between powers of $C_{\ell,\gamma}^{H}$ and $p_{\ell}$ is not straightforward.
In this case, the proof of [87, Theorem 1.4] used the comparison result in
[86] (see Theorem 4.17 later). This is the reason for the assumption
$s<3/2-\sigma_{\gamma}$ and the appearance of $\varepsilon>0$ in (4.18).

(2) Instead of comparing Bessel kernels of $C_{\ell,\gamma}^{H}$ and
$p_{\ell}$, one can compare the Bessel kernels of $C_{\ell,\gamma}^{H}$ and
$p_{\ell}-\gamma/r$. For $\ell=0$, the latter can immediately be derived using
the spectral theorem and recent heat kernel bounds for $|p|-\gamma/|x|$ by
Bogdan et al. [18]. Since for $\ell\geq 1$ the Hardy potential is again an
operator perturbation for $\gamma<1/2$ (by Hardy’s inequality), Bessel kernel
bounds for $p_{\ell}-\gamma/r$ are similar to those for $p_{\ell}$. We expect
that this strategy allows to remove the $\varepsilon>0$ in (4.18) when
$\gamma>(1+\sqrt{2})/4$. It is an open problem to prove that the behavior of
$\varrho_{\ell,C}^{H}$ and $\rho_{C}^{H}$ at $r=0$ for $\gamma>(1+\sqrt{2})/4$
is optimal.

(3) We do not expect Bessel kernel bounds for $p_{\ell}-\gamma/r$ to yield
precise (probably $\gamma$-dependent) bounds for $\varrho_{\ell,C}^{H}$ at the
origin in the cases $\ell=0$ and $\gamma\leq 1/2$, and $\ell\geq 1$ and
$\gamma\leq 2/\pi$, because the Bessel kernel bounds for $p_{\ell}-\gamma/r$
are similar to those for $p_{\ell}$.

(4) The appearance of $\gamma=(1+\sqrt{2})/4$ is technical and comes from the
restriction $\sigma\leq 3/4$ together with the fact that
$\sigma_{(1+\sqrt{2})/4}=3/4$.

#### Larger coupling constants for higher angular momenta

Using a relative to the Fourier–Bessel transform, namely the Mellin transform,
Le Yaouanc et al. [137] showed that the largest admissible coupling constant
associated to $C_{\ell,\gamma}^{H}$ increases as $\ell$ increases.
Independently, Yafaev [248, (2.4), (2.26)] gave an alternative proof of this
fact and showed

$\displaystyle\int_{0}^{\infty}\mathrm{d}k\,k|(\Phi_{\ell}u)(k)|^{2}\geq\frac{2\Gamma\left(\frac{1}{4}(4+2\ell)\right)^{2}}{\Gamma\left(\frac{1}{4}(2+2\ell)\right)^{2}}\int_{0}^{\infty}\frac{|u(r)|^{2}}{r}\,\mathrm{d}r.$
(4.20)

For $d=3$ and $\ell=1$, the largest admissible coupling constant is $\pi/2$,
which compares to the critical value $2/\pi$ when $\ell=0$. We note that
similar inequalities hold in dimensions $d$ other then three and powers
$\alpha\in(0,\min\\{2,d\\})$ of the square root of the Laplacian.

#### Ground state transform

An alternative representation of the operator $(-\Delta)^{1/2}-\gamma/|x|$ in
$L^{2}(\mathbb{R}^{3})$ proceeds via the _ground state transform_. To that
end, recall (4.15). The ground state transform makes use of the fact that the
radial function $x\mapsto|x|^{-\sigma}$ is a _(generalized) ground state_ for
$|p|-\Phi(\sigma)|x|^{-\alpha}$. It states that

$\displaystyle\langle
u,(|p|-\Phi(\sigma)|x|^{-1})u\rangle=\frac{1}{2\pi^{2}}\int_{\mathbb{R}^{3}}\frac{|v(x)-v(y)|^{2}}{|x-y|^{4}}\cdot(|x||y|)^{-\sigma}\,\mathrm{d}x\,\mathrm{d}y$
(4.21)

where $\sigma\in[0,1]$, $u(x)=|x|^{-\sigma}v(x)$ with
$v\in\mathcal{S}(\mathbb{R}^{3}\setminus\\{0\\})$, cf. [83, Proposition 4.1].
Note that for $\alpha=2$ the ground state transform has been known long
before, see, e.g., [190, p. 169] for a textbook treatment when $d=3$. For a
further study of the ground state transform in the fractional case, see [89].

#### Spectrum

Although the eigenvalues $\lambda_{Z,n,\ell}$ ($n\in\mathbb{N}_{0}$) of
$C_{c,Z}^{H}$ are not explicitly known, the inequality $\sqrt{p^{2}+1}-1\leq
p^{2}/2$ and the lower bound of [91, Theorem 2.2] imply the inequalities

$\displaystyle-\frac{Z^{2}}{2(n+\ell+1)^{2}}\geq\lambda_{Z,n,\ell}\geq-\mathrm{const}\
\cdot\frac{Z^{2}}{(n+\ell+1)^{2}},$ (4.22)

where the constant in the second inequality can be chosen independently of
$\gamma\in[0,2/\pi]$. The expression on the left side is just the $n$-th
eigenvalue of the hydrogen operator (1.9) in angular momentum channel $\ell$.
Thus, although the relativistic eigenvalues are smaller than the non-
relativistic ones, their magnitudes in $Z$ are the same and many of their
summability properties with respect to $n$ are similar.

Finally, the spectrum of $C_{\gamma}^{H}$ in $[0,\infty)$ is purely absolutely
continuous, the singular continuous spectrum is empty, and there are no
embedded eigenvalues [109, Theorem 2.3]. In particular, there is no zero
eigenvalue, a fact that we will use later.

#### Physical shortfalls

Although $C_{\gamma}^{H}$ is mathematically well understood, it has a number
of physical deficits. For instance, the restriction $\gamma\leq 2/\pi$ implies
that only atoms with nuclear charge $<88$ can be described. Moreover, the
predicted ground state energies for heavy atoms are much too low. This can
already be anticipated by comparing the ground state energies for hydrogen
with $c=1$ and coupling $\gamma$ close to $2/\pi$, which are $\approx-0.5$ in
the Chandrasekhar model [189, p. 106] and $\approx-0.06$ (cf. (4.37)) in the
Dirac model, respectively.

Although the model is unsuitable for the quantitative description of systems
with strong attractive external Coulomb forces, Chandrasekhar [24] used it
successfully for attractive two-particle Coulomb forces in his Nobel prize
winning estimate on the mass necessary to collapse a star to a white dwarf.
Despite its mathematical simplicity and its success in correctly describing
some qualitative features of relativistic Coulomb systems, it is desirable to
examine models that also lead to quantitatively correct predictions for ground
state properties. Such models are, e.g., based on the Coulomb–Dirac operator,
which we discuss next.

####  4.1.2. Coulomb–Dirac operator

#### Free Dirac operator

In 1928 Dirac [37, 38] derived a Lorentz invariant equation of motion for
quantum mechanical particles with spin moving in an external electromagnetic
field, the so-called Dirac equation. We refer to [17, 231] for comprehensive
treatments. For a free particle, the equation reads

$\displaystyle
i\partial_{t}\psi(t,x)=\left(-ic\underline{\alpha}\cdot\nabla+\beta
c^{2}\right)\psi(t,x)$ (4.23)

with the Dirac matrices
$\underline{\alpha}=(\alpha_{1},\alpha_{2},\alpha_{3})$,

$\displaystyle\alpha_{j}=\left(\begin{array}[]{cc}0_{\mathbb{C}^{2}}&\sigma_{j}\\\
\sigma_{j}&0_{\mathbb{C}^{2}}\end{array}\right),$

the Pauli matrices $\underline{\sigma}:=(\sigma_{1},\sigma_{2},\sigma_{3})$,
and $\beta=\operatorname{diag}(1,1,-1,-1)$. The operator on the right side of
(4.23) is called the free Dirac operator. It acts on states
$\psi_{t}(x)\in\mathbb{C}^{4}$, called Dirac spinors. The underlying Hilbert
space is $L^{2}(\mathbb{R}^{3}:\mathbb{C}^{4})$. The domain on which the free
Dirac operator can be realized as a self-adjoint operator is
$H^{1}(\mathbb{R}^{3}:\mathbb{C}^{4})$. The Foldy–Wouthuysen transform $U_{\rm
FW}$ allows to perform a block diagonalization, whereby the free Dirac
operator takes the form

$\displaystyle U_{\mathrm{FW}}\left(-ic\underline{\alpha}\cdot\nabla+\beta
c^{2}\right)U_{\mathrm{FW}}^{*}=\left(\begin{array}[]{cc}\sqrt{-c^{2}\Delta+c^{4}}&0\\\
0&-\sqrt{-c^{2}\Delta+c^{4}}\end{array}\right).$ (4.26)

This shows that the spectrum equals $(-\infty,-c^{2}]\cup[c^{2},\infty)$.
Physically, this means that states can possess “negative energy” and that
there is an infinitely deep energy reservoir, the so-called Dirac sea. By
adding electromagnetic fields and the charge conjugation operator, one can
interpret states with negative energy as “antiparticles”, i.e., particles with
same mass but opposite charge. Such particles are called positrons.

