upper bound on the norm of the difference by just
integrating $|\psi|^{2}$ over the complement:

$\|\Lambda_{R}\psi-\psi\|^{2}\leq\int_{\mathchoice{\vbox{\hrule
height=0.8pt\hbox{\vrule width=0.8pt,height=6.0pt\kern 6.0pt\vrule
width=0.8pt} \hrule height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}^{\mathcal{N}}(\frac{R}{2})^{c}}|\psi|^{2}\,\leq\int_{\mathchoice{\vbox{\hrule
height=0.8pt\hbox{\vrule width=0.8pt,height=6.0pt\kern 6.0pt\vrule
width=0.8pt} \hrule height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}(\frac{R}{2})^{c}}\mathrm{dens}\,\psi$ (3.6)

Bounding the energy of $\Lambda_{R}\psi$ is a little harder. Using
$\nabla(\Lambda_{R}\psi)={\Lambda}_{R}\nabla\psi+(\nabla{\Lambda}_{R})\psi$
and $|a+b|^{2}\leq 2|a|^{2}+2|b|^{2}$, we find

$\displaystyle{\mathcal{E}}_{0}(\Lambda_{{R}}\psi)$
$\displaystyle\leq{\mathcal{E}}_{0}(\psi)+\int_{\left[\mathchoice{\vbox{\hrule
height=0.8pt\hbox{\vrule width=0.8pt,height=6.0pt\kern 6.0pt\vrule
width=0.8pt} \hrule height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}^{\mathcal{N}}(\frac{R}{2})\right]^{c}}|\nabla\psi|^{2}\,d\underline{x}+2\int_{\left[\mathchoice{\vbox{\hrule
height=0.8pt\hbox{\vrule width=0.8pt,height=6.0pt\kern 6.0pt\vrule
width=0.8pt} \hrule height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}^{\mathcal{N}}(\frac{R}{2})\right]^{c}}|\nabla\Lambda_{R}|^{2}|\psi|^{2}\,d\underline{x}$
$\displaystyle\leq 2{\mathcal{E}}_{0}(\psi)+2\int_{{\mathchoice{\vbox{\hrule
height=0.8pt\hbox{\vrule width=0.8pt,height=6.0pt\kern 6.0pt\vrule
width=0.8pt} \hrule height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}(\frac{R}{2})}^{c}}|\nabla\eta_{R}|^{2}(\mathrm{dens}\,\psi)$
$\displaystyle\leq
2{\mathcal{E}}_{0}(\psi)+\frac{\|\nabla\eta_{1}\|_{\infty}^{2}}{R^{2}}\int_{{\mathchoice{\vbox{\hrule
height=0.8pt\hbox{\vrule width=0.8pt,height=6.0pt\kern 6.0pt\vrule
width=0.8pt} \hrule height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}(\frac{R}{2})}^{c}}\mathrm{dens}\,\psi.$ (3.7)

To the box $\mathchoice{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}(R)$ we associate two closed subspaces of ${\mathcal{H}}$. Let
$\\{\varphi_{1},\varphi_{2},\ldots\\}$ be the single-particle particle-in-a-
box eigenstates in $\mathchoice{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}(R)$. From this single-particle basis, construct a complete
product basis (adding spin and antisymmetrizing as needed) for
${\mathcal{H}}(\mathchoice{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}(R))$, the Hilbert space for $\mathcal{N}$ particles in the box.
${\mathcal{H}}_{R}$ is the subspace spanned by basis vectors with kinetic
energy not exceeding $R$; it is a finite-dimensional Hilbert space
(asymptotically, $\mathrm{dim}{\mathcal{H}}_{R}$ does not grow any faster than
$R^{9\mathcal{N}/2}$). Orthogonal projection
${\mathcal{H}}(\mathchoice{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}(R))\rightarrow{\mathcal{H}}_{R}$ is denoted by $\pi_{R}$. Now,
if $\psi$ has density supported in $\mathchoice{\vbox{\hrule
height=0.8pt\hbox{\vrule width=0.8pt,height=6.0pt\kern 6.0pt\vrule
width=0.8pt} \hrule height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}(R)$, then $\psi\in{\mathcal{H}}(\mathchoice{\vbox{\hrule
height=0.8pt\hbox{\vrule width=0.8pt,height=6.0pt\kern 6.0pt\vrule
width=0.8pt} \hrule height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}(R))$, and if, in addition, ${\mathcal{E}}_{0}(\psi)<\epsilon
R$, then $\|\psi-\pi_{R}\psi\|^{2}<\epsilon$. We use the notation
$\mathrm{Reg}_{R}$ to denote the composition of smooth truncation followed by
projection:

$\mathrm{Reg}_{R}:{\kern-5.0pt}=\pi_{R}\circ\Lambda_{R}.$ (3.8)

Now, if $\psi$ is tight and $R$ is illimited, $\Lambda_{R}\psi\simeq\psi$ is
immediate. The inequality (3.7) shows that also
${\mathcal{E}}_{0}(\Lambda_{R}\psi)\lesssim{\mathcal{E}}_{0}(\psi)$.

####  3.2.3. Tame implies nearstandard

Now we put the constructions of 3.2.2 to work. Two observations are critical.
The constructions were described for $R\in{\mathbb{R}}_{>0}$, but work just as
well for $R\in{{}^{*}}{{\mathbb{R}}}$. Also, since ${\mathcal{H}}_{R}$ is
finite-dimensional, with some finite orthonormal basis
$\\{\varphi_{1},\ldots,\varphi_{n}\\}$, the same is true of
${{}^{*}}{{\mathcal{H}}}_{R}$. The only difference is that
$\phi\in{{}^{*}}{{\mathcal{H}}}_{R}$ has hypercomplex coefficients and may
have illimited norm. But, if $\phi=\sum\alpha_{i}{{}^{*}}{\varphi_{i}}$ is
limited, it is nearstandard with standard part
$\sum{}^{\circ}{\alpha_{i}}\varphi_{i}$.

The translation into standard terms of an external assertion to the effect
that a property implies nearstandardness is a claim about relative
compactness. For instance, the following proposition is essentially a
variation of the Rellich-Kondrashov theorem. We will not, however, explicitly
use the notion of compactness in these notes.

######  Proposition 3.1.

If $\psi\in{{}^{*}}{\mathsf{State}^{\circ}}$ is tame, then $\psi$ is
nearstandard. Furthermore, in that case,
$\mathrm{dens}\,\mathrm{st}\,\psi=\mathrm{st}\,\mathrm{dens}\,\psi$ and
${\mathcal{E}}(\mathrm{st}\,\psi)\leq\mathrm{st}\,{\mathcal{E}}(\psi)$.

###### Proof.

${\mathcal{H}}$ is a complete metric space, hence it suffices to show that
$\psi$ can be approximated to any standard accuracy by a (standard) vector in
${\mathcal{H}}$.

Since $\psi$ is tight, $\Lambda_{R}\psi\simeq\psi$ for illimited $R$.
Furthermore, according to inequality (3.7), $\Lambda_{R}\psi$ also has limited
internal energy. Thus, the projection $\pi_{R}$ will affect it only
infinitesimally: $\mathrm{Reg}_{R}\psi\simeq\psi$.

Now let $\epsilon>0$ be given. The set

$S(\epsilon)=\left\\{a\in{{}^{*}}{{\mathbb{R}}}:\|\mathrm{Reg}_{a}\psi-\psi\|<\epsilon\right\\}$

is internal, but it contains all illimited hyperreals. Thus, it must contain
some limited $R$. $\mathrm{Reg}_{R}\psi$ is a limited vector in
${{}^{*}}{{\mathcal{H}}_{R}}$, and therefore nearstandard by the remarks
immediately preceding the proposition. Thus, $\psi$ is within $2\epsilon$ of a
standard vector. But $\epsilon$ was arbitrary.

The last two statements of the Proposition now follow by Thms. 2.1 and 3.1. ∎

Following the familiar pattern, we now extend Prop. 3.1 to mixed states. This
is not trivial and will involve one of the trickiest pieces of reasoning in
these notes.

######  Corollary 3.1.

If $\gamma\in\mathsf{State}_{1}$ is tame, then $\gamma$ is nearstandard.
Consequently,
$\mathrm{dens}\,\mathrm{st}\,\gamma=\mathrm{st}\,\mathrm{dens}\,\gamma$ and
${\mathcal{E}}(\mathrm{st}\,\gamma)\leq{}^{\circ}{{\mathcal{E}}(\gamma)}$.

###### Proof.

Tackle this in contrapositive form. There are three ways $\gamma$ can be
remote. If $\|\gamma\|_{\mathrm{Tr}}$ is illimited, $\gamma$ is certainly
wild, and if $\|\psi_{i}\|^{2}$ is appreciable for some remote $\psi_{i}$ in
the support of $\gamma$, then $\gamma$ is wild because Prop. 3.1 says that
$\psi_{i}$ is. The third possibility is that $\gamma$ has a tail
$\gamma^{\prime}=\sum_{i\geq
N}c_{i}|\hat{\psi}_{i}\rangle\langle\hat{\psi}_{i}|$ with $c_{i}\simeq 0$ for
$i\geq N$ and $\sum_{i\geq N}c_{i}=M\gg 0$. The following Lemma 3.1 shows that
$\gamma^{\prime}/M$ is wild. Hence, so is $\gamma$. ∎

######  Lemma 3.1.

Suppose the mixed state

$\gamma=\sum
c_{i}|\hat{\psi}_{i}\rangle\langle\hat{\psi}_{i}|\in{{}^{*}}{\mathsf{State}}_{1}$

has all coefficients infinitesimal: $c_{i}\simeq 0$. Then, $\gamma$ is wild.

###### Proof.

Assume to the contrary that $\gamma$ is tame, so that
${\mathcal{E}}(\gamma)=E\ll\infty$ and
$\rho:{\kern-5.0pt}=\mathrm{dens}\,\gamma$ is tight. Define the weight of an
internal set $B\subset{{}^{*}}{{\mathbb{N}}}$ to be $\mathrm{wt}\,B=\sum_{i\in
B}c_{i}$, Since the $c_{i}$ are infinitesimal, if $\mathrm{wt}\,B\gg 0$, then
$|B|$ is illimited. We are going to derive a contradiction as follows. For
some limited $r$, we obtain an illimited set of indices $A(r)$ such that
$\|\mathrm{Reg}_{r}\hat{\psi}_{i}-\hat{\psi}_{i}\|<\epsilon$ for $i\in A(r)$.
Then $\|\mathrm{Reg}_{r}\hat{\psi}_{i}\|>1-\epsilon>1/2$, and since the
$\hat{\psi}_{i}$’s are orthonormal, a quick calculation reveals that
$\langle\mathrm{Reg}_{r}\hat{\psi}_{i}|\mathrm{Reg}_{r}\hat{\psi}_{j}\rangle\leq({2+\epsilon})/\epsilon<3\epsilon$.
Thus, the cosine of the angle between $\mathrm{Reg}_{r}\hat{\psi}_{i}$ and
$\mathrm{Reg}_{r}\hat{\psi}_{j}$ is less than $12\epsilon$. For small
$\epsilon$, this condition is a restriction; a set of directions in the
limited-dimensional space ${\mathcal{H}}_{r}$ satisfying it must be limited,
but $A(r)$ is not. Now we proceed to produce the required $r$ and $A(r)$. Let
standard $\epsilon\in(0,1/12)$ be given, and define for
$R\in{{}^{*}}{{\mathbb{R}}}$, the internal set

$A(R):{\kern-5.0pt}=\left\\{i:\int_{\mathchoice{\vbox{\hrule
height=0.8pt\hbox{\vrule width=0.8pt,height=6.0pt\kern 6.0pt\vrule
width=0.8pt} \hrule height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}(R/2)^{c}}\mathrm{dens}\,\hat{\psi}_{i}\leq\frac{1}{\epsilon}\int_{\mathchoice{\vbox{\hrule
height=0.8pt\hbox{\vrule width=0.8pt,height=6.0pt\kern 6.0pt\vrule
width=0.8pt} \hrule height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}(R/2)^{c}}\rho\,\text{ and
}\,{\mathcal{E}}_{0}(\hat{\psi}_{i})\leq\frac{E}{\epsilon}\right\\}.$

For every $R$, $\mathrm{wt}\,A(R)\geq 1-\epsilon$, for if this failed, there
would be too much mass outside $\mathchoice{\vbox{\hrule
height=0.8pt\hbox{\vrule width=0.8pt,height=6.0pt\kern 6.0pt\vrule
width=0.8pt} \hrule height=0.8pt}}{\vbox{\hrule height=0.8pt\hbox{\vrule
width=0.8pt,height=6.0pt\kern 6.0pt\vrule width=0.8pt} \hrule
height=0.8pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=4.2pt\kern 4.2pt\vrule width=0.6pt} \hrule
height=0.6pt}}{\vbox{\hrule height=0.6pt\hbox{\vrule
width=0.6pt,height=3.0pt\kern 3.0pt\vrule width=0.6pt} \hrule
height=0.6pt}}(R/2)$ or ${\mathcal{E}}(\gamma)$ would exceed $E$.
Consequently, $|A(R)|>(1-\epsilon)/c_{1}\simeq\infty$, independently of $R$.