#### Self-adjoint extensions of the Coulomb–Dirac operator

The one-particle Dirac operator describing the hydrogen atom can initially be
defined on $\mathcal{S}(\mathbb{R}^{3}:\mathbb{C}^{4})$ and is formally given
by the differential operator

$\displaystyle
D_{c,Z}^{H}:=-ic\underline{\alpha}\cdot\nabla+c^{2}\beta-\frac{Z}{|x|}\quad\text{in}\
L^{2}(\mathbb{R}^{3}:\mathbb{C}^{4}).$ (4.27)

Scaling $x\mapsto x/c$ and writing $\gamma:=Z/c$ shows that $D_{c,Z}^{H}$ is
unitarily equivalent to

$\displaystyle
c^{2}\left[-i\underline{\alpha}\cdot\nabla+\beta-\frac{\gamma}{|x|}\right]=:c^{2}D_{1,\gamma}^{H}\equiv
c^{2}D_{\gamma}^{H}.$ (4.28)

Weidmann [244] showed that $D_{\gamma}^{H}$ is essentially self-adjoint on
$\mathcal{S}(\mathbb{R}^{3}\setminus\\{0\\}:\mathbb{C}^{4})$ if and only if
$|\gamma|<\sqrt{3}/2$, see also [231, Theorem 4.4]. For
$\gamma\in[\sqrt{3}/2,1]$ there is a “distinguished” (sometimes called
“physically relevant”) self-adjoint extension of $D_{\gamma}^{H}$. For
$\gamma\in(\sqrt{3}/2,1)$ it was established by Schmincke [197], Wüst [247]
(see also Kalf, Schmincke, Walter, and Wüst [128] for a review of these
results), Nenciu [181], and Klaus and Wüst [133]. According to Schmincke and
Wüst, this realization stands out by the property that all states in the
domain of the Coulomb–Dirac operator had finite potential energy. On the other
hand, Nenciu’s realization was distinguished by the fact that states had
finite kinetic energy. Klaus and Wüst showed that both realizations coincide,
and that the essential spectrum is $(-\infty,1]\cup[1,\infty)$, see [134] (or
[231, p. 117] for a textbook treatment). In summary, the domain
$\operatorname{dom}(D_{\gamma}^{H})$ of the distinguished realization
satisfies
$H^{1}(\mathbb{R}^{3}:\mathbb{C}^{4})\subset\operatorname{dom}(D_{\gamma}^{H})\subset
H^{1/2}(\mathbb{R}^{3}:\mathbb{C}^{4})$, and the quadratic form domain is
$H^{1/2}(\mathbb{R}^{3}:\mathbb{C}^{4})$. In particular, the expectation
values of both kinetic and potential energy are finite in
$\operatorname{dom}(D_{\gamma}^{H})$; this motivates the term “physically
relevant extension”. With the help of the sharp Hardy–Dirac inequality [42],

$\displaystyle\int_{\mathbb{R}^{3}}\frac{|\varphi(x)|^{2}}{|x|}\,\mathrm{d}x\leq\int_{\mathbb{R}^{3}}\left(\frac{|(\underline{\sigma}\cdot\nabla\varphi)(x)|^{2}}{1+|x|^{-1}}+|\varphi(x)|^{2}\right)\,\mathrm{d}x,\quad\varphi\in
H^{1}(\mathbb{R}^{3}:\mathbb{C}^{2}),$ (4.29)

Esteban and Loss [57] constructed a distinguished self-adjoint extension for
$\gamma=1$. States in the domain of this operator need not have finite kinetic
and potential energy separately. We remark that similar results in two
dimensions (where the critical coupling is $\gamma=1/2$) were proved by Warmt
[242]. In this review we will only focus on the three-dimensional case and
$\gamma<1$.

#### Partial wave analysis

Since $D_{\gamma}^{H}$ is spherically symmetric, one can, analogously to the
angular momentum decomposition for spherically symmetric scalar operators,
perform a partial wave decomposition, see, e.g., [59], [7, Section 2.1], [231,
Sections 4.6.3-4.6.5], and [170, Appendix A]. We begin by observing that those
of the spherical $\mathbb{C}^{2}$-spinors

$\Omega_{\ell,m,s}(\omega):=\begin{pmatrix}2s\sqrt{\frac{\ell+\frac{1}{2}+2sm}{2\ell+1}}Y_{\ell,m-\frac{1}{2}}(\omega)\\\
\sqrt{\frac{\ell+\frac{1}{2}-2sm}{2\ell+1}}Y_{\ell,m+\frac{1}{2}}(\omega)\end{pmatrix}$
(4.30)

with $\ell=0,1,2,...$ and $m=-\ell-\frac{1}{2},...,\ell+\frac{1}{2}$, that do
not vanish, form an orthonormal basis of
$L^{2}(\mathbb{S}^{2}:\mathbb{C}^{2})$, see, e.g., [59, (7)]. Moreover, they
are joint eigenfunctions of $L^{2}$, $J^{2}$ ($J=L+S$ being the total angular
momentum), and $J_{3}$ with respective eigenvalues $\ell(\ell+1)$,
$(\ell+s)(\ell+s+1)$, and $m$.

Introducing the spin-orbit operator $K=\beta(J^{2}-L^{2}+1/4)$, there is an
orthonormal basis of eigenvectors $\Phi^{\sigma}_{\kappa,m}$ of
$L^{2}(\mathbb{S}^{2}:\mathbb{C}^{4})$ such that
$J^{2}\Phi^{\sigma}_{\kappa,m}=j_{\kappa}(j_{\kappa}+1)\Phi^{\sigma}_{\kappa,m}$,
$J_{3}\Phi^{\sigma}_{\kappa,m}=m\Phi^{\sigma}_{\kappa,m}$, and
$K\Phi^{\sigma}_{\kappa,m}=\kappa\Phi^{\sigma}_{\kappa,m}$ with the total
angular momentum $j_{\kappa}$ and orbital angular momentum $\ell_{\kappa}$
defined as

$j_{\kappa}:=|\kappa|-\tfrac{1}{2}\ \text{and}\
\ell_{\kappa}:=j_{\kappa}-\tfrac{1}{2}\operatorname{sgn}(\kappa)=|\kappa|-\theta(\kappa),$
(4.31)

the magnetic quantum numbers $m\in\\{-j_{\kappa},...,j_{\kappa}\\}$, the spin-
orbit coupling

$\kappa\in\dot{\mathbb{Z}}:=\mathbb{Z}\setminus\\{0\\},$ (4.32)

and $\sigma\in\\{+,-\\}$. A standard choice is

$\Phi^{+}_{\kappa,m}:=\left(\begin{array}[]{c}{\rm
i}\operatorname{sgn}(\kappa)\Omega_{\ell_{\kappa},m,\frac{1}{2}\operatorname{sgn}(\kappa)}\\\
0\end{array}\right),\ \Phi^{-}_{\kappa,m}:=\left(\begin{array}[]{c}0\\\
-\operatorname{sgn}(\kappa)\Omega_{\ell_{\kappa}+\operatorname{sgn}(\kappa),m,-\frac{1}{2}\operatorname{sgn}(\kappa)}\end{array}\right).$
(4.33)

Using these spinors, we introduce the spaces

$\displaystyle\mathfrak{h}_{\kappa,m}:=$
$\displaystyle\mathrm{span}\\{x\mapsto\tfrac{f^{+}(|x|)}{|x|}\Phi_{\kappa,m}^{+}(\tfrac{x}{|x|})+\tfrac{f^{-}(|x|)}{|x|}\Phi_{\kappa,m}^{-}(\tfrac{x}{|x|}):f^{+},f^{-}\in
L^{2}(\mathbb{R}_{+})\\},$ (4.34) $\displaystyle\mathfrak{h}_{\kappa}:=$
$\displaystyle\bigoplus_{m=-j_{\kappa}}^{j_{\kappa}}\mathfrak{h}_{\kappa,m}.$
(4.35)

These spaces form an orthogonal decomposition of
$L^{2}(\mathbb{R}^{3}:\mathbb{C}^{4})$. Note that Dirac operators with radial
potentials leave the spaces $\mathfrak{h}_{\kappa,m}$ invariant. To see this,
let $f\in\mathfrak{h}_{\kappa,m}\cap H^{1}(\mathbb{R}^{3}:\mathbb{C}^{4})$ and
$g\in\mathfrak{h}_{\kappa^{\prime},m^{\prime}}\cap
H^{1}(\mathbb{R}^{3}:\mathbb{C}^{4})$. Then one has

$\begin{split}&\langle
f,D_{\gamma}^{H}g\rangle_{L^{2}(\mathbb{R}^{3}:\mathbb{C}^{4})}\\\
&\quad=\left\langle\left(\begin{array}[]{c}f^{+}\\\
f^{-}\end{array}\right),\left(\begin{array}[]{cc}1-\frac{\gamma}{r}&-\frac{\mathrm{d}}{\mathrm{d}r}-\frac{\kappa}{r}\\\
\frac{\mathrm{d}}{\mathrm{d}r}-\frac{\kappa}{r}&-1-\frac{\gamma}{r}\end{array}\right)\left(\begin{array}[]{c}g^{+}\\\
g^{-}\end{array}\right)\right\rangle_{L^{2}(\mathbb{R}_{+}:\mathbb{C}^{2})}\delta_{\kappa\kappa^{\prime}}\delta_{mm^{\prime}},\end{split}$
(4.36)

see also [231, (7.105)].