If $R$ is illimited, then by (3.6,3.7) and tightness of $\rho$,
$\Lambda_{R}\hat{\psi}_{i}\simeq\hat{\psi}_{i}$ and
${\mathcal{E}}_{0}(\Lambda_{R}\hat{\psi}_{i})\ll\infty$ for $i\in A(R)$. And,
this implies $\|\mathrm{Reg}_{R}\hat{\psi}_{i}-\hat{\psi}_{i}\|\simeq 0$. As a
result, the internal set

$C:{\kern-5.0pt}=\\{R\in{{}^{*}}{{\mathbb{R}}}:\forall i\in
A(R),\,\|\mathrm{Reg}_{R}\hat{\psi}_{i}-\hat{\psi}_{i}\|<\epsilon\\}$

contains all illimited hyperreals. By underflow, $C$ contains some
$r\ll\infty$. $A(r)$ is the hyperfinite set discussed in the first paragraph.
∎

##  4\. Internal energy as function of density ($F^{\circ}$ and $F$)

###  4.1. Constrained energy minimization

We now move away from talking directly, or at least primarily, about states,
instead addressing them through the mediation of density. The discussion of
the Levy-Lieb constrained search formulation in 1.2 introduced the idea of
minimizing internal energy over all states with given density. However, it is
now clear that we want to consider two different internal energy functions of
density, depending upon whether all states or only pure states are considered.
Thus, we define

$\displaystyle F^{\circ}(\rho)$
$\displaystyle=\inf\left\\{{\mathcal{E}}(\psi):\psi\in\mathsf{State}^{\circ},\,\mathrm{dens}\,\psi=\rho\right\\}$
$\displaystyle F(\rho)$
$\displaystyle=\inf\left\\{{\mathcal{E}}(\gamma):\gamma\in\mathsf{State},\,\mathrm{dens}\,\gamma=\rho\right\\}.$
(4.1)

Important questions about these functions are these: For which densities they
are finite? Are there useful upper/lower bounds in terms of explicit functions
of density? Are $F^{\circ}$ and $F$ lower semicontinuous? Are the infima in
their definitions actually realized (are they actually minima)? Using the
results of 2.3 and 2.4, we can immediately settle the last question.

######  Theorem 4.1.

The infima in the definitions of $F^{\circ}$ and $F$ are realized.

###### Proof.

Let standard $\rho$ be given and assume $F^{\circ}(\rho)<+\infty$, as
otherwise there is nothing to prove. Let $\psi\in{{}^{*}}{{\mathcal{H}}}$ be
such that ${\mathcal{E}}(\psi)\simeq{{}^{*}}{F^{\circ}}({{}^{*}}{\rho})$.
Since $\psi$ is thus tame, it is nearstandard by Thm. 3.1 so
$\mathrm{st}\,\psi$ exists, and according to Thm. 2.1,
${\mathcal{E}}(\mathrm{st}\,\psi)\lesssim{\mathcal{E}}(\psi)\simeq
F^{\circ}(\rho)$. Since $F^{\circ}(\rho)$ and
${\mathcal{E}}(\mathrm{st}\,\psi)$ are both standard, they are equal.

The proof for $F$ is entirely analogous, just substituting Cor. 2.1 for Thm.
2.1, and Cor. 3.1 for Thm. 2.1. ∎

###  4.2. Sobolev norms and kinetic energy bounds

In this section, we derive some important lower bounds on $F^{\circ}$ and $F$.

####  4.2.1. Basic inequality

Starting from (3.1), we deduce

$\displaystyle\nabla\rho(x)$ $\displaystyle\leq
2\mathcal{N}\int_{x_{1}=x}|\psi^{*}\nabla_{1}\psi|^{2}\,\leq
2\mathcal{N}\left(\int_{x_{1}=x}|\psi|^{2}\,\right)^{1/2}\left(\int_{x_{1}=x}|\nabla_{1}\psi|^{2}\,\right)^{1/2}$
$\displaystyle=2\rho(x)^{1/2}\left(\int_{x_{1}=x}|\nabla_{1}\psi|^{2}\,\right)^{1/2}.$
(4.2)

The subscript on $\nabla$ indicates that differentiation is only with respect
to $x_{1}$.

####  4.2.2. Two bounds

There are two useful things we can do with this result. First, squaring,
dividing through by $\rho$ and noting that
$|\nabla\rho|^{2}/\rho=|\nabla\sqrt{\rho}|^{2}$,

$\|\nabla\sqrt{\rho}\|_{2}^{2}\leq\frac{4}{\mathcal{N}}{\mathcal{E}}_{0}(\psi).$
(4.3)

To extend this to a mixed state $\gamma=\sum|\psi_{i}\rangle\langle\psi_{i}|$,
abbreviate $\rho=\mathrm{dens}\,\gamma$, $\rho_{i}=\mathrm{dens}\,\psi_{i}$
and use $\nabla\rho=2\sqrt{\rho}\,\,\nabla\sqrt{\rho}$ and the Cauchy-Schwartz
inequality to obtain

$\left|\sqrt{\rho}\,\,\nabla\sqrt{\rho}\right|=\left|\sum_{i}\sqrt{\rho_{i}}\,\,\nabla\sqrt{\rho_{i}}\right|\leq\left(\sum_{i}\rho_{i}\right)^{1/2}\left(\sum_{i}\left|\nabla\sqrt{\rho_{i}}\right|^{2}\right)^{1/2}.$

Since $\rho=\sum_{i}\rho_{i}$, divide through by $\sqrt{\rho}$ and square to
get
$\left|\nabla\sqrt{\rho}\right|^{2}\leq\sum_{i}\left|\nabla\sqrt{\rho_{i}}\right|^{2}$.
Integrating and inserting (4.3) yields

$\|\nabla(\mathrm{dens}\,\gamma)^{1/2}\|_{2}^{2}\leq\frac{4}{\mathcal{N}}{\mathcal{E}}_{0}(\gamma).$
(4.4)

For given $\rho$, this applies to any $\gamma$ with
$\mathrm{dens}\,\gamma=\rho$ to give

$\|\nabla\sqrt{\rho}\|_{2}^{2}\leq\frac{4}{\mathcal{N}}F(\rho).$ (4.5)

Since we wish to work with density rather than the square root of density, the
inequality (4.4) is not always convenient. Alternatively, returning to (4.2),
integrating and applying the Cauchy-Schwartz inequality,

$\int|\nabla\rho|\,\leq
2\left(\int\rho\,\right)^{1/2}\left(\frac{1}{\mathcal{N}}\int|\nabla\psi|^{2}\,\right)^{1/2},$

which yields $\|\nabla\rho\|_{1}\leq 2\sqrt{{\mathcal{E}}_{0}(\psi)}$. This is
extended to mixed states by a method similar to what gave (4.4), with the
result

$\|\nabla\mathrm{dens}\,\gamma\|_{1}\leq 2\sqrt{{\mathcal{E}}_{0}(\gamma)}.$
(4.6)

Again, we turn this into a bound for $F$, as

$\|\nabla\rho\|_{1}\leq 2\sqrt{F(\rho)}.$ (4.7)

###  4.3. Effective domain

In this section we will show that

$\boxed{\mathrm{dom}\,F^{\circ}=\mathrm{dom}\,F={\mathcal{J}}_{\mathcal{N}}:{\kern-5.0pt}=\left\\{\rho\in\mathsf{Dens}_{\mathcal{N}}\,:\,\sqrt{\rho}\in
H^{1}({\mathbb{R}})^{3}\right\\}}$ (4.8)

is the effective domain of both $F^{\circ}$ and $F$. That
${\mathcal{J}}_{\mathcal{N}}\supseteq\mathrm{dom}\,F\supseteq\mathrm{dom}\,F^{\circ}$
follows from the results of 4.2 and the fact that $F\leq F^{\circ}$. So what
needs to be shown is
${\mathcal{J}}_{\mathcal{N}}\subset\mathrm{dom}\,F^{\circ}$. This is done by
an explicit construction.

####  4.3.1. Transporting densities and wavefunctions

Suppose a density $\rho_{0}$ is given on some domain $\Omega$ and we wish to
pull back the mass in $\Omega$ along a bijection
$\varphi:{{\mathbb{R}}}^{3}\rightarrow\Omega$. That is, we want the density
$\rho$ on ${{\mathbb{R}}}^{3}$ such that

$\int_{A}\rho=\int_{\varphi(A)}\rho_{0}$

for every measurable $A\subset{{\mathbb{R}}}^{3}$. The solution to this
problem is very familiar: with $J(x)=|\partial\varphi/\partial x|$ denoting
the Jacobian determinant and abbreviating $y=\varphi(x)$,

$\rho(x)=\rho_{0}(\varphi(x))\,J(x)=\rho_{0}(y)J(x).$ (4.9)

What if, instead, we are given a wavefunction $\psi_{0}$ on
$\Omega^{\mathcal{N}}$ with $\mathrm{dens}\,\psi_{0}=\rho_{0}$ and wish to
find $\psi$ with $\mathrm{dens}\,\psi=\rho$? A solution is nearly as easy.
First extend the map $\varphi$ to the $\mathcal{N}$-particle configuration
space as

$\underline{y}=\underline{\varphi}(\underline{x})=(\varphi(x_{1}),\varphi(x_{2}),\ldots\varphi(x_{\mathcal{N}})).$

Then, we can take

$\psi(\underline{x})=\psi_{0}(\underline{y})\,\Big{|}\frac{\partial\underline{\varphi}}{\partial\underline{x}}\Big{|}^{\frac{1}{2}}=\psi_{0}(\underline{y})\prod_{i}J(x_{i})^{1/2}.$
(4.10)

Naturally, one wants to bound $\|\psi\|_{H^{1}({{\mathbb{R}}}^{3})}$ in terms
of $\|\psi_{0}\|_{H^{1}(\Omega)}$, but we shall not pursue that problem in
general, but move on to the special case that interests us.

####  4.3.2. Wavefunction construction for
$\rho\in{\mathcal{J}}_{\mathcal{N}}$

We now construct[1, 9, 10] a wavefunction $\psi$ with $\mathrm{dens}\,\psi$
equal to given $\rho$ in ${\mathcal{J}}_{\mathcal{N}}$, and calculate a bound
on its internal energy. Using 4.3.1 as inspiration, we find a map
$\varphi:{{\mathbb{R}}}^{3}\rightarrow\Omega$ onto region $\Omega$ and a type
of density $\rho_{0}$ on that region for which an appropriate Slater
determinant can be easily found. We leave spin coordinates out of
consideration, so implicitly all $\sigma$ are supposed to take the maximum
value.

Take $\Omega=[0,1]\times{{\mathbb{R}}}\times{{\mathbb{R}}}$, and assume
$\rho_{0}$ has the property

$\int\rho_{0}(y^{1},y^{2},y^{3})\,dy^{2}\,dy^{3}={\mathcal{N}}.$ (4.11)

Then, it is readily apparent that the one-particle wavefunctions
[$y\equiv(y^{1},y^{2},y^{3})$]

$\phi_{0,k}(y):{\kern-5.0pt}=\left(\frac{\rho_{0}(y)}{\mathcal{N}}\right)^{1/2}e^{2\pi
iky^{1}}$ (4.12)

indexed by integer $k$ are orthonormal over $\Omega$. Since the density for
each $\phi_{k}$ is $\rho_{0}(y)/\mathcal{N}$, a Slater determinant of
$\phi_{k}$’s for $\mathcal{N}$ distinct $k$’s will give density $\rho_{0}$.

All that is now required is a mapping of ${\mathbb{R}}^{3}$ onto $\Omega$
which carries the given $\rho$ on to a $\rho_{0}$ satisfying (4.11). With the
definition

$J(x^{1}):{\kern-5.0pt}=\frac{1}{\mathcal{N}}\int\rho(x)\,dx^{2}\,dx^{3},$
(4.13)

one immediately checks that the following works.

$\displaystyle\varphi(x)$
$\displaystyle=(y^{1},y^{2},y^{3})=(y^{1},x^{2},x^{3}),$ $\displaystyle y^{1}$
$\displaystyle=\int_{-\infty}^{x^{1}}J(z^{1})\,dz^{1}.$ (4.14)

Mass is being redistributed only along the first direction, at a varying rate
so that (4.11) is satisfied.

Of course, the notation was chosen because $J$ actually is the Jacobian of
$\varphi$:

$J(x^{1})=\frac{dy^{1}}{dx^{1}}=\left|\frac{\partial y}{\partial
x}\right|=\frac{\rho(x)}{\rho_{0}(y)}.$ (4.15)

According to (4.9), the $\phi_{0,k}$ of (4.12) are now transported to
${\mathbb{R}}^{3}$ as

$\phi_{k}(x)=\phi_{0,k}(y)\sqrt{J(x)}=\sqrt{\frac{\rho(x)}{\mathcal{N}}}\,\,e^{2\pi
iky^{1}}.$ (4.16)

Without further calculation, the general construction assures us that the
$\phi_{k}$ are orthonormal on ${{\mathbb{R}}}^{3}$ and that a Slater
determinant $\psi=|\phi_{k_{1}},\ldots,\phi_{k_{\mathcal{N}}}|$ of them yields
the desired density $\rho$. As illustrated here, $y$ may be regarded as a
function of $x$ or vice-versa, and either one as a coordinatization of
${\mathbb{R}}^{3}$ or of $\Omega$ as convenient, because the correspondence
between $x$ and $y$ is bijective.

The wavefunction now being constructed, all that remains is to bound on the
kinetic energy of $\psi$ by bounding the kinetic energy of each $\phi_{k}$:

${\mathcal{E}}_{0}(\phi_{k})=\frac{1}{\mathcal{N}}\|\nabla\sqrt{\rho}\|_{2}^{2}+\frac{(2\pi
k)^{2}}{\mathcal{N}}\int{\rho(x)}\left(\frac{dy^{1}}{dx^{1}}\right)^{2}\,dx.$
(4.17)

The second term here will occupy us with a bit of Cauchy-Scwhartz
calisthenics. Using (4.15),

$\int\left(\frac{dy^{1}}{dx^{1}}\right)^{2}\rho(x)\,dx=\int{\rho_{0}(y)}J^{2}\,dy=\mathcal{N}\int_{0}^{1}J^{2}\,dy^{1}.$
(4.18)

Estimate the integrand in the final expression as

$\displaystyle J(x^{1})^{2}$
$\displaystyle=\left(\int_{-\infty}^{x_{1}}{J(z)}^{\frac{1}{2}}\left(J(z)^{-\frac{1}{2}}\frac{d{J(z)}}{dz}\right)dz\right)^{2}\leq\left(\int
J\,dx^{1}\right)\int{J}^{-1}\left|\frac{d{J}}{dx^{1}}\right|^{2}dx^{1}$
$\displaystyle=\int J^{-1}\left|\frac{d{J}}{dx^{1}}\right|^{2}dx^{1}.$ (4.19)

The first equality is just the fundamental theorem of calculus, the inequality
is Cauchy-Schwartz, and the final equality follows from $\int J\,dx^{1}=\int
dy^{1}=1$. To evaluate the final integral, Again, we go to work on the
integrand with the Cauchy-Schwartz inequality, to obtain

$\left|\frac{dJ}{dx^{1}}\right|^{2}=\left(\frac{1}{\mathcal{N}}\int
2\sqrt{\rho(x)}\left|\frac{\partial\sqrt{\rho(x)}}{\partial
x^{1}}\right|\,dx^{2}\,dx^{3}\right)^{2}\leq\frac{4}{\mathcal{N}^{2}}{J(x^{1})}\int|\nabla\sqrt{\rho}|^{2}\,dx^{2}\,dx^{3}.$

Inserting this into (4.19),

$J(x^{1})^{2}\leq\int
J^{-1}\left|\frac{d{J}}{dx^{1}}\right|^{2}dx^{1}\leq\frac{4}{\mathcal{N}^{2}}\|\nabla\sqrt{\rho}\|_{2}^{2}$

Finally, returning to (4.17) via (4.18) results in

${\mathcal{E}}_{0}(\phi_{k})\leq\frac{1}{\mathcal{N}}\left(1+\left(\frac{4\pi
k}{\mathcal{N}}\right)^{2}\right)\|\nabla\sqrt{\rho}\|_{2}^{2}.$

This is for one of the orbitals entering our Slater determinant, so summing
over $\mathcal{N}$ values of $|k|<\mathcal{N}$,

$F^{\circ}(\rho)\leq{\mathcal{E}}_{0}(\psi)\leq(1+16\pi^{2})\|\nabla\sqrt{\rho}\|_{2}^{2}.$
(4.20)

###  4.4. ${\mathcal{J}}_{\mathcal{N}}\subset L^{1}\cap L^{3}\cap W^{1,1}$

Combining the upper bound on $F^{\circ}(\rho)$ from (4.20) with the lower
bound of (4.5) yields

$\boxed{c\|\sqrt{\rho}\|_{H^{1}}^{2}\leq F(\rho)\leq F^{\circ}(\rho)\leq
c^{\prime}\|\sqrt{\rho}\|_{H^{1}}^{2}}$ (4.21)

for some $\mathcal{N}$-dependent constants $c$ and $c^{\prime}$. The
demonstration of (4.8) is thus achieved, with some extra information.

Characterizations in terms of norms of $\rho$ rather than of $\sqrt{\rho}$ can
also be useful. Inequality (4.7) shows that
${\mathcal{J}}_{\mathcal{N}}\subset W^{1,1}$, where $W^{1,1}$ is the Sobolev
space with norm $\|f\|_{W^{1,1}}=\int(|f|+|\nabla f|)\,dx$. Also[1], the
Sobolev inequality

$3\left(\frac{\pi}{2}\right)^{4/3}\|f\|_{6}^{2}\leq\|\nabla f\|_{2}^{2},$

with $\sqrt{\rho}$ substituted for $f$ yields

$\|\rho\|_{3}\leq cF(\rho),$ (4.22)

for some constant $c$. Thus, ${\mathcal{J}}_{\mathcal{N}}\subset L^{3}$, as
well. The inequality (4.22) will be applied to the consideration of potentials
in §6.4.

###  4.5. Internal energy is lower semicontinuous on
$\mathsf{Dens}_{\mathcal{N}}$

####  4.5.1. Topologies on $\mathsf{Dens}_{\mathcal{N}}$

The subject of this section, lower semicontinuity of $F^{\circ}$ and $F$ on
$\mathsf{Dens}_{\mathcal{N}}$, is important to the Fenchel conjugacy with the
ground-state energy $E$ discussed in §6, among other things. But, lower
semicontinuous with respect to what topology on $\mathsf{Dens}_{\mathcal{N}}$?
In §3.1, we showed that
$\mathrm{dens}\,:\mathsf{State}\rightarrow\mathsf{Dens}$ is continuous with
respect to the $L^{1}$ topology on $\mathsf{Dens}$. We have not made use of
that result, yet. Continuity of $\mathrm{dens}\,$ thus holds for any topology
weaker than $L^{1}$. That is, so long as the topology is not so weak that it
fails to be Hausdorff,
$\mathrm{dens}\,\mathrm{st}\,\gamma=\mathrm{st}\,\mathrm{dens}\,\gamma$ for
nearstandard $\gamma$. (Recall that a Hausdorff topology is such that for
$x\not=y$, one can find neighborhoods $U$ of $x$ and $V$ of $y$, such that
$U\cap V=\emptyset$.) The topology is implicit in the standard part operation,
which is not well-defined for a non-Hausdorff topology. Continuity of
$\mathrm{dens}\,$ is important for the proof that $F^{\circ}$ is lsc, but
there is another condition that we need to impose.

######  Definition 4.1.

A topology $\tau$ on $\mathsf{Dens}_{\mathcal{N}}$ is weak-but-not-leaky if it
is weaker than, or equivalent to, $L^{1}$, yet Hausdorff and such that all
nearstandard points are tight.

An obvious example of such a topology is the weak-$L^{1}$ topology induced by
the seminorms $\rho\mapsto\int f\rho$ for $f\in L^{\infty}$. A disadvantage of
the weak-$L^{1}$ topology is that it is not metrizable. In §5, we will
construct a metrizable weak-but-not-leaky topology on
$\mathsf{Dens}_{\mathcal{N}}$, the weak-${\mathfrak{P}}$ topology, which is
even weaker than weak-$L^{1}$.

####  4.5.2. Pure-state internal energy

######  Theorem 4.2.

$F^{\circ}:(\mathsf{Dens}_{\mathcal{N}},\tau)\rightarrow\overline{{\mathbb{R}}}$
is lsc, if $\tau$ is weak-but-not-leaky.

###### Proof.

Let $\rho\in\mathsf{Dens}_{\mathcal{N}}$ be given and
$\rho^{\prime}\in{{}^{*}}{\mathsf{Dens}_{\mathcal{N}}}$ be $\tau$-near $\rho$.
Assume $F^{\circ}(\rho^{\prime})\ll\infty$, since otherwise there is nothing
to show. $\rho^{\prime}$ is tight by definition of weak-but-not-leaky, so
there is a tame $\psi^{\prime}$ with
$\mathrm{dens}\,\psi^{\prime}=\rho^{\prime}$, and
${\mathcal{E}}(\psi^{\prime})\simeq F^{\circ}(\rho^{\prime})$. By Prop. 3.1
and 4.5.1, $\psi^{\prime}$ is nearstandard, with
${\mathcal{E}}(\mathrm{st}\,\psi^{\prime})\lesssim{\mathcal{E}}(\psi^{\prime})\simeq
F^{\circ}(\rho^{\prime})$, and
$\mathrm{dens}\,\mathrm{st}\,\psi^{\prime}=\mathrm{st}\,\mathrm{dens}\,\psi^{\prime}=\mathrm{st}\,\rho^{\prime}=\rho$.
∎

####  4.5.3. Mixed-state internal energy

######  Definition 4.2.

If $A$ is a subset of a vector space, then its convex hull is defined as

$\mathrm{co}\,A:{\kern-5.0pt}=\left\\{\sum_{i=1}^{n}\alpha_{i}x_{i}\,:\,\sum_{i=1}^{n}\alpha_{i}=1\,\,\text{
and }\,\,x_{1},\ldots,x_{n}\in A\right\\}.$ (4.23)

The closed convex hull of $A$, denoted $\overline{\mathrm{co}}\,A$ is the
closure of $\mathrm{co}\,A$ in whatever topology is being considered on the
ambient vector space.

The goal now is to extend the result of 4.5.2 from $F^{\circ}$ to $F$ by
showing that
$\mathrm{epi}\,F=\overline{\mathrm{co}}\,\mathrm{epi}\,F^{\circ}$. For, lower
semicontinuity (convexity) of $F$ is equivalent to closure (resp., convexity)
of its epigraph.

######  Theorem 4.3.

$\mathrm{epi}\,F=\overline{\mathrm{co}}\,\mathrm{epi}\,F^{\circ}$ in
$\mathsf{Dens}_{\mathcal{N}}\times{{\mathbb{R}}}$, with any weak-but-not-leaky
topology on $\mathsf{Dens}_{\mathcal{N}}$.

###### Proof.

To show that
$\mathrm{epi}\,F=\overline{\mathrm{co}}\,\mathrm{epi}\,F^{\circ}$, we
establish inclusion in both directions.

(a) $\mathrm{epi}\,F\subseteq\overline{\mathrm{co}}\,\mathrm{epi}\,F^{\circ}$:
It suffices to show that, if $\mathrm{dens}\,\gamma=\rho$, then
$(\rho,{\mathcal{E}}(\gamma))\in\overline{\mathrm{co}}\,\mathrm{epi}\,F^{\circ}$.
From the second normal form for $\gamma$, define

$w_{n}=\sum_{i=1}^{n}c_{i},\quad\gamma_{n}=\sum_{i=1}^{n}\frac{c_{i}}{w_{n}}|\hat{\psi}_{i}\rangle\langle\hat{\psi}_{i}|.$

Then,

${\mathcal{E}}(\gamma_{n})=\sum_{i=1}^{n}\frac{c_{i}}{w_{n}}{\mathcal{E}}(\hat{\psi}_{i})\to{\mathcal{E}}(\gamma),\quad\mathrm{dens}\,\gamma_{n}=\sum_{i=1}^{n}\frac{c_{i}}{w_{n}}\mathrm{dens}\,\hat{\psi}_{i}\to\mathrm{dens}\,\gamma.$

$(\mathrm{dens}\,\gamma_{n},{\mathcal{E}}(\gamma_{n}))$ is in
$\mathrm{co}\,\mathrm{epi}\,F^{\circ}$, because
$(\mathrm{dens}\,\hat{\psi}_{i},{\mathcal{E}}(\hat{\psi}_{i}))\in\mathrm{epi}\,F^{\circ}$.
Therefore,
$(\mathrm{dens}\,\gamma,{\mathcal{E}}(\gamma))\in\overline{\mathrm{co}}\,\mathrm{epi}\,F^{\circ}$,
as required.

(b) $\mathrm{epi}\,F\supseteq\overline{\mathrm{co}}\,\mathrm{epi}\,F^{\circ}$:
Let $(\rho,\lambda)\in\overline{\mathrm{co}}\,\mathrm{epi}\,F^{\circ}$ be
given (standard). If $\lambda=+\infty$, there is nothing to show, so assume
$\lambda<+\infty$. By definition of closed convex envelope, there is
$N\in{{}^{*}}{{\mathbb{N}}}$ and collections of coefficients and orthonormal
∗-pure-states such that

$\sum_{i\leq N}c_{i}=1,\quad\rho\simeq\sum_{i\leq
N}c_{i}\,\mathrm{dens}\,(\psi_{i}),\quad\lambda\simeq\sum_{i\leq
N}c_{i}\,{\mathcal{E}}(\psi_{i}).$ (4.24)

But this shows that
$\gamma=\sum_{i=1}^{N}c_{i}|\psi_{i}\rangle\langle\psi_{i}|\in{{}^{*}}{\mathsf{State}}_{1}$
satisfies $\mathrm{dens}\,\gamma\simeq\rho$ and
${\mathcal{E}}(\gamma)\simeq\lambda$. Since $\gamma$ is thus tame, Cor. 3.1
implies that ${\mathcal{E}}(\mathrm{st}\,{\gamma})\leq\lambda$ and
$(\mathrm{dens}\,\mathrm{st}\,{\gamma},{\mathcal{E}}(\mathrm{st}\,{\gamma}))\in\mathrm{epi}\,F$.
∎

This of course immediately gives us lower-semicontinuity.