#### Spectrum

The Coulomb–Dirac operator $D_{\gamma}^{H}$ has no embedded eigenvalues (Kalf
[129]) and no singular continuous spectrum in $[0,\infty)$ (Vogelsang [241],
and Richard and Tiedra de Aldecoa [193]). The lowest eigenvalue is
$\lambda_{1}=\sqrt{1-\gamma^{2}}$ and one has
$\lim_{k\to\infty}\lambda_{k}=1$. The eigenvalues of $D_{\gamma}^{H}$ are
explicitly known and given by

$\displaystyle\lambda_{n,\kappa}=\left(1+\tfrac{\gamma^{2}}{\left(n+\sqrt{\kappa^{2}-\gamma^{2}}\right)^{2}}\right)^{-1/2}.$
(4.37)

They only depend on
$(\kappa,n)\in(-\mathbb{N}\times\mathbb{N})\cup(\mathbb{N}\times\mathbb{N}_{0})$.
Sommerfeld [225] anticipated these eigenvalues in the framework of the old
relativistic theory of quanta even before the Coulomb–Dirac operator was
written down. Darwin [32], Gordon [96], and Pidduck [185] solved the
eigenvalue equation for the Coulomb–Dirac operator only 12 years later; see
also Bethe’s [17] or Thaller’s [231, Section 7.4] textbooks for comprehensive
treatments and Mawhin and Ronveaux [167] for interesting historical comments.
In particular, one has the bounds

$-\frac{\gamma^{2}}{2(n+\ell+1)^{2}}\geq\lambda_{n,\kappa}-1\geq-\mathrm{const}\
\cdot\frac{\gamma^{2}}{(n+\ell+1)^{2}},$

see also [100, Lemma 1].

######  Remark 4.3.

If the Coulomb potential $\gamma/|x|$ is replaced by a more general measurable
Hermitian $4\times 4$-matrix-valued function
$V:\mathbb{R}^{3}\to\mathbb{C}^{4\times 4}$ with $0\leq
V(x)\leq\gamma/|x|\otimes\mathbf{1}_{\mathbb{C}^{4}}$, then the eigenvalues of
$-i\underline{\alpha}\cdot\nabla+\beta-V$ can be computed using analogues of
the classical Courant–Fischer min-max principle, (cf. [191, Theorems
XIII.1-2]) originally written down by Talman [230] and Datta and Devaiah [33],
mathematically established by Esteban and Séré [58] and by [99, 98], and
further developed by Dolbeault et al. [41, 42], Morozov and Müller [173],
Müller [177], Esteban et al. [55, 56], and Schimmer et al. [196]. The latter
turned the principle around and used it to define the Hamiltonian, since the
maximization leads to the Hardy type inequality (4.29).

#### Hydrogen eigenvectors and density

Similarly, the eigenvectors $\psi_{n,\kappa,m}^{D}$ of the eigenvalue equation
$D_{\gamma}^{H}\psi_{n,\kappa,m}^{D}=\lambda_{n,\kappa}\psi_{n,\kappa,m}^{D}$
are also well known, see, e.g., Pidduck [185] (in terms of Laguerre
polynomials) or Gordon [96] and Darwin [32] (in terms of hypergeometric
confluent functions). For nice pictorial representations, see, e.g., White
[245] and, for textbook references, see Bethe [17, Formula (9.37)] and Thaller
[232, p. 427]. The _three-dimensional_ density for a Bohr atom for a given
$\gamma\in(0,1)$ in spin-orbit channel $\kappa\in\dot{\mathbb{Z}}$ is

$\displaystyle\rho_{\kappa,D}^{H}(x):=\sum_{n=\theta(-\kappa)}^{\infty}\sum_{m=-j_{\kappa}}^{j_{\kappa}}\sum_{\sigma=1}^{4}|\psi_{n,\kappa,m}^{D}(x,\sigma)|^{2},\quad
x\in\mathbb{R}^{3},$ (4.38)

and the total, three-dimensional hydrogenic density is

$\rho_{D}^{H}(x):=\sum_{\kappa\in\dot{\mathbb{Z}}}\rho_{\kappa,D}^{H}(x),\quad
x\in\mathbb{R}^{3}.$ (4.39)

These quantities are indeed well defined, as the following theorem
demonstrates. To state it, let

$\displaystyle\Sigma_{\gamma}:=1-\sqrt{1-\gamma^{2}}\in[0,1]\quad\text{for}\
\gamma\in[0,1].$ (4.40)

######  Theorem 4.4 ([170, Theorem 2.3]).

Let $1/2<s\leq 3/4$, if $\gamma\in(0,\sqrt{15}/4)$ and
$1/2<s<3/2-\Sigma_{\gamma}$, if $\gamma\in[\sqrt{15}/4,1)$. Then for all
$\kappa\in\dot{\mathbb{Z}}$ there is a constant $A_{s,\gamma}>0$ such that for
all $x\in\mathbb{R}^{3}\setminus\\{0\\}$ one has

$\rho_{\kappa,D}^{H}(x)\leq A_{s,\gamma}\frac{|\kappa|^{1-4s}}{|x|^{2}}\\\
\times\left[\left(\frac{|x|}{|\kappa|}\right)^{2s-1}\mathbf{1}_{\\{|x|\leq|\kappa|\\}}+\left(\frac{|x|}{|\kappa|}\right)^{4s-1}\mathbf{1}_{\\{|\kappa|\leq|x|\leq|\kappa|^{2}\\}}+|\kappa|^{4s-1}\mathbf{1}_{\\{|x|\geq|\kappa|^{2}\\}}\right].$

Moreover, for any $\varepsilon>0$ there are constants
$A_{\gamma,\varepsilon},A_{\gamma}>0$ such that for all
$x\in\mathbb{R}^{3}\setminus\\{0\\}$ one has

$\displaystyle\rho_{D}^{H}(x)\leq\begin{cases}A_{\gamma}|x|^{-3/2}&\text{if}\
\gamma\in(0,\sqrt{15}/4]\\\
A_{\gamma,\varepsilon}\left(|x|^{-2\Sigma_{\gamma}-\varepsilon}\mathbf{1}_{\\{|x|\leq
1\\}}+|x|^{-3/2}\mathbf{1}_{\\{|x|>1\\}}\right)&\text{if}\
\gamma\in(\sqrt{15}/4,1)\end{cases}.$ (4.41)

The proof of this theorem is similar to that of Theorem 4.1.

#### Instability

In the Chandrasekhar model we call an atom “unstable”, if the coupling
constant is so large that the operator is unbounded from below. Recall that
for $\gamma<1$ the Coulomb–Dirac operator $D_{\gamma}^{H}$ has a
“distinguished” self-adjoint extension with the property that the expectation
values of both kinetic and potential energy are finite in
$\operatorname{dom}(D_{\gamma}^{H})$. _Instability_ for the Coulomb–Dirac
operator refers to the fact that all self-adjoint extensions of the
Coulomb–Dirac operator have infinitely many eigenfunctions with infinite
expectation value of the potential energy. This situation occurs when
$\gamma>1$, i.e., in the case when the lowest eigenvalue of $D_{\gamma}^{H}$
has hit the lower threshold of the continuous spectrum. See, e.g., Hogreve
[111, Theorem 2.1.(iii)] and the references therein for details and Thaller
[231, p. 218] for an overview.

#### Brown–Ravenhall operator

According to Dirac, the “vacuum”, i.e., the situation in which no (negatively
charged) electrons with positive energies are present, is described by a
completely filled negative energy continuum of the Dirac operator (the so-
called “Dirac sea”), whereas only solutions to the free Dirac equation with
positive kinetic energy should be regarded as “physical electrons”. Dirac
proposed the following interpretation [40, p. 362]:

> “ _The most stable states for an electron (the states of lowest energy) are
> those with negative energy and very high velocity. All the electrons in the
> world will tend to fall into these states with emission of radiation. The
> Pauli exclusion principle, however, will come into play and prevent more
> than one electron going into any one state. Let us assume there are so many
> electrons in the world that all the most stable states are occupied, or,
> more accurately, that all the states of negative energy are occupied except
> perhaps a few of small velocity. Any electrons with positive energy will now
> have very little chance of jumping into negative-energy states and will
> therefore behave like electrons are observed to behave in the laboratory. We
> shall have an infinite number of electrons in negative-energy states, and
> indeed an infinite number per unit volume all over the world, but if their
> distribution is exactly uniform we should expect them to be completely
> unobservable. Only the small departures from exact uniformity, brought about
> by some of the negative-energy states being unoccupied, can we hope to
> observe._ ”

Shortly after Dirac’s equation was formulated, Breit [19, 20, 21] derived a
relativistic wave equation for helium using the quantum electrodynamics of
Heisenberg and Pauli, which reads

$\displaystyle i\frac{\partial\psi(x_{1},x_{2})}{\partial t}$
$\displaystyle=\left[(-i\underline{\alpha}\cdot\nabla+\beta)^{(1)}+(-i\underline{\alpha}\cdot\nabla+\beta)^{(2)}-\frac{\gamma}{|x_{1}|}-\frac{\gamma}{|x_{2}|}+\frac{1}{|x_{1}-x_{2}|}\right.$
(4.42a)
$\displaystyle\left.-\frac{1}{2|x_{1}-x_{2}|}\left(\underline{\alpha}^{(1)}\cdot\underline{\alpha}^{(2)}+\frac{\underline{\alpha}^{(1)}\cdot(x_{1}-x_{2})\cdot\underline{\alpha}^{(2)}\cdot(x_{1}-x_{2})}{|x_{1}-x_{2}|^{2}}\right)\right]\psi(x_{1},x_{2}).$
(4.42b)

As is explained in [22, p. 552] “the superscripts (1) and (2) indicate
operation on the coordinate or spinor components of the first or second
electron respectively in the sixteen-component wave function
$\psi(x_{1},x_{2})$”. Up to the single term in (4.42b), Breit’s equation
coincides with the naive extension of Dirac’s equation for two particles.
Brown and Ravenhall [22] observed that the energies predicted by Breit’s
equation did not match the experimentally measured values very well and
explained their observation as follows [22, pp. 552–553]:

> “ _Because of the negative-energy states, equation ( 4.42) is in fact
> meaningless. This can be seen by constructing a solution of [the Dirac
> equation for two electrons in absence of electron-electron repulsion] and
> then turning on the inter-electron interaction slowly. The system can make
> real transitions to states where one electron has a large negative energy
> and the other electron is in the positive-energy continuum; thus equation
> (4.42) has no stationary solutions if interpreted in this way._”

In the language of spectral theory, the spectrum of a two-particle
Coulomb–Dirac operator occupies the whole real axis (already without electron-
electron repulsion), and all eigenvalues are embedded. This phenomenon is
sometimes called _Brown–Ravenhall disease_ [229, 186, 192]. As is well known
from the non-relativistic theory, these are likely to turn into resonances
when the electron-electron repulsion is turned on, which leads to unphysical
consequences, such as the instability of the atom. We conclude this discussion
by mentioning that despite these physical deficits, Oelker [182] recently
showed that the single-particle Dirac operator may be extended to a self-
adjoint multi-particle operator.