######  Corollary 4.1.

$F:(\mathsf{Dens}_{\mathcal{N}},\tau)\rightarrow\overline{{\mathbb{R}}}$ is
lower semicontinuous if $\tau$ is weak-but-not-leaky.

##  5\. Weak-${\mathfrak{P}}$ topology on $\mathsf{Dens}_{\mathcal{N}}$

The aim in this section is to construct a convenient, metrizable, weak-but-
not-leaky topology on $\mathsf{Dens}_{\mathcal{N}}$ which is called the
weak-${\mathfrak{P}}$ topology. Lower semicontinuity of $F$ with respect to it
immediately implies the same for any $\|\cdot\|_{p}$, $1\leq p<\infty$.
Although we are interested in only $\mathsf{Dens}_{\mathcal{N}}$, the
weak-${\mathfrak{P}}$ metric will be defined on all of $\mathsf{Dens}$.

###  5.1. Hierarchy of partitions

Let ${\mathfrak{P}}^{0}$ denote a partition of ${{\mathbb{R}}}^{3}$ into a
regular grid of cubes of side length $1$, with corners at integer coordinates.
To make it a genuine partition, we agree that each cell includes its front,
left and bottom faces, but not the back, right or top. Now, divide each cell
of ${\mathfrak{P}}^{0}$ into $2^{3}$ cubes of side length $2^{-1}$; these
latter cubes are the cells of ${\mathfrak{P}}^{1}$. Continue subdividing so
that each cell of ${\mathfrak{P}}^{n}$ is the union of $2^{3}$ cells of
${\mathfrak{P}}^{n+1}$ to obtain an infinite hierarchy of partitions

${\mathfrak{P}}^{0}<{\mathfrak{P}}^{1}<\cdots.$

A set $A$ is ${\mathfrak{P}}^{n}$-measurable if it is the union of cells of
${\mathfrak{P}}^{n}$, ${\mathfrak{P}}^{n}$-finitely-measurable if it is a
finite such union, and simply ${\mathfrak{P}}$-measurable if a finite union of
cells. In this last case, $A$ is actually ${\mathfrak{P}}^{m}$-finitely-
measurable, where $m$ is the maximal rank of members of $A$, but the
terminology allows us to avoid mentioning $m$.

We define a linear projection $\pi_{n}$ from integrable functions to
${\mathfrak{P}}^{n}$-measurable integrable functions as follows. For a
function $f\in L^{1}({{\mathbb{R}}}^{3})$, $\pi_{n}f$ is constant over each
cell of ${\mathfrak{P}}^{n}$, with value on a cell equal to the average of $f$
over that cell. Thus, $f$ and $\pi_{n}f$ have the same integral over each
cell.

####  5.1.1. a metric for the weak-${\mathfrak{P}}$ topology

######  Definition 5.1.

For $A$ a ${\mathfrak{P}}^{n}$-finitely-measurable set, the continuous
seminorm $p_{A}$ on $L^{1}({{\mathbb{R}}}^{3})$ is defined as

$p_{A}(f):{\kern-5.0pt}=\int_{A}|\pi_{n}f|\,\leq\|f\|_{1}.$ (5.1)

The weak-${\mathfrak{P}}$ topology is simply the topology generated by these
seminorms. However, not all these seminorms are needed. The smaller collection
$\\{p_{\Omega}:\Omega\in{\mathfrak{P}}\\}$ suffices. That is,
$f_{n}\stackrel{{\scriptstyle\mathrm{wk}-{\mathfrak{P}}}}{{\to}}f$ if and only
if $p_{\Omega}(f-f_{n})\to 0$ for every $\Omega\in{\mathfrak{P}}$. We can go
even further and use just $\\{p_{C_{n}},n=0,1,\ldots\\}$ where $C_{n}$ is the
largest ${\mathfrak{P}}^{n}$-measurable set inside the ball $B_{0}(n)$ of
radius $n$ centered at $0$. Finally, if a metric is preferred,

$d_{\mathfrak{P}}(f,g)=\sum_{n=0}^{\infty}\frac{p_{C_{n}}(f-g)}{2^{n+2}}$
(5.2)

generates the weak-${\mathfrak{P}}$ topology. Note that for $f,g\in L^{1}$,
$d_{\mathfrak{P}}(f,g)\leq\|f-g\|_{1}/2$.

###  5.2. How weak is it?

An example shows an important way in which weak-${\mathfrak{P}}$ is weaker
than the $L^{1}$ topology. Let $f$ equal $1$ on some cell
$\Omega\in{\mathfrak{P}}^{0}$, and $0$ elsewhere. One can easily construct a
function $f_{n}$ which is zero off $\Omega$, and inside $\Omega$ equal to $2$
on half the ${\mathfrak{P}}^{n}$ cells and to $0$ on the other half in such a
way that $\pi_{m}f_{n}\equiv 0$ for $m<n$. Then,
$f_{n}\stackrel{{\scriptstyle\text{wk-}{\mathfrak{P}}}}{{\to}}f$, but
$\|f_{n}-f\|_{p}=1$ for all $n$ and all $1\leq p<\infty$. On the other hand,
since $|\int_{\Omega}f\,|\leq\|f\|_{p}|\Omega|^{1/p^{\prime}}$, $L^{p}$
convergence implies wk-${\mathfrak{P}}$ convergence. Thus, on
$\mathsf{Dens}_{\mathcal{N}}$, weak-${\mathfrak{P}}$ is weaker than $L^{p}$
($1\leq p<\infty$).

The $f_{n}$ of the example became more and more oscillatory as $n$ increased.
In our application, that sort of behavior implies ever increasing kinetic
energy.

###  5.3. Approximation by projection

####  5.3.1. Weak-${\mathfrak{P}}^{n}$ is weak-but-not-leaky

Recall that $C_{n}$ is the union of ${\mathfrak{P}}^{n}$-cells contained fully
within the ball of center $0$ and radius $n$, and $1_{C_{n}}$ is the indicator
function of $C_{n}$, equal to 1 on $C_{n}$ and to zero on its complement
$C_{n}^{c}$.

######  Lemma 5.1.

Let $g\in L^{1}({\mathbb{R}}^{3})$. Then, both $1_{C_{n}}\,\Pi_{n}g$ and
$\Pi_{n}g$ tend to $g$ in $L^{1}$ as $n\to\infty$.

###### Proof.

Consider first a continuous function with bounded support, $f\in
C_{c}({\mathrm{R}}^{3})$. It is easy to see that $\|f-\Pi_{n}f\|_{1}\to 0$.
For, being compactly supported, $f$ is uniformly continuous, so given
$\epsilon>0$, there is $\delta>0$ such that $|f(x+y)-f(x)|<\epsilon/M$
whenever $|x-y|<\delta$, where $M$ is the Lebesgue measure of the support of
$f$. But then the average of $f$ over a region $A$ with diameter less than
$\delta$ differs by less than $\epsilon/M$ from its value at any point in $A$.
Therefore, if $n$ is large enough that $\sqrt{3}2^{-n}<\delta$, it follows
that $\|f-\Pi_{n}f\|_{1}<\epsilon$.

Now, given $g\in L^{1}({\mathbb{R}}^{3})$, and $\epsilon>0$, There is $f\in
C_{c}({{\mathbb{R}}}^{3})$ with $\|g-f\|_{1}<\epsilon/3$, because continuous
functions with bounded support are dense in $L^{1}({{\mathbb{R}}})$. Then, by
the triangle inequality,

$\displaystyle\|g-1_{C_{n}}\,\Pi_{n}\rho\|_{1}$
$\displaystyle\leq\|\rho-f\|_{1}+\|f-\Pi_{n}f\|_{1}+\|\Pi_{n}(f-\rho)\|_{1}+\|\Pi_{n}1_{C_{n}^{c}}\,g\|_{1}$
$\displaystyle\leq
2\|\rho-f\|_{1}+\|f-\Pi_{n}f\|_{1}+\|1_{C_{n}^{c}}\,g\|_{1}.$

The final line follows because $\Pi_{n}$ is an $L^{1}$ contraction. In the
final line, the second term tends to zero by the previous paragraph and the
last term simply because $g$ is integrable. Therefore,
$\limsup_{n\to\infty}\|g-1_{C_{n}}\,\Pi_{n}g\|_{1}<\epsilon$. The proof for
$\Pi_{n}g$ is even easier, as the final term in the displayed inequality is
absent in that case. ∎

This lemma immediately implies that the weak-${\mathfrak{P}}$ topology is
weak-but-not-leaky, as shown in the following corrolary. However, it may be
skipped without loss since a stronger result is proven independently in Thm.
5.1.

######  Corollary 5.1.

The weak-${\mathfrak{P}}$ topology on $\mathsf{Dens}_{\mathcal{N}}$ is weak-
but-not-leaky.

###### Proof.

That the weak-${\mathfrak{P}}$ topology is weaker than $L^{1}$ is trivial.
Lemma 5.1 shows that it is Hausdorff. There only remains to show that if
$\rho\in\mathsf{Dens}_{\mathcal{N}}$ and $\rho^{\prime}$ is
$d_{\mathfrak{P}}$-near $\rho$, that $\rho^{\prime}$ is tight. But,
$\int_{C_{n}}|\Pi_{n}(\rho^{\prime}-\rho)|\simeq 0$ implies that
$\int_{C_{n}}\rho^{\prime}\simeq\int_{C_{n}}\rho$. Since both integrate to
$\mathcal{N}$, $\int_{C_{n}^{c}}\rho^{\prime}\simeq\int_{C_{n}^{c}}\rho$ for
all limited $n$. But $\int_{C_{n}^{c}}\rho^{\prime}$ is decreasing in $n$ and
$\int_{C_{n}}\rho\to 0$. ∎

###  5.4. Weak-${\mathfrak{P}}$ with internal energy control is as good as
$L^{1}$

The next theorem shows that if a sequence $\rho_{n}$ converges with respect to
$d_{{\mathfrak{P}}}$, then the only way it can avoid converging with respect
to $L^{1}$ is to have divergent internal energy.

####  5.4.1. Poincaré inequality[11, 12, Acosta+Duran03, 14]

Let $\Omega$ be a convex bounded region with diameter $\mathrm{diam}(\Omega)$.
Then, if $f$ is in $W^{1,1}(\Omega)$ (it and its distributional gradient
$\nabla f$ are integrable over $\Omega$), and denoting the mean of $f$ over
$\Omega$ by $\langle{f}\rangle_{\Omega}$, the Poincaré inequality we need is

$\int_{\Omega}|f-\langle{f}\rangle_{\Omega}|\,d{x}\leq\frac{\pi}{2}\,\mathrm{diam}(\Omega)\int_{\Omega}|\nabla
f|\,d{x}.$ (5.3)

Applying this inequality to each cell of ${\mathfrak{P}}^{n}$ and summing the
results yields

$\|\rho-\Pi_{n}\rho\|_{1}\leq\frac{\pi\sqrt{3}}{2^{n+1}}\|\nabla\rho\|_{1}\leq\frac{c}{2^{n}}F(\rho)^{1/2},\quad\text{for
}\rho\in\mathsf{Dens}_{\mathcal{N}}.$ (5.4)

Always, of course, $\|\rho-\Pi_{n}\rho\|_{1}\leq 2\mathcal{N}$.

####  5.4.2. A dichotomy

In preparation for Thm. 5.1, we derive a useful inequality. For any measurable
set $B$ and $\rho,\rho^{\prime}\in\mathsf{Dens}_{\mathcal{N}}$,

$\|\rho-\rho^{\prime}\|_{1}=\|(\rho-\rho^{\prime})1_{B}\|_{1}+\|(\rho-\rho^{\prime})1_{B^{c}}\|_{1},$

Now we estimate the second term on the right-hand side.

$\displaystyle\|(\rho-\rho^{\prime})1_{B^{c}}\|_{1}$
$\displaystyle\leq\int_{B^{c}}\rho\,dx+\int_{B^{c}}\rho^{\prime}\,dx\leq
2\int_{B^{c}}\rho\,dx+\left|\int_{B^{c}}(\rho^{\prime}-\rho)\,dx\right|$
$\displaystyle=2\int_{B^{c}}\rho\,dx+\left|\int_{B}(\rho^{\prime}-\rho)\,dx\right|\leq
2\int_{B^{c}}\rho\,dx+\|(\rho-\rho^{\prime})1_{B}\|_{1}.$

The first inequality is a consequence of the fact that densities are non-
negative. The equality in the second line follows because both densities
$\rho$ and $\rho^{\prime}$ have the same total integral ($\mathcal{N}$), and
the final inequality is just pulling the absolute value inside the integral.
Plugging this last result into the previous display,

$\rho,\rho^{\prime}\in\mathsf{Dens}_{\mathcal{N}}\Longrightarrow\|\rho-\rho^{\prime}\|_{1}\leq
2\|(\rho-\rho^{\prime})1_{B}\|_{1}+2\int_{B^{c}}\rho\,dx.$ (5.5)

######  Theorem 5.1.

Suppose $\rho_{n}\stackrel{{\scriptstyle wk-{\mathfrak{P}}}}{{\to}}\rho$ in
$\mathsf{Dens}_{\mathcal{N}}$. Then, either $\rho_{n}\stackrel{{\scriptstyle
L^{1}}}{{\to}}\rho$ or $\liminf_{n}F^{\circ}(\rho_{n})=+\infty$.

###### Proof.

The equivalent nonstandard phrasing of the lemma is: if $\rho^{\prime}$ is
$d_{\mathfrak{P}}$-near $\rho$, then either it is $L^{1}$-near $\rho$, or it
has illimited internal energy. There is nothing to show unless
${{}^{*}}{F^{\circ}}(\rho^{\prime})\ll\infty$, so assume that is so.

The inequality (5.5) plays a key role, and a triangle inequality,

$\|\rho-\rho^{\prime}\|_{1}\leq\|\rho-\Pi_{n}\rho\|_{1}+\|\Pi_{n}(\rho-\rho^{\prime})\|_{1}+\|\rho^{\prime}-\Pi_{n}\rho^{\prime}\|_{1},$
(5.6)

will be used much as in Lemma 5.1. We proceed by showing that all three terms
on the right-hand side are infinitesimal for some
$N\in{{}^{*}}{{\mathbb{N}}}$. By definition of the weak-${\mathfrak{P}}$
topology, $\int_{C_{n}}|\Pi_{n}(\rho-\rho^{\prime})|\,dx\,\,\simeq 0$ for
every limited $n$ [recall the definition of the $C_{n}$ from (5.2)].
Robinson’s lemma (§B.6) implies existence of an illimited $N$, fixed in the
following discussion, such that that continues to hold for all $n\leq N$.

1st term of (5.6): $\rho$ is standard and $N$ illimited. By Lemma 5.1,
$\|\rho-\Pi_{N}\rho\|_{1}\simeq 0$.

2nd term: Apply (5.5) with $C_{N}$ for $B$, and note that
$\int_{C_{N}^{c}}\rho\,\simeq 0$ ($\rho$ is standard). Thus,
$\|\Pi_{N}(\rho-\rho^{\prime})\|_{1}\simeq 0$.

3rd term: $\rho^{\prime}$ has limited internal energy. Therefore, (5.4)
implies $\|\rho^{\prime}-\Pi_{N}\rho^{\prime}\|_{1}\simeq 0$. ∎

Internal energy functionals are not upper semicontinuous because by putting in
short-wavelength wiggles which are a negligible change in $L^{p}$ (for any
$p$) norm can drive the internal energy arbitrarily high. The inequality (5.1)
and Thm. 5.4 show the flip-side of this phenomenon: under a bound on internal
energy, we really only need to know a density down to a certain spatial
resolution. Below that scale, it must be almost flat.

##  6\. Ground-state energy, or ‘energy floor’ ($E$)

At long last, we are ready to begin considering external one-body potentials.
A one-body potential $v$ induces the $\mathcal{N}$-body external potential
$\underline{v}(\underline{x})=\sum_{i=1}^{\mathcal{N}}v(x_{i})$. The usual way
of defining the ground-state energy is

$E(v)=\inf\left\\{{\mathcal{E}}(\psi)+\langle\psi|\underline{v}\psi\rangle\,:\,\psi\in\mathsf{State}^{\circ}\right\\}.$

Of course, the term $\langle\psi|\underline{v}\psi\rangle$ only depends on the
state through its density, so the minimization can done in two stages:

$E(v)=\inf\left\\{F^{\circ}(\rho)+\langle
v,\rho\rangle\,:\,{\rho\in\mathsf{Dens}_{\mathcal{N}}}\right\\}.$ (6.1)

Here, we have introduced the notational abbreviation

$\langle v,\rho\rangle:{\kern-5.0pt}=\int v\,\rho.$ (6.2)

This is common in situations of pairing between elements of a Banach space and
its dual, but we will use it even in cases which may yield $\langle
v,\rho\rangle=+\infty$. Now, if the external potential $v$ is bounded, then
$\int v\rho$ is well-defined for all $\rho\in\mathsf{Dens}_{\mathcal{N}}$ and
the minimization in (6.1) makes sense. But if it is unbounded, we must proceed
more cautiously.

####  6.0.1. on the term “ground-state energy”

The term “ground-state energy” for $E(v)$ is customary in the DFT literature.
But, unless the infimum is actually a minimum, there is no genuine ground
state. $v\equiv 0$ is the simplest example of a potential for which there is
no minimum. A term, such as ‘energy floor’, which does not imply the existence
of a ground state might therefore be more appropriate. Mostly, we shall avoid
the issue by simply writing ‘$E$’.

####  6.0.2. Habitability and stability

######  Definition 6.1.

An external potential $v$ is habitable if $E(v)<+\infty$ and stable if
$E(v)>-\infty$.

Needless to say, we are interested in external potentials that are both
habitable and stable, but, ensuring these two conditions involves very
differing considerations.

####  6.0.3. some notation

$\langle v,\rho\rangle$ is not a function; it is the value of the function
$x\mapsto\langle v,x\rangle$ at the argument $\rho$. To deal with this
awkwardness, we write $\langle v,\cdot\rangle$ for the function
$\rho\mapsto\langle v,\rho\rangle$. The notation

$v^{+}:{\kern-5.0pt}=v\vee 0,\,\,\,v^{-}:{\kern-5.0pt}=(-v)\vee 0$

for the positive and negative part of an external potential, such that
$v=v^{+}-v^{-}$ and $\langle v,\cdot\rangle=\langle v^{+},\cdot\rangle-\langle
v^{-},\cdot\rangle$, will also be useful in this section.

###  6.1. Bounded external potentials

We begin by restricting our attention to external potentials in $L^{\infty}$,
for which we use the notation ‘$\mathsf{Bdd}$’.

If $w\in\mathsf{Bdd}$, then $\langle w,\cdot\rangle$ is $L^{1}$ continuous,
since $L^{\infty}$ is the Banach space dual of $L^{1}$. That is, $|\langle
w,\rho\rangle-\langle
w,\rho^{\prime}\rangle|\leq\|w\|_{\infty}\|\rho-\rho^{\prime}\|_{1}$. And
since $F$ is $L^{1}$ lsc, so too is $F+\langle w,\cdot\rangle$. However, Cor.
4.1 established that $F$ is lsc with respect to much weaker topologies, so we
should try to show the same for $F+\langle w,\cdot\rangle$. The full class of
weak-but-not-leaky topologies is beyond our reach, but the
weak-${\mathfrak{P}}$ topology can be handled.

######  Proposition 6.1.

For $w\in\mathsf{Bdd}$, $F+\langle
w,\cdot\rangle:(\mathsf{Dens}_{\mathcal{N}},\text{wk-}{\mathfrak{P}})\to\overline{{\mathbb{R}}}$
is lower semicontinuous.

###### Proof.

Take $\rho$ in $\mathsf{Dens}_{\mathcal{N}}$ and $\rho^{\prime}$
wk-${\mathfrak{P}}$-near $\rho$. According to Lemma 5.1, either
$F(\rho^{\prime})\simeq\infty$, or $\rho^{\prime}$ is $L^{1}$-near $\rho$. In
the former case, of course $F(\rho^{\prime})\gtrsim F(\rho)$, because $|\int
w\rho^{\prime}|\leq\mathcal{N}\|w\|_{\infty}$ is bounded. In the latter, the
conclusion follows by $L^{1}$ lsc of $F$ and $L^{1}$ continuity of $\langle
w,\cdot\rangle$. ∎

####  6.1.1. Fenchel conjugate and biconjugate[15, 16, 17, 18]

We take a break to do some general geometric analysis. A proper (not
identically $+\infty$) function $f:X\rightarrow\overline{{\mathbb{R}}}$ on a
vector space $X$ is given, along with a vector space $Y$ of linear functionals
$X\rightarrow{\mathbb{R}}$. The topology on $X$ is that induced by $Y$,
denoted $\sigma(X,Y)$. By definition, then, a set is a neighborhood of the
origin if and only if it contains an intersection
$\cap_{n}\\{x:|\langle\xi_{n},x\rangle|<\epsilon_{n}\\}$, with $\xi_{n}\in Y$
and $\epsilon_{n}>0$. In the $\sigma(X,Y)$ topology, a point $x_{0}$ is
outside a closed convex set $A$ if and only if there is $(\zeta,c)\in
Y\times{\mathbb{R}}$ such that
$\langle\zeta,x\rangle>c>\langle\zeta,x_{0}\rangle$ for all $x\in A$, that is,
$x_{0}$ can be strictly separated from $A$ by a linear functional in $Y$. Here
is an explanation for that. Forgetting about $A$’s convexity for the moment,
there must be a collection $\\{(\xi_{i},c_{i}):1\leq i\leq n\\}$ and
$\epsilon>0$ such that $c_{i}=\xi(x_{0})$, and for every $x\in A$, there is
some $i$ such that $|\langle\xi_{i},x\rangle-c_{i}|>\epsilon$. Arrange the
$\xi_{i}$’s into a map
$\underline{\xi}=(\xi_{1},\ldots,\xi_{n}):X\rightarrow{\mathbb{R}}^{n}$ into a
finite-dimensional Euclidean space. Then, the image of $A$,
$\underline{\xi}(A)$ is outside the $\epsilon$ ball around
$\underline{c}=(c_{1},\ldots,c_{n})$. But if $A$ is convex, then so is its
image in ${\mathbb{R}}^{n}$. We can apply finite-dimensional separation
properties in ${\mathbb{R}}^{n}$ to obtain a linear combination $\sum
a_{i}\xi_{i}$ which achieves the desired separation.

The function defined on $Y$ by

$f^{*}(\zeta):{\kern-5.0pt}=\sup\left\\{\langle\zeta,x\rangle-f(x):x\in
X\right\\},$ (6.3)

is called, among other names, the Fenchel conjugate. Up to signs, the
relationship between $f$ and $f^{*}$ is essentially the same as that between
$F$ and $E$. Note that

$\forall x\in X.\,\,f(x)\geq
c+\langle\zeta,x\rangle\,\,\,\Leftrightarrow\,\,\,-f^{*}(\zeta)\geq c.$

Defining the half-space indexed by $(\zeta,c)\in Y\times{\mathbb{R}}$ by

$H(\zeta,c):{\kern-5.0pt}=\left\\{(x,z)\in X\times{\mathbb{R}}:z\geq
c+\langle\zeta,x\rangle\right\\},$ (6.4)

we see that $\mathrm{epi}\,f\subseteq H(\zeta,c)$ if and only if
$-f^{*}(\zeta)\geq c$. (Recall that the epigraph of $f$ is the set
$\mathrm{epi}\,f=\left\\{(x,z)\in X\times{\mathbb{R}}:f(x)\leq z\right\\}$ of
points on or above the graph of $f$.)

The proposition we are aiming for is this: with $\hat{f}$ the function
satisfying $\mathrm{epi}\,\hat{f}=\overline{\mathrm{co}}\,\mathrm{epi}\,f$,

$\hat{f}(x)=\sup\left\\{\langle\zeta,x\rangle-f^{*}(\zeta):{\zeta\in
Y}\right\\}=f^{**}(x).$ (6.5)

An equivalent statement is that $\overline{\mathrm{co}}\,\mathrm{epi}\,f$ is
equal to the intersection of all $H(\zeta,c)$ that contain $\mathrm{epi}\,f$,
and it is in this form that we prove it, by showing that any point not in
$\mathrm{epi}\,f$ falls outside some such half-space. We prove this assuming
that $f\geq 0$ (adequate for our purposes), and later comment on the general
case. Suppose $(x_{0},z_{0})\not\in\overline{\mathrm{co}}\,\mathrm{epi}\,f$.
Then, there is $(\zeta,\beta)\in Y\times{\mathbb{R}}$, $c\in{\mathbb{R}}$ and
$\epsilon>0$ such that

$\forall(x,z)\in\mathrm{epi}\,f.\,\,\langle\zeta,x\rangle+\beta
z>c+\epsilon>c>\langle\zeta,x_{0}\rangle+\beta z_{0}.$

Since we can always increase $z$ without leaving $\mathrm{epi}\,f$, $\beta\geq
0$. If $x_{0}\in\mathrm{dom}\,f$, then $(x_{0},z)\in\mathrm{epi}\,f$ for large
enough $z$, so $\beta>0$ in that case. If $\beta>0$, dividing through by
$\beta$, we see that $\mathrm{epi}\,f$ is in $H(-\zeta/\beta,c/\beta)$, but
$(x_{0},z_{0})$ is not. The other case is $\beta=0$. Then,

$c+\langle-\zeta,x\rangle<0<c+\langle-\zeta,x_{0}\rangle,\,\,\forall(x,z)\in\mathrm{epi}\,f.$

Thus, since $f\geq 0$, $f(x)>Mc+\langle-M\zeta,x\rangle$ for any $M>0$. But,
for large enough $M$, we will certainly have
$z_{0}<Mc+\langle-M\zeta,x_{0}\rangle$. That completes the proof assuming
$f\geq 0$. If that is not the case, since $f$ is proper, there is some
$x_{0}\in\mathrm{dom}\,f$. The $\beta$ we get for this $x_{0}$ cannot be zero,
as remarked earlier. Hence, $f+\langle\zeta/\beta,\cdot\rangle-c/\beta\geq 0$.
If the proposition is true for this tilted and shifted function, it is also
true for $f$.