To remedy the above serious defects, Brown and Ravenhall proposed to only
allow states with positive energy with respect to the free Dirac operator,
i.e., they restricted the Hilbert space of admissible states to

$\displaystyle\mathfrak{h}_{c,0}:=\Lambda_{c,0}(L^{2}(\mathbb{R}^{3}:\mathbb{C}^{4})):=\mathbf{1}_{(0,\infty)}(-ic\underline{\alpha}\cdot\nabla+c^{2}\beta)(L^{2}(\mathbb{R}^{3}:\mathbb{C}^{4})).$
(4.43)

The energy of an electron in the Brown–Ravenhall picture is then given by

$\displaystyle\langle\psi,(D_{c,Z}^{H}-c^{2})\psi\rangle,\quad\psi\in\Lambda_{c,0}\mathcal{S}(\mathbb{R}^{3}:\mathbb{C}^{4}).$
(4.44)

The work [59] showed that this energy is bounded from below, if and only if
$\gamma=Z/c\leq\gamma_{B}$ with

$\displaystyle\gamma_{B}:=\frac{2}{\pi/2+2/\pi},$ (4.45)

which implies $Z<125$. For such $\gamma$ the energy form can be extended to a
closed quadratic form in $\mathfrak{h}_{c,0}$ with form domain
$\mathfrak{h}_{c,0}\cap\mathcal{S}(\mathbb{R}^{3}:\mathbb{C}^{4})$. The
resulting self-adjoint operator constructed according to Friedrichs is called
Brown–Ravenhall operator and is denoted by $B_{c,Z}$. By scaling $x\mapsto
x/c$ the Brown–Ravenhall operator is seen to be unitarily equivalent to
$c^{2}B_{1,\gamma}$ with $\gamma=Z/c$. We write $B_{\gamma}:=B_{1,\gamma}$.

Although we will not use it in this review, we record the following convenient
representation of the Brown–Ravenhall operator as a self-adjoint operator in
$L^{2}(\mathbb{R}^{3}:\mathbb{C}^{2})$. For
$E_{c}(\xi):=\sqrt{c^{2}|\xi|^{2}+c^{4}}$ with $\xi\in\mathbb{R}^{3}$ and

$\displaystyle\varphi_{j}(\xi):=\sqrt{\frac{E_{1}(\xi)+(-1)^{j}}{2E_{1}(\xi)}},\quad
j\in\\{0,1\\},\quad\xi\in\mathbb{R}^{3},$ (4.46)

we define the $\mathbb{C}^{2\times 2}$-valued functions

$\displaystyle\Phi_{0}(\xi):=\varphi_{0}(|\xi|)\mathbf{1}_{\mathbb{C}^{2}},\quad\Phi_{1}(\xi):=\varphi_{1}(|\xi|)\frac{\underline{\sigma}\cdot\xi}{|\xi|},\quad\xi\in\mathbb{R}^{3}.$
(4.47)

Then the map

$\displaystyle\begin{split}\bm{\Phi}_{c}:L^{2}(\mathbb{R}^{3}:\mathbb{C}^{2})\to
L^{2}(\mathbb{R}^{3}:\mathbb{C}^{4}),\quad
u\mapsto\left(\begin{array}[]{c}\Phi_{0}(-i\nabla/c)u\\\
\Phi_{1}(-i\nabla/c)u\end{array}\right)\end{split}$ (4.48)

maps $L^{2}(\mathbb{R}^{3}:\mathbb{C}^{2})$ unitarily onto
$\mathfrak{h}_{c,0}$, cf. [59]. Therefore, $B_{\gamma}$ in
$\mathfrak{h}_{c,0}$ is unitarily equivalent to the operator

$\displaystyle
E_{c}(-i\nabla)-c^{2}-\mathcal{T}_{c}\left(\frac{Z}{|x|}\right)\quad\text{in}\
L^{2}(\mathbb{R}^{3}:\mathbb{C}^{2})$ (4.49)

with the “twisted potential”

$\displaystyle\begin{split}\mathcal{T}_{c}\left(V\right):&=\left(\begin{array}[]{c}\Phi_{0}(-i\nabla/c)\\\
\Phi_{1}(-i\nabla/c)\end{array}\right)V\left(\begin{array}[]{c}\Phi_{0}(-i\nabla/c)\\\
\Phi_{1}(-i\nabla/c)\end{array}\right)\\\
&=\Phi_{0}\left(\frac{-i\nabla}{c}\right)V\Phi_{0}\left(\frac{-i\nabla}{c}\right)+\Phi_{1}\left(\frac{-i\nabla}{c}\right)V\Phi_{1}\left(\frac{-i\nabla}{c}\right)\end{split}$
(4.50)

for any $V:\mathbb{R}^{3}\to\mathbb{C}^{4}$, whenever meaningfully defined.
From a technical point of view the representation (4.49) is important, e.g.,
in [59, 23, 91, 168]. (Figuratively speaking, the transformation
$\mathcal{T}_{c}$ mollifies the Coulomb singularity and ensures the lower
boundedness of $B_{\gamma}$ for coupling constants greater than $2/\pi$. This
transpires, e.g., in [91, Lemma 2.7] and [82, Lemma 5.2].)

In [59] the authors showed that the Brown–Ravenhall operator $B_{\gamma}$ is
bounded from below by $-\gamma(\pi/4-1/\pi)-1$ when $\gamma\leq\gamma_{B}$. In
fact, Tix [235, 236] proved the stronger lower bound $-\gamma_{B}$.

If $\gamma<\gamma_{B}$, the essential spectrum of the Brown–Ravenhall operator
$B_{\gamma}$ is $[0,\infty)$ and the singular continuous spectrum is empty
[59, Theorem 2]. Moreover, there are no embedded eigenvalues, and the spectrum
in $[0,\infty)$ is purely absolutely continuous [7, Theorem 3.4.1]. As in the
Chandrasekhar case, the eigenvalues $\lambda_{Z,n,j,\ell}$ of $B_{c,Z}$ in
channel $(j,\ell)$ satisfy the bounds

$-\frac{Z^{2}}{2(n+\ell+1)^{2}}\geq\lambda_{Z,n,j,\ell}\geq-\mathrm{const}\
\cdot\frac{Z^{2}}{(n+\ell+1)^{2}},$

where the constant in the second inequality can be chosen independently of
$\gamma$. The lower bound is due to [91, Theorem 2.1], while the upper bound
follows from the fact that the non-relativistic kinetic energy dominates the
relativistic one. In particular, the Brown–Ravenhall eigenvalues are smaller
than those of $D_{c,Z}^{H}-c^{2}$ due to the min-max-principle for operators
with spectral gaps, cf. [99, 98].

#### Furry operator

Naturally, the projection onto the positive spectral subspace of the free
Dirac operator is not the only possibility to get rid of the positronic part
of the wave functions. Furry and Oppenheimer [93] proposed rather to project
onto the positive spectral subspace of the Coulomb–Dirac operator, i.e., the
Hilbert space of admissible states is

$\mathfrak{h}_{c,Z}:=\Lambda_{c,Z}(L^{2}(\mathbb{R}^{3}:\mathbb{C}^{4})):=\mathbf{1}_{(0,\infty)}(D_{c,Z}^{H})(L^{2}(\mathbb{R}^{3}:\mathbb{C}^{4})).$

Since $D_{\gamma}^{H}$ can be realized as a self-adjoint operator with form
domain $H^{1/2}(\mathbb{R}^{3}:\mathbb{C}^{4})$ by Nenciu’s method [181] when
$Z/c=\gamma<\gamma_{F}$ with

$\displaystyle\gamma_{F}:=1,$ (4.51)

we have

$\Lambda_{c,Z}(\mathcal{S}(\mathbb{R}^{3}:\mathbb{C}^{4}))\subseteq
H^{1/2}(\mathbb{R}^{3}:\mathbb{C}^{4})$

and dense in $\mathfrak{h}_{c,Z}$. Therefore, the quadratic form

$\displaystyle\langle\psi,(D_{c,Z}^{H}-c^{2})\psi\rangle,\quad\psi\in\Lambda_{c,Z}\mathcal{S}(\mathbb{R}^{3}:\mathbb{C}^{4})$
(4.52)

is well-defined and bounded from below when $\gamma\in(0,1)$. By Friedrichs,
there is a corresponding self-adjoint operator. The quadratic form domain of
this operator is
$H^{1/2}(\mathbb{R}^{3}:\mathbb{C}^{4})\cap\mathfrak{h}_{c,Z}$. This operator
is called Furry operator and denoted by $F_{c,Z}$. Scaling $x\mapsto x/c$
shows that the Furry operator is unitarily equivalent to $c^{2}F_{1,\gamma}$
with $\gamma=Z/c$ and we write $F_{\gamma}:=F_{1,\gamma}$ in the rescaled
picture.

#### Mittleman operator

Although, at this point, the choice of the projections seems to be arbitrary
and only justifiable by the comparison of the results with the measured
quantities this is – according to Mittleman [171] not the case: the optimal
projection and optimal ground state should be obtained by a minimax-principle,
namely the infimum over the states in a class of fermionic Hilbert spaces
defined by the positive spectral subspace of some Dirac operator
$-i\underline{\alpha}\cdot\nabla+mc^{2}\beta-\varphi$ followed by a supremum
over a suitable class of potentials $\varphi$. However, this has not been
implemented on a mathematical level. In fact there some elementary no-go
results [9, 11] that a potential mathematical implementation has to
circumvent. For a more detailed review of Mittleman’s principle and
references, see, e.g., [54, Section 4.5].