####  6.1.2. application to $F$

Recall that $\mathrm{epi}\,F=\overline{\mathrm{co}}\,\mathrm{epi}\,F^{\circ}$
(Thm. 4.3), where the closure is with respect to any weak-but-not-leaky
topology on $\mathsf{Dens}_{\mathcal{N}}$, such as $L^{1}$ or
weak-${\mathfrak{P}}$. Now, the weak-${\mathfrak{P}}$ topology is generated by
the indicator functions of cells of ${\mathfrak{P}}$. Thus, to apply 6.1.1 we
take $Y$ to be

$\mathsf{Potl}^{f}({\mathfrak{P}}):{\kern-5.0pt}=\mathrm{span}\left\\{1_{\Omega}\,:\,\Omega\in{\mathfrak{P}}\right\\},$
(6.6)

the vector space of functions spanned by the indicator functions of cells of
${\mathfrak{P}}$. A potential in $\mathsf{Potl}^{f}({\mathfrak{P}})$ is
nonzero on only a finite number of cells. Then,

$F(\rho)=\sup\left\\{E(v)-\langle
v,\rho\rangle:{v\in\mathsf{Potl}^{f}({\mathfrak{P}})}\right\\}.$ (6.7)

If we call a set of external potentials $F$-sufficient just in case the
supremum over the functions $E(v)-\langle v,\cdot\rangle$ is $F$, then the
display says that $\mathsf{Potl}^{f}({\mathfrak{P}})$ is $F$ sufficient. Since
always

$E(v)\leq F(\rho)+\langle v,\rho\rangle,$ (6.8)

any larger class of external potentials is also $F$-sufficient. We are not
claiming that $\mathsf{Potl}^{f}({\mathfrak{P}})$ is a minimal $F$-sufficient
class, but it is certainly pretty small.

######  Remark 6.1.

Starting from $L^{1}$ lower-semicontinuity of $F$, one may similarly show the
more-traditional, but weaker, result that $L^{\infty}$ is $F$-sufficient. The
proof uses the Hahn-Banach principle in the form of the assertion that the
closures of a convex set relative to the initial and weak topology are the
same.

###  6.2. $[0,+\infty]$-valued external potentials

We now move from the relatively modest space $\mathsf{Bdd}$ of bounded
external one-body potentials to a much larger class
$\mathsf{Potl}^{\uparrow}$. In contrast to the perturbations we consider
later, it is not a vector space, but only a convex cone. This class consists
of all measurable external potentials with values in the non-negative extended
reals:

$\mathsf{Potl}^{\uparrow}=({\mathbb{R}}^{3}\rightarrow[0,+\infty]).$ (6.9)

Why deal with such a large class of potentials? First, they have practical use
in modeling various sorts of confining potentials, such as a harmonic
oscillator potential or even an infinite cubic-well. Neither of these is in
any $L^{p}$ space, and a nice feature of our treatment is that such confining
potentials are integrated into the density functional picture without any need
to go back to talking explicitly about wavefunctions. Second, if we are going
to deal with external potentials unbounded above, there is hardly any extra
work to go all the way, for what we do. Third, and closely related to the
second point, the differing physical significances of unboundedness above and
below for an external potential is brought into the starkest possible relief.

A potential $w\in\mathsf{Potl}^{\uparrow}$ can be approximated density-wise by
bounded potentials. That is, with the truncations

$(w\wedge n)(x):{\kern-5.0pt}=(\text{if }w(x)<n\text{ then }w(x)\text{ else
}n),$ (6.10)

we have

$\langle w\wedge n,\rho\rangle\nearrow\langle w,\rho\rangle,\,\,\text{as
}{n\to\infty}.$

Thus,

$F(\rho)+\langle w\wedge n,\rho\rangle\nearrow F(\rho)+\langle w,\rho\rangle.$

The point is that if we can show that $\rho\mapsto F(\rho)+\langle w\wedge
n,\rho\rangle$ is lower semicontinuous for each $n$, then so is the supremum
(§2.3.1). But, we’ve already done that.

######  Theorem 6.1.

For $w\in\mathsf{Potl}^{\uparrow}$, $F+\langle
w,\cdot\rangle:(\mathsf{Dens}_{\mathcal{N}},\text{wk-}{\mathfrak{P}})\to\overline{{\mathbb{R}}}$
is lower semicontinuous.

###### Proof.

By Prop. 6.1 and the preceding discussion. ∎

####  6.2.1. sufficient condition for habitability of
$\mathsf{Potl}^{\uparrow}$ external potentials

######  Proposition 6.2.

$w\in\mathsf{Potl}^{\uparrow}$ is habitable if $\int_{U}w<\infty$ for some
open set $U\in{\mathbb{R}}^{3}$.

###### Proof.

Since $U$ contains an open ball, one can easily find a smooth $\rho$ with
support in $U$, bounded by $\|\rho\|_{\infty}<\infty$. By inequality (4.21),
$F(\rho)<\infty$, and $\langle
w,\rho\rangle\leq\|\rho\|_{\infty}\int_{U}w<\infty$. ∎

###  6.3. $F$-norm, and relatively $F$-small perturbations

An unbounded negative external potential poses a danger of instability. For
such an external potential, we can find a sequence $\rho_{n}$ of densities
such that $\langle v,\rho_{n}\rangle\to-\infty$. Intuitively, this is achieved
by squeezing the density more and more tightly into a bottomless potential
well. To assure that the total energy cannot be driven to $-\infty$ in this
way, we make sure that the kinetic energy cost of such a squeezing eventually
dominates. This motivates the following definition.

######  Definition 6.2.

The $F$-norm of a potential $v$ is

$\|v\|_{F}:{\kern-5.0pt}=\inf\left\\{a\in(0,\infty]:\forall\rho\in\mathrm{dom}\,F.\,\langle|v|,\rho\rangle\leq
aF(\rho)\right\\}.$ (6.11)

A potential $v$ is $F$-small if $\|v\|_{F}<1$, and relatively $F$-small if it
has a decomposition $v=v_{1}+v_{2}$, with $v_{1}$ $F$-small and
$\|v_{2}\|_{\infty}<\infty$. The set of all $F$-small potentials is denoted
$\mathsf{FSmall}$, and ${\mathsf{FSmall}}^{+}:{\kern-5.0pt}=\\{v\geq
0:v\in\mathsf{FSmall}\\}$.

Thus, the set of relatively $F$-small potentials is
$\mathsf{FSmall}+\mathsf{Bdd}$. But, note that the decomposition of a
potential into $\mathsf{FSmall}$ and $\mathsf{Bdd}$ pieces is not unique.

The appelation ‘$F$-norm’ is justified because the norm conditions
$\|v+w\|_{F}\leq\|v\|_{F}+\|w\|_{F}$, $\|av\|_{F}\leq|a|\|v\|_{F}$ ($a$ real)
and $\|v\|_{F}\Rightarrow v\equiv 0$ are satisfied. These imply that
$\|\cdot\|_{F}$ is convex. A subset $S$ of a vector space is called balanced
if $x\in S$ implies $-x\in S$.

######  Lemma 6.1.

$\mathsf{FSmall}$ is convex and balanced.

With that, we are ready to define our big space of potentials.

######  Definition 6.3.

$\displaystyle\mathsf{Potl}$
$\displaystyle:{\kern-5.0pt}=\mathsf{Potl}^{\uparrow}+\mathsf{Bdd}+\mathsf{FSmall}$
$\displaystyle=\mathsf{Potl}^{\uparrow}+\mathsf{Const}-\mathsf{FSmall}^{+}.$
(6.12)

Note that neither of the exhibited decompositions is unique, but the second is
closer to being so.

######  Theorem 6.2.

Every external potential in $\mathsf{Potl}$ is stable.

###### Proof.

Let $v$ be decomposed as
$v=w+u-u^{\prime}\in\mathsf{Potl}^{\uparrow}+\mathsf{Const}-\mathsf{FSmall}^{+}$.
Then, since $w+u$ is bounded below, so is $\langle w+u,\rho\rangle$
independently of $\rho\in\mathsf{Dens}_{\mathcal{N}}$. Also,

$F(\rho)-\langle u^{\prime},\rho\rangle\geq(1-\|u^{\prime}\|_{F})F(\rho)>0.$

∎

###  6.4. $L^{3/2}+L^{\infty}$

Conceptually, the $F$-norm is the right thing. Unfortunately, its computation
poses significant problems. However, if we have a tractable upper bound, we
will have sufficient conditions for an external potential to be stable. This
is where Lieb’s space of potentials $L^{\infty}+L^{3/2}=\mathsf{Bdd}+L^{3/2}$
comes in. Recall the inequality (4.22): $\|\rho\|_{3}\leq cF(\rho)$. This does
not say that an $L^{3/2}$ potential is $F$-small. But, if we chop off a big
enough bounded part, the remainder will be so.

######  Lemma 6.2.

$L^{3/2}\subseteq\mathsf{Bdd}+\mathsf{FSmall}$.

###### Proof.

$v^{+}$ and $v^{-}$ can be treated separately, so, without loss, assume $v\geq
0$. Then, $v\wedge n\to v$ in $L^{3/2}$ norm, so there is an $n$ such that
$v^{\prime}=v-(v\wedge n)$ satisfies $\|v^{\prime}\|_{3/2}<a/c$, with $a<1$.
But then, $|\langle
v^{\prime},\rho\rangle|\leq\|v^{\prime}\|_{3/2}\|\rho\|_{3}<aF(\rho)$ for
every $\rho\in\mathsf{Dens}_{\mathcal{N}}$. (The first inequality is
Hölder’s). And since $|v\wedge n|\leq n$, $v=(v\wedge n)+v^{\prime}$ is the
required decomposition. ∎

###  6.5. Concavity and continuity

This section treats concavity and continuity properties of $E$.

####  6.5.1. concavity

A function $f$ is concave if $-f$ is convex. So $E$ is concave if its domain
is convex and for all $v$ and $w$ in its domain,

$\forall\alpha\in(0,1).\,\,E(\alpha v+(1-\alpha)w)\geq\alpha
E(v)+(1-\alpha)E(w).$

The formula (6.1) expresses $E$ as the infimum of a collection of functions
$F(\rho)+\langle\cdot,\rho\rangle$ indexed by densities. If we phrase this in
terms of subgraphs ($\mathrm{sub}\,f$, the subgraph of $f$, is the region on
or below the graph of $f$),

$\mathrm{sub}\,E=\bigcap_{\rho\in\mathsf{Dens}_{\mathcal{N}}}\mathrm{sub}\,\left(F(\rho)+\langle\cdot,\rho\rangle\right).$

If every function on the right is concave, then their subgraphs are convex,
hence the intersection of their subgraphs is convex, and therefore $E$ is also
concave. We therefore need to verify the condition that for each
$\rho\in\mathsf{Dens}_{\mathcal{N}}$, $v,w\in\mathsf{Potl}$ and $0<\alpha<1$,

$F(\rho)+\langle\alpha
v+(1-\alpha)w,\rho\rangle\geq\alpha\left[F(\rho)+\langle
v,\rho\rangle\right]+(1-\alpha)\left[F(\rho)+\langle w,\rho\rangle\right].$

Decompose $v=v^{\prime}+v^{\prime\prime}$ with
$v^{\prime}\in\mathsf{Potl}^{\uparrow}+\mathsf{Const}$ and
$v^{\prime\prime}\in\mathsf{FSmall}$, and similarly,
$w=w^{\prime}+w^{\prime\prime}$. Now,

$\langle\alpha v^{\prime}+(1-\alpha)w^{\prime},\rho\rangle=\alpha\langle
v^{\prime},\rho\rangle+(1-\alpha)\langle w^{\prime},\rho\rangle,$

so we need show only

$F(\rho)-\langle\alpha
v^{\prime\prime}+(1-\alpha)w^{\prime\prime},\rho\rangle\geq\alpha\left[F(\rho)-\langle
v^{\prime\prime},\rho\rangle\right]+(1-\alpha)\left[F(\rho)-\langle
w^{\prime\prime},\rho\rangle\right].$

But, since $v^{\prime\prime}$ and $w^{\prime\prime}$ are $F$-small, if either
potential term on the right-hand side is infinite, then $F(\rho)=+\infty$, so
both sides are infinite. Thus, the following is now established.

######  Proposition 6.3.

$E:\mathsf{Potl}\rightarrow\overline{{\mathbb{R}}}$ is concave.

####  6.5.2. A locally lower-bounded convex function is locally Lipschitz

Convexity of a function is an essentially two-dimensional affair, and it
allows us to usefully apply elementary geometry to function spaces. A simple
drawing shows that slopes of secant lines to a convex function
$f:{\mathbb{R}}\rightarrow{\mathbb{R}}$ are non-decreasing: $z>y\geq x>w$
implies $\frac{f(z)-f(y)}{z-y}\geq\frac{f(x)-f(w)}{x-w}$. From this
observation, we immediately deduce the following.

######  Lemma 6.3.

Let $f:[-a,a]\rightarrow{\mathbb{R}}$ with $f(0)=0$ be convex and bounded
above by $M$. Then $f$ is bounded below by $-M$. Futhermore, it satisfies
$|f(x)|\leq(M/a)|x|$ (locally Lipschitz at $0$ with Lipschitz contant $M/a$).

We have phrased this lemma in terms of a convex function. Of course, it has an
analog for a concave function: locally upper-bounded implies locally
Lipschitz.

####  6.5.3. “Perturbative” continuity of $E$

For $w\in\mathsf{Potl}$, we think of $w+v$ for
$v\in\mathsf{Bdd}+\mathsf{FSmall}$ as a perturbation of the “background” $w$
and write

$E_{w}(v)=E(w+v).$ (6.13)

For fixed background $w$, $E_{w}(v)$ is locally Lipschitz as a function of
$v$.

######  Theorem 6.3.

If $w\in\mathsf{Potl}$ is habitable, then
$E_{w}(\cdot):\mathsf{Bdd}+\mathsf{FSmall}\rightarrow{\mathbb{R}}$ is locally
Lipschitz at zero.

###### Proof.

Given $\epsilon>0$, choose $\rho_{0}$ such that

$F(\rho_{0})+\langle w,\rho_{0}\rangle<E(w)+\epsilon.$ (6.14)

Then, for $v\in\mathsf{Bdd}$ and $v^{\prime}\in\mathsf{FSmall}$,

$\displaystyle E_{w}(v+v^{\prime})\leq F(\rho_{0})+\langle
w,\rho_{0}\rangle+\langle v+v^{\prime},\rho_{0}\rangle$
$\displaystyle<E(w)+\epsilon+\mathcal{N}\|v\|_{\infty}+F(\rho_{0})\|v\|_{F}.$

Thus, $E_{w}$ is locally bounded above, and Lemma 6.3 finishes the proof. ∎

This brings us to the end of our investigation of the functionals $F$ and $E$.
In 6.1.2 we just scratched the surface of the relationship between them, and
have not even touched the vexed $v$-representability problem of finding a
potential with given ground-state density. Nevertheless, it is time to quit,
for now.

##  Appendix A Selective functional analysis refresher

This appendix contains a refresher on some basic notions of functional
analysis. The reader for whom this is completely new is likely to feel the
need for additional resources.

###  A.1. Lebesgue ($L^{p}$) and Sobolev ($H^{1}$, $W^{1,1}$) spaces

We recall here the definitions of $L^{p}$ norms. For any measurable function
$f:{{\mathbb{R}}}^{n}\rightarrow{\mathbb{R}}$,

$\displaystyle\|f\|_{L^{p}}^{p}$
$\displaystyle=\|f\|_{p}^{p}:{\kern-5.0pt}=\int|f(x)|^{p}\,dx,\quad
0<p<\infty$ $\displaystyle\|f\|_{L^{\infty}}$
$\displaystyle=\|f\|_{\infty}:{\kern-5.0pt}=\inf\\{a\geq
0:\mu_{\mathrm{Leb}}(|f|\leq a)>0\\}.$ (A.1)

In the second line, $\mu_{\mathrm{Leb}}$ denotes Lebesgue measure. For a
continuous function, $\|\cdot\|_{\infty}$ is just the supremum of the absolute
value. The complicated business with $\mu_{\mathrm{Leb}}$ is needed so that
changing a function on a set of measure zero does not change its norm. The
space $L^{p}({{\mathbb{R}}}^{n})$ consists of the functions with finite
$L^{p}$ norm. Actually, this is a white lie. None of the $\|\cdot\|_{p}$ can
distinguish functions which differ on sets of measure zero, so we really
should say that the elements are equivalence classes of functions rather than
functions, tout court. But nobody does.

Sobolev spaces involve also norms of derivatives. These are really
distributional derivatives, and for our purposes, it is simplest to think in
terms of Fourier transform. With

$\hat{f}(q)=\lim_{R\to\infty}\int_{|x|\leq R}e^{iq\cdot
x}f(x)\,\frac{dx}{(2\pi)^{n/2}}$

the Fourier transform of $f$, the $H^{1}$ norm of $f$ is

$\|f\|_{H^{1}}^{2}:{\kern-5.0pt}=\int(1+|q|^{2})|\hat{f}(q)|^{2}\,{dq}.$

If $f$ is actually differentiable, then this is the same as

$\|f\|_{H^{1}}^{2}:{\kern-5.0pt}=\int(|f|^{2}+|\nabla
f|^{2})\,dx=\|f\|_{2}^{2}+\|\nabla f\|_{2}^{2},$

where the derivative is understood in the classical sense. Otherwise $\nabla
f$ is a distributional derivative. The other Sobolev space used in these notes
is $W^{1,1}$:

$\|f\|_{W^{1,1}}:{\kern-5.0pt}=\int(|f|+|\nabla f|)\,dx.$

The superscripts in ‘$W^{1,1}$’ indicate maximum derivative order and the
power to which things are raised in the integral. $H^{1}$ is also called
$W^{1,2}$. The reader will see that these are just the beginning of a family
of function spaces. Many sources are available for more information, for
example [19, 20, 21, 11, 22].

###  A.2. the concept of ‘topology’

‘Topology’ is a badly overloaded word, largely due to a tendency to deploy it
in contexts (turning coffee cups into a doughnuts) where a more precise term
would be better. As we use it here, a topology is essentially a notion of
convergence. In a metric space, a sequence $(x_{n})$ converges to $x$ if the
distances $d(x_{n},x)\to 0$ in ${\mathbb{R}}$. What is involved here are
certain standards of closeness, technically called neighborhoods. The ball
$B_{x}(a)$ of radius $a$ about $x$ is one neighborhood of $x$ and the ball
$B_{x}(b)$ is another. If $a>b$, the second standard of closeness is strictly
more demanding than the first; $B_{x}(b)\subset B_{x}(a)$. We could consider
“lopsided” neighborhoods which do not nest. The point for a metric space is
that we do not need such things; the notion of convergence is completely and
cleanly captured by the centered balls. But, such a simple structure is not
always obtainable. Consider the space of bounded functions
${\mathbb{R}}\rightarrow{\mathbb{R}}$ where we say that $f_{n}\to f$ if and
only if $f_{n}(x)\to f(x)$ for each $x$. Example: $f_{n}(x)=(\text{ if
}\,x>n\,\text{ then }\,1\,\text{ else }\,0)$ converges to the function $0$,
which is identically equal to zero. $\\{g\,:\,|g(a)|<\epsilon\\}$ is one
neighborhood of $0$ and $\\{g\,:\,|g(b)|<\epsilon\\}$ is another. The two are
not comparable, and we cannot rejigger to eliminate this complication. But,
there is a third neighborhood, $\\{g\,:\,|g(x)|<\epsilon\,\text{ for
}\,x\in\\{a,b\\}\\}$ which is contained in both. This illustrates the general
pattern. One way to specify the topology of a topological space is to give,
for each point $x$, a neighborhood base ${\mathcal{B}}_{x}$ such that if
$A,B\in{\mathcal{B}}_{x}$, then there is $C\in{\mathcal{B}}_{x}$ satisfying
$C\subseteq A$, $C\subseteq B$. Often, given such a system of neighborhood
bases, we can find a metric which has the same concept of convergence. This is
done in §5.1.1, for example. In fact, no non-metrizable topology is essential
anywhere in these notes, so the reader loses nothing by reading every
occurence of “topology” in the text as “notion of convergence associated to
some metric”.

##  Appendix B Infinitesimals

###  B.1. seriously

Infinitesimals have a long history in mathematics. Leibniz based his version
of the calculus on them, and Euler made masterful use of infinitesimals and
their reciprocals, infinitely large quantities. But the logical status of old-
style infinitesimals was ever shaky, and they were banished from mathematical
discourse during the 19th century, replaced by the $\epsilon$-$\delta$ method.
Eventually, infinitesimals were revived in a rigorous form, primarily through
the work of Abraham Robinson in the 1960’s, giving us modern infinitesimal
analysis, or nonstandard analysis[23]. Although there are alternate axiomatic
treatments[24, 25, 26], we follow the model theoretic approach pioneered by
Robinson[23, 27, 28, 29, 30, 31, 32, 33]. The idea is to extend a conventional
mathematical universe ${{\mathbb{U}}}$ to a nonstandard universe
${{}^{*}}{{{\mathbb{U}}}}$ which has all the same formal properties as
${{\mathbb{U}}}$ does, and yet is chock full of ideal elements. That an
element is ideal, for example infinitesimal, is an external judgement not
available to the internal object language. The interplay of internal and
external is responsible for the power of infinitesimal methods. We give here
an informal exposition. The previously-cited references give details of the
customary construction of nonstandard universes by means of ultraproducts over
set-theoretic superstructures. Relieved of the burden of proving anything, we
describe matters instead in a type-theoretic way inspired by modern functional
programming languages such as ML and Haskell. A good strategy for the reader
might be to go through just B.2–B.4, and then return when necessary. The only
nonstandard argument which is not elementary is contained in the proof of
Lemma 3.1.

###  B.2. base datatypes

We begin with the base datatypes of Booleans
$\mathsf{Bool}=\\{\mathrm{T},\mathrm{F}\\}$, natural numbers ${\mathbb{N}}$
and real numbers ${\mathbb{R}}$. These types, and their elements, are denizens
of the mathematical universe ${{\mathbb{U}}}$. The nonstandard universe
${{}^{*}}{{\mathbb{U}}}$ has ${{}^{*}}{\mathsf{Bool}}$, which is actually
identical to $\mathsf{Bool}$, the hypernaturals ${{}^{*}}{{\mathbb{N}}}$
extending ${\mathbb{N}}$ and the hyperreals ${{}^{*}}{{\mathbb{R}}}$ extending
the reals. ${{}^{*}}{{\mathbb{N}}}$ and ${{}^{*}}{{\mathbb{R}}}$ have all the
order and arithemetic properties of ${\mathbb{N}}$ and ${\mathbb{R}}$. But
these sets have additional elements; the new elements in
${{}^{*}}{{\mathbb{N}}}$ are larger than all naturals and some of the
hyperreals are infinitesimal. With the understanding of implicit coercion from
${\mathbb{N}}$ to ${\mathbb{R}}$ and from ${{}^{*}}{{\mathbb{N}}}$ to
${{}^{*}}{{\mathbb{R}}}$, we have the following grab-bag of terminology and
ideas. The essential image is of a number line with a “halo” of nonstandard
numbers around every standard real and then extended on the far left and far
right with “infinitely large” numbers.

######  Definition B.1.

Let $x,y\in{{}^{*}}{{\mathbb{R}}}$.

  1. a.

$x$ is standard if $x\in{\mathbb{R}}$. Otherwise, $x$ is nonstandard.

  2. b.

$x$ is limited if $|x|<n$ for some $n\in{\mathbb{N}}$; otherwise $x$ is
illimited. There is no smallest illimited hypernatural or hyperreal.

  3. c.

$x$ is infinitesimal if $|x|<1/n$ for all $n\in{\mathbb{N}}$; otherwise $x$ is
appreciable.

  4. d.

If $x-y$ is infinitesimal, then $x$ and $y$ are infinitely close, or simply
near. Notation: $x\simeq y$.

  5. e.

If $x$ is limited, then $x\simeq y$ for a unique $y\in{\mathbb{R}}$. $y$ is
the standard part of $x$, denoted by ${}^{\circ}{x}$ or $\mathrm{st}\,x$.

  6. f.

If $x$ is standard, the collection of all hyperreals near $x$ makes up the
halo (or monad) of $x$. Halos of distinct reals are disjoint.

  7. g.

$x\gg y$ means $x>y$ but $x\not\simeq y$. $x\ll\infty$ is a synonym for “$x$
is limited”.