For a variational principle inspired by Mittleman, see, e.g., [5], and for
works connecting Mittleman’s principle and the Dirac–Fock equations, see,
e.g., [9, 10, 11].

###  4.2. Many-particle operators

The results for the above-discussed single-particle operators allow to define
many-particle operators.

####  4.2.1. Chandrasekhar operator

Chandrasekhar molecules with $q=2$ are described by

$\displaystyle
C_{N,V}:=\sum_{\nu=1}^{N}\left(\sqrt{-c^{2}\Delta_{\nu}+c^{4}}-c^{2}-V(x_{\nu})\right)+\sum_{1\leq\nu<\mu\leq
N}\frac{1}{|x_{\nu}-x_{\mu}|}+U$ (4.53)

for $\gamma\leq\gamma_{C}=2/\pi$ and $V$ and $U$ as in (1.2) and (1.3),
respectively. (Technically, $C_{N,V}$ is defined as the Friedrichs extension
of the corresponding quadratic form with form domain
$\mathcal{S}(\mathbb{R}^{3N}:\mathbb{C}^{2^{N}})$.)

The ground state energy of a Chandrasekhar molecule is

$\displaystyle
E_{c,C}(N,\underline{Z},\underline{R}):=\inf\operatorname{spec}(C_{N,V}).$
(4.54)

In the neutral, atomic case $K=1$, $\underline{R}=0$, $N=Z$, the ground state
energy $E_{c,C}(Z):=E_{c,C}(Z,Z,0)$ is an eigenvalue [139].

For an associated ground state $\psi$ of $C_{N,V}$ (or an approximate ground
state on the Thomas–Fermi or Scott scale), we define the associated three-
dimensional one-particle ground state densities

$\displaystyle\rho_{C}(x):=N\sum_{\sigma=1}^{2}\int_{\Gamma^{N-1}}|\psi(x,\sigma;y_{2},...,y_{N})|^{2}\,\mathrm{d}y_{2}...\mathrm{d}y_{N},\quad
x\in\mathbb{R}^{3}.$ (4.55)

Similarly as in the Schrödinger case, we define the one-dimensional angular
momentum resolved version of $\rho_{C}$ for any $\ell\in\mathbb{N}_{0}$ by

$\displaystyle\varrho_{\ell,C}(r):=Nr^{2}\sum_{\sigma=1}^{2}\sum_{m=-\ell}^{\ell}\int_{\Gamma^{N-1}}\left|\int_{\mathbb{S}^{2}}\overline{Y_{\ell,m}(\omega)}\psi(r\omega,\sigma;y_{2},...,y_{N})\right|^{2}\,\mathrm{d}y_{2}...\mathrm{d}y_{N}$
(4.56)

for $r>0$. Note that

$\displaystyle\int_{\mathbb{S}^{2}}\rho_{C}(r\omega)\,\mathrm{d}\omega=r^{-2}\sum_{\ell\geq
0}\varrho_{\ell,C}(r),\quad r>0.$ (4.57)

####  4.2.2. Brown–Ravenhall and Furry operators

The energy of an atom in the Brown–Ravenhall or Furry pictures is given by

$\displaystyle\mathcal{E}_{c,Z,N}[\psi]:=\langle\psi,\left(\sum_{\nu=1}^{N}\left(D_{c,Z}-c^{2}\right)_{\nu}+\sum_{1\leq\nu<\mu\leq
N}\frac{1}{|x_{\nu}-x_{\mu}|}\right)\psi\rangle,$ (4.58)

whenever
$\psi\in\bigwedge_{\nu=1}^{N}\Lambda_{c,\\#}\mathcal{S}(\mathbb{R}^{3}:\mathbb{C}^{4})$
with $\\#\in\\{0,Z\\}$. The quadratic form is bounded from below if
$\gamma\leq\gamma_{B}$ in the Brown–Ravenhall case and if $\gamma\leq 1$ in
the Furry case by the previous discussion of the one-particle operators. The
resulting self-adjoint operators constructed according to Friedrichs are
called Brown–Ravenhall and Furry operator, respectively. If $N=Z$, we drop the
third index in the above energy form and write
$\mathcal{E}_{c,Z}[\psi]:=\mathcal{E}_{c,Z,Z}$.

The ground state energy of a Brown–Ravenhall or Furry atom is

$\displaystyle
E_{c,B}(N,Z):=\inf\\{\mathcal{E}_{c,Z,N}[\psi]:\,\psi\in\Lambda_{c,0}\mathcal{S}(\mathbb{R}^{3}:\mathbb{C}^{4})\\},$
(4.59) $\displaystyle
E_{c,F}(N,Z):=\inf\\{\mathcal{E}_{c,Z,N}[\psi]:\,\psi\in\Lambda_{c,Z}\mathcal{S}(\mathbb{R}^{3}:\mathbb{C}^{4})\\}.$
(4.60)

We record that Morozov and Vugalter [174] (see also Morozov [172], Jakubaßa-
Amundsen [123] for HVZ theorems), and Matte and Stockmeyer [166] proved that
for $N=Z$, the Brown–Ravenhall and Furry ground state energies
$E_{B/F}(Z):=E_{B/F}(Z,Z)$ are eigenvalues.

For an associated ground state $\psi$ of $\mathcal{E}_{c,Z,N}$ (or an
approximate ground state on the Thomas–Fermi or Scott scale) in either the
Brown–Ravenhall or Furry picture, we define the associated one-particle ground
state densities

$\displaystyle\rho_{B/F}(x):=N\sum_{\sigma=1}^{4}\int_{\Gamma^{N-1}}|\psi(x,\sigma;y_{2},...,y_{N})|^{2}\,\mathrm{d}y_{2}...\mathrm{d}y_{N},\quad
x\in\mathbb{R}^{3}.$ (4.61)

As in the Chandrasekhar case, we define a spin-orbit resolved version of
$\rho_{B/F}$. More precisely, for given spin-orbit coupling
$\kappa\in\dot{\mathbb{Z}}$ (recall (4.31)-(4.32)), we define

$\displaystyle\rho_{\kappa,B/F}(x):=\frac{N}{4\pi}\sum_{\sigma\in\\{+,-\\}}\sum_{m=-j_{\kappa}}^{j_{\kappa}}\int_{\Gamma^{N-1}}\mathrm{d}y\,\left|\sum_{\tau=1}^{4}\int_{\mathbb{S}^{2}}\,\mathrm{d}\omega\,\overline{\Phi_{\kappa,m}^{\sigma}(\omega,\tau)}\psi(|x|\omega,\tau,y)\right|^{2}$
(4.62)

for $x\in\mathbb{R}^{3}$. Here $\Phi^{\sigma}_{\kappa,m}$ are the spherical
Dirac spinors (4.33). Note that

$\displaystyle\frac{1}{4\pi}\int_{\mathbb{S}^{2}}\rho_{B/F}(|x|\omega)\mathrm{d}\omega=\sum_{\kappa\in\dot{\mathbb{Z}}}\rho_{\kappa,B/F}(x),\quad
x\in\mathbb{R}^{3}.$ (4.63)

######  Remarks 4.5.

(1) The Brown–Ravenhall and Furry operators are examples of so-called “no-
pair” operators, i.e., Schrödinger operators that can _formally_ be derived
from quantum electrodynamics by _neglecting the creation of electron-positron
pairs_ [228]. These operators are popular among quantum chemists, as they
provide decent numerical results which are in good accordance with
experimentally measured data. For instance, the Scott correction in the Furry
picture (Formula (4.80) in Theorem 4.11) coincides astonishingly well with
experimental data (see, e.g., [135]), see [100, Section 6] and [186, 192] for
textbook treatments.

(2) The Scott correction is also believed to be true when the mean field in
the sense of Mittleman [171] is taken into account. However, this is so far
only known in the Hartree–Fock approximation when the involved projection is
given by the Dirac–Fock operator, see Fournais et al. [79].

In the following subsections we summarize results concerning the asymptotic
expansion of the ground state energies and convergence of the one-particle
ground state densities for the above-introduced models in the atomic and
molecular cases. In particular we review the elements of the proof of the
_relativistic strong Scott conjecture_ for Chandrasekhar atoms in [87, 85].

###  4.3. Atoms without magnetic fields

As far as we know, the results below have only been proved in the neutral case
$N=Z$. We believe that they also hold for ions.

####  4.3.1. Thomas–Fermi scale

In all of the above relativistic models the leading order of the asymptotic
expansion of the ground state energy is non-relativistic. These results
indicate that electrons whose distances to the nucleus are of order $Z^{-1/3}$
still behave non-relativistically, although they are being sucked into the
nucleus. The following theorem was proved by Sørensen [184] (Chandrasekhar),
by [23] and [91] (Brown–Ravenhall), and by [100] (Furry, as a consequence of
the Scott correction). Recall the critical coupling constants
$\gamma_{C},\gamma_{B},\gamma_{F}$ in (4.4), (4.45), (4.51).