Any formal statement $S$ we can make about $\mathsf{Bool}$, ${\mathbb{N}}$,
and ${\mathbb{R}}$ using Boolean operations (conjunction $\wedge$, disjunction
$\vee$ and negation $\neg$), natural and real constants and arithmetic
operations ($0$,$1$,$+$,$\cdot$,$<$,$|\cdot|$) and bounded quantifiers
($\forall$ and $\exists$), is turned into a statement ${{}^{*}}{S}$ about
$\mathsf{Bool}$, ${{}^{*}}{{\mathbb{N}}}$ and ${{}^{*}}{{\mathbb{R}}}$ by
putting stars on the constants and parameters. Leibniz’ Principle, which we
require, says that $S$ and ${{}^{*}}{S}$ have the same truth value. For
example, every natural is either even or odd, hence the same is true of a
hypernatural, even if it is illimited. Every hypernatural has a successor.
There is a third hyperreal strictly between any two distinct hyperreals. There
is no real number squaring to $-1$, hence neither is there such a hyperreal.
Since we identify reals, naturals and Booleans with their $*$-transforms,
‘${{}^{*}}{(-1)}$’ is simply ‘$-1$’.

###  B.3. limits and continuity

With individual functions added to our object language, we can discuss
nonstandard characterizations of sequence limits and function continuity.
These are bread-and-butter applications of infinitesimal methods. A real
sequence $(x_{n})$ is really a function ${\mathbb{N}}\to{\mathbb{R}}$, and has
an extension $({{}^{*}}{x}_{n})$ with $n$ running over
${{}^{*}}{{\mathbb{N}}}$. We want to show the equivalence of “$x_{n}\to a$”
and “${{}^{*}}{x}_{n}\simeq a$ for $n\simeq\infty$”.

The formalization of $x_{n}\to a$ is (${{\mathbb{R}}}_{>0}$ abbreviates the
reals greater than zero) $\forall\epsilon\in{{\mathbb{R}}}_{>0}\,\exists
n\in{\mathbb{N}}.\,m>n\Rightarrow|x_{m}-a|<\epsilon$. Actually, we do not want
to put $\epsilon$ and $n$ in the formal statement, but keep them at the
metalinguistic level. Then we have simply,

$\forall m>n(\epsilon)\Rightarrow|x_{m}-a|<\epsilon,$

which is true in ${{\mathbb{U}}}$. Transfer says (recall, $n$ is the same as
${{}^{*}}{n}$ and similarly for $\epsilon$)

$\forall m>{n(\epsilon)}\Rightarrow|{{}^{*}}{x}_{m}-{a}|<\epsilon.$

Now, if $m\simeq\infty$, then certainly $m>n(\epsilon)$. And that’s true no
matter how small $\epsilon$ is, so $x_{m}\simeq a$. For the other direction,
suppose we know that $m\simeq\infty$ implies $x_{m}\simeq a$. Then, given
standard $\epsilon>0$, the statement

$\exists
n\in{{}^{*}}{{\mathbb{N}}}.\,m>n\Rightarrow|{{}^{*}}{x}_{m}-a|<\epsilon$

is true in ${{}^{*}}{{\mathbb{U}}}$; any illimited $n$ will serve. Backward
transfer gives us

$\exists n\in{{\mathbb{N}}}.\,m>n\Rightarrow|{x}_{m}-a|<\epsilon.$

Since this procedure is valid for any $\epsilon\in{\mathbb{R}}_{>0}$, we
conclude that $x_{n}\to a$.

Continuity of a function $f:{\mathbb{R}}\rightarrow{\mathbb{R}}$ at $a$ has an
infinitesimal characterization with a very similar flavor. It is “$x\simeq a$
implies ${{}^{*}}{f}(x)\simeq{{}^{*}}{f}(a)$”, and is demonstrated by a very
similar technique. Ordinarily, we might drop the stars on ‘${{}^{*}}{f}(x)$’
and ‘${{}^{*}}{f}(a)$’, as there is neither ambiguity nor possibility for
confusion (Not even externally does ‘$a\simeq b$’ make sense for
$a,b\in{\mathbb{R}}$.)

###  B.4. metric, function, and topological spaces

Any metric space $X$ is extended in much the same way as ${\mathbb{R}}$. For
example, consider the space $\ell^{2}$ of square summable sequences
$x:{\mathbb{N}}\rightarrow{\mathbb{R}}$, with the norm
$\|x\|_{2}^{2}=\sum_{n}|x_{n}|^{2}$. Then ${{}^{*}}{\ell^{2}}$ consists of
sequences ${{}^{*}}{{\mathbb{N}}}\rightarrow{{}^{*}}{{\mathbb{R}}}$.
$x\in\ell^{2}$ has an extension ${{}^{*}}{x}\in{{}^{*}}{\ell^{2}}$; it is a
standard element. Now, since $\sum|x_{n}|^{2}$ converges, for any standard
$\epsilon>0$, there is $n(\epsilon)$ such that $\sum_{m\geq
n(\epsilon)}|x_{m}|^{2}<\epsilon$. So using Transfer, we conclude that
$\sum_{m\geq N}|x_{m}|^{2}\simeq 0$ for any illimited $N$. This is important
to the nature of nearstandard elements of ${{}^{*}}{\ell^{2}}$. If
$y\in{{}^{*}}{\ell^{2}}$ has a standard part, then
$\sum_{n\in{{}^{*}}{{\mathbb{N}}}}|(\mathrm{st}\,y)_{n}-y_{n}|^{2}\simeq 0$.
This implies that $(\mathrm{st}\,y)_{n}\simeq{}^{\circ}{(}y_{n})$ for limited
$n$, as well as $\sum_{n\in{\mathbb{N}}}|{}^{\circ}{(}y_{n})|^{2}\ll\infty$
and $\sum_{n\geq N}|y_{n}|^{2}\simeq 0$ for any illimited $N$. We see, then,
that there are basically two ways $y\in{{}^{*}}{\ell^{2}}$ can fail to be
nearstandard (not nearstandard is also called remote). First, it could have an
illimited norm, $\|y\|\approx\infty$. Second, it could have appreciable weight
at infinity. The first has as analog for ${{}^{*}}{{\mathbb{R}}}$, but the
second does not. This is the nonstandard manifestation of the fact that
bounded subsets of ${\mathbb{R}}$ are relatively compact, but bounded subsets
of $\ell^{2}$ are not. The element $y_{n}=\delta_{nN}$ ($n$ illimited) is a
remote element with norm one.

If $d$ is a metric for $X$ in ${\mathbb{U}}$, then ${{}^{*}}{d}$ is a metric
for ${{}^{*}}{X}$. Each standard point ${{}^{*}}{x}\in{{}^{*}}{X}$ is
surrounded by a halo (or ‘monad’) of nonstandard points at infinitesimal
distance from it. If $d(y,{{}^{*}}{x})\simeq 0$, then $x$ is the standard part
of $y$, written either ${}^{\circ}{y}$ or $\mathrm{st}\,y$, as convenient.
${{}^{*}}{X}$ may well have points which are not near any standard point.
These are remote. Indeed, if there are points in $X$ arbitrarily far from $x$
($\forall n\in{\mathbb{N}}\,\exists y\in X.\,d(y,x)>n$), then there are points
in ${{}^{*}}{X}$ an illimited distance from ${{}^{*}}{x}$. These are
necessarily remote, since any putative standard part must be a finite distance
from $x$. A function $f:X\to{\mathbb{R}}$ is continuous at $x\in X$ if and
only if ${{}^{*}}{f}$ maps the halo of $x$ into the halo of $f(x)$. All these
points are exemplified by the space $\ell^{2}$ of the previous paragraph.

We touch briefly on the generalization to an arbitrary topological space in
${\mathbb{U}}$. There is only one essential difference. Without a metric,
there is no numerical measure of “infinitely close”. Instead, the halo of a
standard point $x$ must be defined as $\cap{{}^{*}}{U}$, where the
intersection runs over all the standard open neighborhoods of $x$. One can
check that for a metric space this matches the definition we have already
given.

###  B.5. $*$-transform and Leibniz’ Principle

To do much interesting mathematics, we need higher-order types, such as power
sets and function spaces, that are so far lacking from the language. If $A$
and $B$ are datatypes in ${\mathbb{U}}$, then so is the product type $A\times
B$ consisting of all ordered pairs $(x,y)$ for $x\in A$ and $y\in B$, the sum
(disjoint union) type $A+B$, and the type $[A\rightarrow B]$ of all functions
from $A$ to $B$. With these datatype constructors, along with function
abstraction and application, we can fill out the universe ${\mathbb{U}}$ and
extend the $*$-transform to a map
${{\mathbb{U}}}\stackrel{{\scriptstyle*}}{{\rightarrow}}{{}^{*}}{{\mathbb{U}}}$
obeying the Transfer Principle.

  1. a.

The range of $*$ is the standard objects. These are raw materials from which
the other, internal, objects of ${{}^{*}}{{\mathbb{U}}}$ are obtained by
ordinary mathematical operations (de novo function space construction is not
ordinary).

  2. b.

Individual Booleans, naturals and reals are urelements with no internal
structure; we identify them with their $*$-transforms.

  3. c.

If $x\in A$, then ${{}^{*}}{x}\in{{}^{*}}{A}$. But if $A$ is infinite,
${{}^{*}}{A}$ contains nonstandard members as well (example: illimited
hypernaturals).

  4. d.

$*$ respects product and sum construction, ${{}^{*}}{(A\times
B)}={{}^{*}}{A}\times{{}^{*}}{B}$, ${{}^{*}}{(A+B)}={{}^{*}}{A}+{{}^{*}}{B}$.
, tupling, ${{}^{*}}(x,y)=({{}^{*}}{x},{{}^{*}}{y})$, and function
application, ${{}^{*}}{(f(x))}={{}^{*}}{f}({{}^{*}}{x})$.

  5. e.

Only the function space constructor has an abnormal interpretation in
${{}^{*}}{{\mathbb{U}}}$. ${{}^{*}}[A\rightarrow B]$ is only a subset of
$[{{}^{*}}{A}\rightarrow{{}^{*}}{B}]$, which we write as
$[{{}^{*}}{A}\dashrightarrow{{}^{*}}{B}]$. Members of
$[{{}^{*}}{A}\dashrightarrow{{}^{*}}{B}]$ are internal functions.

  6. f.

The power set constructor, ${\mathcal{P}}(\cdot)$, is basically syntactic
sugar for $[\,\cdot\rightarrow\mathsf{Bool}]$. Other set operations are
likewise abbreviations, which we use freely.

  7. g.

${{}^{*}}{\mathbb{U}}$ consists of internal objects, anything we can describe
using the resources of the formal language and beginning with standard
objects. In particular, a standard set is internal, any member of a (member of
a $\cdots$ of a) standard set is internal. If $c\in{{}^{*}}{{\mathbb{R}}}$ is
infinitesimal, then it is not standard (not of the form ${{}^{*}}{b}$), but it
is internal, and so is the subset $[0,c]$ of ${{}^{*}}{{\mathbb{R}}}$.

  8. h.

If $C$ and $D$ are internal subsets of ${{}^{*}}{A}$ and ${{}^{*}}{B}$,
respectively, $[C\dashrightarrow D]$ is obtained in the normal way by
selection and restriction. Also, the internal function space constructor
respects currying: if $A$, $B$ and $C$ are internal, then
${A}\dashrightarrow[{B}\dashrightarrow{C}]=({A}\times{B})\dashrightarrow{C}$.

  9. i.

The symbol ‘$\simeq$’, the predicate ‘limited’, the standard part operation
and the other elements of Definition B.1 are not part of the object language.
Most especially, neither $\\{x\in{{}^{*}}{{\mathbb{N}}}:x\ll\infty\\}$ nor
$\\{x\in{{}^{*}}{{\mathbb{R}}}:x\simeq 0\\}$ is internal. They are external.
Because we can formally express the least-number principle (any nonempty
subset of ${\mathbb{N}}$ has a least element), Transfer implies that it holds
for internal subsets of ${{}^{*}}{{\mathbb{N}}}$. But, there is no smallest
illimited hypernatural. Consistency thus demands that these things be external
sets.

###  B.6. overspill

‘Limited’ and ‘infinitesimal’ are the most important external concepts. We
illustrate a typical pattern of use. Suppose a sequence
$x:{{}^{*}}{{\mathbb{N}}}\rightarrow{{}^{*}}{{\mathbb{R}}}$ has the property
that $nx_{n}<1$ for every limited $n$. Then, the set
$B=\\{m\in{{}^{*}}{{\mathbb{N}}}:\forall n<m,nx_{n}<1\\}$ is evidently
internal. Therefore, $B$ cannot consist of only the limited hypernaturals,
since that is an external set. There must be some illimited $m$ in $B$. Thus
the property $nx_{n}<1$ spills over into the illimited hypernaturals.
Similarly, if a formally expressible property holds for all appreciable
hyperreals, it must hold for some infinitesimal ones. Now suppose an internal
sequence $(x_{n})$ satisfies $x_{n}\simeq 0$ for all limited $n$. Can we use
overspill? Since “$\simeq$” is external, it is not evidently applicable, but
there is a clever trick discovered by Robinson. The given condition implies
$nx_{n}<1$ for limited $n$. And for illimited $n$, $nx_{n}<1$ implies
$x_{n}\simeq 0$. Thus the property $x_{n}\simeq 0$ actually does extend into
the illimited hypernaturals. This result, known as Robinson’s Lemma, is used
in 5.4.2.

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