######  Theorem 4.6.

Let $N=Z$ and $E^{\rm TF}(Z)$ denote the Thomas–Fermi energy of a neutral
Thomas–Fermi atom with $q=2$. Then

$\displaystyle\lim_{\begin{subarray}{c}Z,c\to\infty\end{subarray}}\frac{E_{c,C}(Z)-E^{\rm
TF}(Z)}{Z^{7/3}}=0\quad\text{for fixed}\ \frac{Z}{c}\leq\gamma_{C},$ (4.64)
$\displaystyle\lim_{\begin{subarray}{c}Z,c\to\infty\end{subarray}}\frac{E_{c,B}(Z)-E^{\rm
TF}(Z)}{Z^{7/3}}=0\quad\text{for fixed}\ \frac{Z}{c}\leq\gamma_{B},$ (4.65)
$\displaystyle\lim_{\begin{subarray}{c}Z,c\to\infty\end{subarray}}\frac{E_{c,F}(Z)-E^{\rm
TF}(Z)}{Z^{7/3}}=0\quad\text{for fixed}\ \frac{Z}{c}<\gamma_{F}.$ (4.66)

######  Remarks 4.7 (Elements in the proof of Theorem 4.6).

We make some remarks on Sørensen’s proof [184] for the Chandrasekhar case.
Since $\sqrt{p^{2}+1}-1\leq p^{2}/2$, it suffices to prove the lower bound,
whose proof is similar to Lieb’s simplified proof of Theorem 3.1, see [144,
Theorem 5.1]. It can be split into the following steps.

  1. (1)

Reduce the linear many-particle problem to estimating the non-linear one-
particle quadratic form

$\displaystyle\sum_{\nu=1}^{N}\langle
m_{\nu},\left[\sqrt{-c^{2}\Delta+c^{4}}-c^{2}-\frac{Z}{|x|}+\rho_{Z}^{\rm
TF}\ast\frac{1}{|\cdot|}\right]m_{\nu}\rangle$ (4.67)

with the Thomas–Fermi density $\rho_{Z}^{\rm TF}$ and orthonormal orbitals
$\\{m_{\nu}\\}_{\nu=1}^{N}$ from below with the help of a correlation
inequality (e.g., by Lieb and Oxford [183, 142, 149] or that of [160]).

  2. (2)

Due to the (non-perturbative) Coulomb singularity at the origin, one localizes
position space into the regions $|x|\lesssim Z^{-o}$ and $|x|\gtrsim Z^{-o}$
with $o\in(1/3,2/3)$. Guided by the proof of Theorem 3.1 (Lieb [144, Theorem
5.1]) and the intuition that electrons on distances $Z^{-o}$ behave non-
relativistically, it is expected that the electrons in the region $|x|\gtrsim
Z^{-o}$ lead to the TF energy, while the contribution from $|x|\lesssim
Z^{-o}$ is $o(Z^{7/3})$. The localization errors can be controlled at the end
of the argument with the help of an arbitrarily small amount
($Z^{-\varepsilon}$) of kinetic energy.

  3. (3)

The contribution of the electrons in $\\{|x|\lesssim Z^{-o}\\}$, where the
Coulomb singularity is located, can be controlled with the help of the
following strengthening of one of Daubechies’ inequalities [34],

$\displaystyle\operatorname{Tr}\left(|p|-\frac{2/\pi}{|x|}-V\right)_{-}\lesssim\int_{\mathbb{R}^{3}}V(x)_{+}^{4}\,\mathrm{d}x$
(4.68)

by [83] and the inequality $\sqrt{p^{2}+1}-1\geq|p|-1$. (Inequality (4.68) is
often called Hardy–Lieb–Thirring inequality because of the homogeneity of the
“unperturbed Hardy operator” $|p|-\frac{2/\pi}{|x|}$.) Here, $V$ is a bounded
function, supported on $\\{|x|\lesssim Z^{-o}\\}$. Using (4.68) one computes
the contribution to the energy to be $\mathcal{O}(Z^{2+3(1-o)})$, which is
more than $Z^{7/3}$. For this reason another localization on the length scale
$Z^{-i}$ is necessary. The Hardy–Lieb–Thirring inequality for $V$ being
supported on $\\{|x|\lesssim Z^{-i}\\}$ then produces an $o(Z^{7/3})$ error
when $i\in(8/9,1)$. The additionally introduced localization error can be
controlled by an $\varepsilon$ of kinetic energy, too.

  4. (4)

The energy contribution of electrons located in the intermediate region
$\\{Z^{-i}\lesssim|x|\lesssim Z^{-o}\\}$ can be controlled using Daubechies’
inequality,

$\displaystyle\operatorname{Tr}\left(\sqrt{-\Delta+1}-1-V(x)\right)_{-}\lesssim\int_{\mathbb{R}^{3}}(V_{+}^{5/2}(x)+V_{+}^{4}(x))\,\mathrm{d}x.$
(4.69)

Note that $|x|^{-1}\notin L^{4}(\mathbb{R}^{3})$.

  5. (5)

Electrons in the region $\\{|x|\gtrsim Z^{-o}\\}$ are expected to generate the
TF energy. Here semiclassical analysis is used. Roughly speaking, one compares
the quantum energy

$\displaystyle\sum_{\nu=1}^{N}\langle m_{\nu},\mathbf{1}_{\\{|x|\gtrsim
Z^{-o}\\}}\left[\sqrt{-c^{2}\Delta+c^{4}}-c^{2}-\frac{Z}{|x|}+\rho_{Z}^{\rm
TF}\ast\frac{1}{|\cdot|}\right]\mathbf{1}_{\\{|x|\gtrsim
Z^{-o}\\}}m_{\nu}\rangle$ (4.70)

to the classically expected energy

$\displaystyle\iint\limits_{|q|\gtrsim
Z^{-o}}\left(\sqrt{c^{2}p^{2}+c^{4}}-c^{2}-\frac{Z}{|q|}+\rho_{Z}^{\rm
TF}\ast\frac{1}{|\cdot|}(q)\right)_{-}\,\frac{\mathrm{d}p\,\mathrm{d}q}{(2\pi)^{3}}.$
(4.71)

The latter leads to the _non-relativistic_ TF energy, since
$\sqrt{c^{2}p^{2}+c^{4}}-c^{2}\sim p^{2}/2$ for $|p|\lesssim Z^{o}\ll Z$
(since $|x|\gtrsim Z^{-o}$ with $o\in(1/3,2/3)$). To that end a phase-space
localization using coherent states [144, Theorem 5.1] is used. In fact, merely
the localization errors coming from the phase-space localization force the
position localization to the scale $\lesssim Z^{-1/3}$.

Theorem 4.6 is accompanied by the following convergence results for the ground
state densities.

######  Theorem 4.8 ([169, 168]).

Let $N=Z$ and $\rho_{1}^{\rm TF}$ denote the hydrogenic Thomas–Fermi minimizer
with $\int\rho_{1}^{\rm TF}(x)\,\mathrm{d}x=1$ and $q=2$. Then

$\displaystyle\lim_{\begin{subarray}{c}Z,c\to\infty\end{subarray}}Z^{-2}\rho_{C}(Z^{-1/3}\cdot)=\rho_{1}^{\rm
TF}\quad\text{for fixed}\ \frac{Z}{c}\leq\gamma_{C},$ (4.72)
$\displaystyle\lim_{\begin{subarray}{c}Z,c\to\infty\end{subarray}}Z^{-2}\rho_{B}(Z^{-1/3}\cdot)=\rho_{1}^{\rm
TF}\quad\text{for fixed}\ \frac{Z}{c}\leq\gamma_{B},$ (4.73)
$\displaystyle\lim_{\begin{subarray}{c}Z,c\to\infty\end{subarray}}Z^{-2}\rho_{F}(Z^{-1/3}\cdot)=\rho_{1}^{\rm
TF}\quad\text{for fixed}\ \frac{Z}{c}<\gamma_{F}.$ (4.74)

In all three formulae the convergence holds in Coulomb norm (see (2.5)-(2.6))
with convergence rate $\mathcal{O}(Z^{-3/16})$. In the Chandrasekhar case the
convergence also holds when both sides are integrated against any $U\in
L^{5/2}\cap L^{4}(\mathbb{R}^{3})$ and in the Brown–Ravenhall case when in
addition $U\in|\cdot|^{-1}L^{\infty}$ is Lipschitz.

######  Remarks 4.9.

(1) The proof of the convergence in Coulomb norm uses an observation of
Fefferman and Seco [66], together with the energetic expansion of the ground
state energy. For an error term $\mathcal{O}(Z^{a})$ to the leading TF energy
with $a<7/3$ the convergence rate is $\mathcal{O}(Z^{(a-7/3)/2})$. In view of
the energetic results [90, 223, 91, 100] (Scott correction), we have
$a=47/24$.

(2) The proof of weak convergence uses the proof of the energetic results
(Theorem 4.6) together with a linear response argument. We will flesh out the
details in the discussion of the densities on the Scott scale (Subsection
4.3.3).

####  4.3.2. Scott scale

As explained in the introduction, electrons in proximity of the nucleus are
expected to generate relativistic effects that should be visible in the ground
state energy and density on the spatial scale $Z^{-1}$. In fact, the Scott
correction is relativistically lowered. The precise amount depends on the sum
of the differences of the non-relativistic and the relativistic hydrogen
eigenvalues. To that end, let $\lambda_{n}^{S}$, $\lambda_{n}^{C}$,
$\lambda_{n}^{B}$, and $\lambda_{n}^{F}$ denote the $\gamma$-dependent
eigenvalues of the non-relativistic operator $S_{\gamma}^{H}$ in (1.9), of the
Chandrasekhar operator $C_{\gamma}^{H}$ in (4.2) (with $q=2$), of the
Brown–Ravenhall operator $B_{\gamma}$. and of the Furry operator $F_{\gamma}$.
(The Furry eigenvalues coincide of course with those (4.37) of
$D_{\gamma}^{H}-1$.) We introduce the spectral shifts

$\displaystyle[0,\gamma_{C}]\ni\gamma\mapsto s_{C}(\gamma)$
$\displaystyle:=\gamma^{-2}\sum_{n\geq
0}\left(\lambda_{n}^{S}-\lambda_{n}^{C}\right)\geq 0,$ (4.75)
$\displaystyle[0,\gamma_{B}]\ni\gamma\mapsto s_{B}(\gamma)$
$\displaystyle:=\gamma^{-2}\sum_{n\geq
0}\left(\lambda_{n}^{S}-\lambda_{n}^{B}\right)\geq 0,$ (4.76)
$\displaystyle[0,\gamma_{F}]\ni\gamma\mapsto s_{F}(\gamma)$
$\displaystyle:=\gamma^{-2}\sum_{n\geq
0}\left(\lambda_{n}^{S}-\lambda_{n}^{F}\right)\geq 0,$ (4.77)

and record the following observation.

######  Proposition 4.10.

Let $\\#\in\\{C,B,F\\}$. Then the functions $s_{\\#}$ on their respective
domains are continuous and monotone decreasing, and obey $s_{\\#}(0)=0$.

###### Proof.

For $s_{C}$ this is proved in [223, Theorems 1.1, 1.4, Corollary 1.6]. See
also [90, p. 552], where it is shown that $s_{C}$ is monotone decreasing and
finite. For $s_{F}$ the claim can be inferred from the explicitly known
eigenvalues $\lambda_{n}^{S}$ and $\lambda_{n}^{F}$, respectively. The
continuity and monotonicity of $s_{B}$ follow from the explicit knowledge of
the Schrödinger and Coulomb–Dirac eigenvalues, and the inequality
$\lambda_{n}^{F}\geq\lambda_{n}^{B}$. ∎

The following theorem concerning the energy asymptotics of Chandrasekhar atoms
was proved independently by Solovej et al. [223], and the work [90] using
different techniques. The result for Brown–Ravenhall atoms was proved by [91],
and that for Furry atoms by [100].

######  Theorem 4.11.

Let $N=Z$ and $E^{\rm TF}(Z)$ denote the Thomas–Fermi energy of a neutral
Thomas–Fermi atom with $q=2$. Then

$\displaystyle\lim_{\begin{subarray}{c}Z,c\to\infty\end{subarray}}\frac{E_{c,C}(Z)-\left[E^{\rm
TF}(Z)+\left(\frac{1}{2}-s_{C}(\gamma)\right)Z^{2}\right]}{Z^{2}}$
$\displaystyle=0\quad\text{for fixed}\ \frac{Z}{c}\leq\gamma_{C},$ (4.78)
$\displaystyle\lim_{\begin{subarray}{c}Z,c\to\infty\end{subarray}}\frac{E_{c,B}(Z)-\left[E^{\rm
TF}(Z)+\left(\frac{1}{2}-s_{B}(\gamma)\right)Z^{2}\right]}{Z^{2}}$
$\displaystyle=0\quad\text{for fixed}\ \frac{Z}{c}\leq\gamma_{B},$ (4.79)
$\displaystyle\lim_{\begin{subarray}{c}Z,c\to\infty\end{subarray}}\frac{E_{c,F}(Z)-\left[E^{\rm
TF}(Z)+\left(\frac{1}{2}-s_{F}(\gamma)\right)Z^{2}\right]}{Z^{2}}$
$\displaystyle=0\quad\text{for fixed}\ \frac{Z}{c}<\gamma_{F}.$ (4.80)

In all of the above limits the error term can be quantified and is
$\mathcal{O}(Z^{47/24})$.

######  Remark 4.12.

It is believed that these results also hold for ions (at least as long as the
ionization degree is sufficiently small), since the electrons on length scales
$\mathcal{O}(Z^{0})$ should not disturb the energy generated by electrons on
length scales $\mathcal{O}(Z^{-1})$. However, a rigorous proof is lacking.

The energetic results on the Scott scale are accompanied by recent results
[87, 85, 170] for the density in the Chandrasekhar and Furry cases.

######  Theorem 4.13 ([87, 85, 170]).

Let

$\displaystyle U\in\mathcal{D}:=$ $\displaystyle\left\\{W\in L_{\rm
loc}^{1}(\mathbb{R}_{+}):\,\forall\varepsilon>0\,\exists a>0\,\forall
r>0:\right.$ $\displaystyle\qquad\qquad\qquad\quad\left.|W(r)|\leq
a\left(r^{-1}\mathbf{1}_{\\{r\leq
1\\}}+r^{-\frac{3}{2}-\varepsilon}\mathbf{1}_{\\{r\geq 1\\}}\right)\right\\}$
(4.81)

be arbitrary. Then the following statements hold.

  1. (1)

(Convergence for fixed angular momentum/spin-orbit coupling) Let
$\ell\in\mathbb{N}_{0}$ and $\kappa\in\dot{\mathbb{Z}}$ be fixed. Then

$\displaystyle\lim_{\begin{subarray}{c}Z,c\to\infty\end{subarray}}\int_{0}^{\infty}c^{-3}\varrho_{\ell,C}(c^{-1}r)U(r)\,\mathrm{d}r$
$\displaystyle=\int_{0}^{\infty}\varrho_{\ell,C}^{H}(r)U(r)\,\mathrm{d}r\quad\text{for
fixed}\ \frac{Z}{c}<\gamma_{C},$ (4.82)
$\displaystyle\lim_{\begin{subarray}{c}Z,c\to\infty\end{subarray}}\int_{\mathbb{R}^{3}}c^{-3}\rho_{\kappa,F}(c^{-1}x)U(|x|)\,\mathrm{d}x$
$\displaystyle=\int_{\mathbb{R}^{3}}\rho_{\kappa,D}^{H}(x)U(|x|)\,\mathrm{d}x\quad\text{for
fixed}\ \frac{Z}{c}<\gamma_{F}.$ (4.83)

  2. (2)

(Convergence of total density) We have

$\displaystyle\lim_{\begin{subarray}{c}Z,c\to\infty\end{subarray}}\int_{\mathbb{R}^{3}}c^{-3}\rho_{C}(c^{-1}x)U(|x|)\,\mathrm{d}x$
$\displaystyle=\int_{\mathbb{R}^{3}}\rho_{C}^{H}(x)U(|x|)\,\mathrm{d}x\quad\text{for
fixed}\ \frac{Z}{c}<\gamma_{C},$ (4.84)
$\displaystyle\lim_{\begin{subarray}{c}Z,c\to\infty\end{subarray}}\int_{\mathbb{R}^{3}}c^{-3}\rho_{F}(c^{-1}x)U(|x|)\,\mathrm{d}x$
$\displaystyle=\int_{\mathbb{R}^{3}}\rho_{D}^{H}(x)U(|x|)\,\mathrm{d}x\quad\text{for
fixed}\ \frac{Z}{c}<\gamma_{F}.$ (4.85)

######  Remarks 4.14.

(1) For the sake of clarity we restricted attention to the above class
$\mathcal{D}$ of test functions, although the results actually hold for a
substantially larger class.

(2) However, the exemplary test function class (4.13) is believed to be
optimal.

  1. (a)

Due to Kato’s inequality, we cannot expect (at least not for $\ell=0$)
(4.82)-(4.85) to hold for test functions, whose singularity is worse than
$|x|^{-1}$.

  2. (b)

The $|x|^{-3/2-\varepsilon}$ decay seems optimal in view of the expected
$|x|^{-3/2}$-decay of $\rho_{C/F}^{H}$, see Theorems 4.1 and 4.4.

(3) In view of Theorems 4.1 and 4.4, a transition between the length scales
$Z^{-1}$ and $Z^{-1/3}$ is again clearly visible.

####  4.3.3. Elements in the proof of Theorem 4.13

The rest of this subsection is concerned with explaining the key elements of
the proof of Theorem 4.13 in the Chandrasekhar case, i.e., the limits (4.82)
and (4.84). For simplicity we set $q=1$ here. We begin with the argument to
prove (4.82) for a fixed angular momentum channel.

We follow the lines of Lieb and Simon [152], Baumgartner [12], and [117] by
employing a linear-response argument. Let $\lvert\psi\rangle\langle\psi\rvert$
be a ground state density matrix of the atomic many-particle operator $C_{Z}$
(see (4.53)), and define, for $U\in\mathcal{D}$ and in slight abuse of
notation, the perturbed operator

$\displaystyle
C_{Z,\lambda}:=C_{Z}-\lambda\sum_{\nu=1}^{Z}c^{2}U(c|x_{\nu}|)\Pi_{\ell,\nu}\quad\text{in}\
\bigwedge_{\nu=1}^{Z}L^{2}(\mathbb{R}^{3}).$ (4.86)

Here $\Pi_{\ell}$ is the orthogonal projection in $L^{2}(\mathbb{R}^{3})$ onto
the $\ell$-th angular momentum channel defined by

$\Pi_{\ell}:=\sum_{m=-\ell}^{\ell}|Y_{\ell,m}\rangle\langle Y_{\ell,m}|,$

and $\Pi_{\ell,\nu}$ acts as $\Pi_{\ell}$ with respect to the $\nu$-th
particle. Since the singularity of $U$ is Coulombic, $C_{Z,\lambda}$ is
realized as a self-adjoint operator by Kato’s inequality if $\gamma<2/\pi$ and
$|\lambda|$ is sufficiently small. Moreover, since (4.82) is linear in $U$ we
may assume $U\geq 0$ and $\lambda>0$ without loss of generality.

By the linear response argument, the Scott correction (Theorem 4.11), and
scaling $x\mapsto x/c$, we have

$\displaystyle\begin{split}&\varlimsup\limits_{Z\to\infty}\int_{0}^{\infty}c^{-3}\varrho_{\ell,C}(r/c)U(r)\,\mathrm{d}r\\\
&\quad=\varliminf\limits_{\lambda\searrow
0}\varlimsup\limits_{Z\to\infty}\operatorname{Tr}_{\bigwedge_{\nu=1}^{Z}L^{2}(\mathbb{R}^{3})}\left[\frac{|\psi\rangle\langle\psi|(C_{Z}-C_{Z,\lambda})}{\lambda
c^{2}}\right]\\\ &\quad\leq(2\ell+1)\cdot\varliminf\limits_{\lambda\searrow
0}\frac{\operatorname{Tr}_{L^{2}(\mathbb{R}_{+})}(C_{\ell,\gamma}^{H}-\lambda
U(r))_{-}-\operatorname{Tr}_{L^{2}(\mathbb{R}_{+})}(C_{\ell,\gamma}^{H})_{-}}{\lambda}\end{split}$
(4.87)

with $C_{\ell,\gamma}^{H}$ as in (4.6). To compute the right side of (4.87) we
have two options.

  1. (1)

Find a majorant to apply the dominated convergence theorem to interchange
$\liminf_{\lambda\searrow 0}$ and $\operatorname{Tr}$. Then apply standard
perturbation theory, i.e., the classical Hellmann–Feynman theorem for a single
eigenvalue. This lead to the shorter proof in [85].

  2. (2)

Compute the derivative with respect to $\lambda$ directly. This lead to the
longer, original proof in [87].

Here we shall present the arguments of the longer proof, as we believe that it
better unearths the involved mathematics of the relativistic strong Scott
conjecture. Besides, it allows us to popularize a generalization of the
classical Hellmann–Feynman theorem that may be of independent interest in the
analysis of many-particle problems.

First we state this generalized Hellmann–Feynman theorem with “natural”
assumptions on the perturbation. However, this version is not applicable to
our problem. Afterwards we state a generalization with weaker assumptions,
which suffices for our purposes. Recall that an operator $B$ is called
_relatively form trace class_ with respect to a self-adjoint operator $A$ that
is bounded from below, if $(A+M)^{-1/2}B(A+M)^{-1/2}$ is trace class for some
(and hence any) large enough $M>0$.

######  Theorem 4.15 ([87, Theorem 3.1]).

Assume that $A$ is a self-adjoint operator in some Hilbert space with $A_{-}$
trace class. Assume that $B$ is a non-negative operator in the same Hilbert
space, and relatively form trace class with respect to $A$. Then the one-sided
derivatives of

$\lambda\mapsto S(\lambda):=\operatorname{Tr}(A-\lambda B)_{-}$

satisfy

$\operatorname{Tr}B\mathbf{1}_{(-\infty,0)}(A)=D^{-}S(0)\leq
D^{+}S(0)=\operatorname{Tr}B\mathbf{1}_{(-\infty,0]}(A).$ (4.88)

In particular, $S$ is differentiable at $\lambda=0$, if and only if $B|_{\ker
A}=0$.

######  Remarks 4.16.

(1) The relative form trace class assumption implies that the expression on
the right of (4.88), and consequently also that on the left, is finite.

(2) By the variational principle it follows that $S$ is convex. Thus, $S$ has
left and right sided derivatives (cf. [217, Theorem 1.26]).

(3) If $\inf\sigma_{\mathrm{ess}}(A)>0$ or $A-\lambda B$ has only finitely
many negative eigenvalues, then the derivative and the trace can be
interchanged and the result follows from the classical Hellman–Feynman
theorem. The point is that the formulae remain valid even when the bottom of
the essential spectrum is zero, so that perturbation theory is not directly
applicable.

In our application, $A$ is the Chandrasekhar operator $C_{\ell,\gamma}^{H}$
(which has no zero eigenvalue) and $B=U$ in
$L^{2}(\mathbb{R}_{+},\mathrm{d}r)$. Thus, (4.82) would follow from Theorem
4.15, if one could verify its assumptions. By Kato’s inequality (using
$\gamma<2/\pi$), one can replace $A$ by the kinetic energy $C_{\ell}$. In this
case, the relative trace class condition can be formulated explicitly using
the Fourier–Bessel transform. But since $(k+1)^{-1}\notin
L^{1}(\mathbb{R}_{+},\mathrm{d}k)$, this shows that $C_{\ell,\gamma}^{H}$
cannot satisfy the relative trace class assumption, no matter how nice the
perturbation $B=U$ is.

For this reason, we proved and used a generalization of Theorem 4.15, where
the relative trace class assumption is stated with respect to $(A+M)^{2s}$ for
some $s>1/2$. We will state this generalization in Theorem 4.18 in a moment.
Using the following special case of [86, Theorem 1.1], the assumption of
Theorem 4.18 can be recast as an assumption involving $C_{\ell}$ instead of
$C_{\ell,\gamma}^{H}$.

######  Theorem 4.17 ([86, Theorem 1.1]).

Let $2/\pi\geq\gamma\geq 0$, $0<s<3/2-\sigma_{\gamma}$ (with $\sigma_{\gamma}$
as in (4.16)), and $V\in L_{\rm loc}^{1}(\mathbb{R}^{3})$ satisfying
$-\gamma/|x|\leq V\leq 0$. Then we have the quadratic form inequality

$\displaystyle|p|^{2s}\lesssim(|p|+V)^{2s}.$ (4.89)

We are now ready to state the generalization of Theorem 4.15.

######  Theorem 4.18 ([87, Theorem 3.2]).

Assume that $A$ is self-adjoint with $A_{-}$ trace class. Assume that $B$ is
non-negative and relatively form bounded with respect to $A$. Assume that
there are $1/2<s\leq 1$ such that for some $M>-\inf\operatorname{spec}A$,

$(A+M)^{-s}B(A+M)^{-s}\qquad\text{is trace class}$ (4.90)

and

$\limsup_{\lambda\to 0}\left\|(A+M)^{s}(A-\lambda B+M)^{-s}\right\|<\infty.$
(4.91)

Then the conclusions in Theorem 4.15 are valid.

Thus, to conclude the proof of (4.82), we are left with showing the
assumptions of Theorem 4.18. Since the test functions $U$ in the above
formulation of Theorem 4.13 are bounded by a multiple of the Coulomb
potential, the condition (4.91) can be verified easily using Theorem 4.17. We
now verify (4.90) with $A=C_{\ell,\gamma}^{H}$ and $B=U$. Letting
$\|\cdot\|_{2}$ denote the Hilbert–Schmidt norm, we have, for any $1/2<s\leq
1$,

$\displaystyle\|U^{1/2}(C_{\ell,\gamma}^{H}+M)^{-s}\|_{2}^{2}\lesssim\|U^{1/2}(C_{\ell}+M)^{-s}\|_{2}^{2}$
$\displaystyle\quad=\int_{0}^{\infty}\mathrm{d}r\
U(r)\int_{0}^{\infty}\mathrm{d}k\
\frac{krJ_{\ell+1/2}(kr)^{2}}{(\sqrt{k^{2}+1}-1+M)^{2s}}\lesssim\|U\|_{\mathcal{K}_{s}^{(0)}},$

where we used Theorem 4.17 and the Fourier–Bessel transform. Here,

$\displaystyle\|W\|_{\mathcal{K}_{s}^{(0)}}:=\sup_{R\geq
1/2}\left[\int_{0}^{R}\left(\frac{r}{R}\right)^{2s-1}|W(r)|\,\mathrm{d}r+\int_{R}^{\infty}|W(r)|\,\mathrm{d}r\right].$

Clearly, functions $U\in\mathcal{D}$, the test function space (4.13), satisfy
$\|U\|_{\mathcal{K}_{s}^{(0)}}<\infty$. This concludes the sketch of the
arguments in the proof of (4.82).

To prove the convergence of the total density $c^{-3}\rho_{C}(x/c)$, we also
need to interchange the $\ell$-summation with the limits $Z\to\infty$ and
$\lambda\to 0$. This is done using the dominated convergence theorem. To apply
it, it suffices to prove that there is an $\varepsilon>0$ such that

$\displaystyle\operatorname{Tr}(C_{\ell}-V-\lambda
U(r))_{-}-\operatorname{Tr}(C_{\ell}-V)_{-}\lesssim\lambda(\ell+1/2)^{-2-\varepsilon},$
(4.92)

whenever $0\leq V\leq\gamma/r$. In our application, $V$ is closely related to
the Thomas–Fermi potential. For the proof of (4.92), see [87, Section 5].

###  4.4. Molecules without magnetic fields

We immediately present the energetic result for Chandrasekhar molecules on the
Scott scale, which was proved by Solovej et al. [223].

######  Theorem 4.19.

Let $q\in\mathbb{N}$, $\underline{z}=(z_{1},...,z_{K})\in(0,1)^{K}$ with
$\sum_{\kappa=1}^{K}z_{\kappa}=1$ and
$\underline{r}=(r_{1},...,r_{K})\in\mathbb{R}^{3K}$ with
$\min_{k\neq\ell}|r_{k}-r_{\ell}|>r_{0}$ for some $r_{0}>0$. Define
$\underline{Z}=(Z_{1},...,Z_{K})=|\underline{Z}|z$ and
$R=|\underline{Z}|^{-1/3}r$ for $|\underline{Z}|>0$. Let $E^{\rm
TF}(\underline{z},\underline{r})$ be the Thomas–Fermi energy of the
unconstrained problem (2.9) and $E_{c,C}(N,\underline{Z},\underline{R})$ be
the ground state energy with nuclear configuration $\underline{Z}$ and
$\underline{R}$. Then the following statements hold:

  1. (1)

The function

$\displaystyle\left[0,\frac{2}{\pi}\right]\ni\gamma\mapsto\mathcal{S}_{C}(\gamma):=\lim_{\kappa\to
0}\left(\iint\left[\frac{p^{2}}{2}-\frac{1}{|v|}+\kappa\right]_{-}\,\frac{\mathrm{d}p\,\mathrm{d}v}{(2\pi)^{3}}-\operatorname{Tr}[H_{C}(\gamma)+\kappa]_{-}\right)$
(4.93)

with

$\displaystyle
H_{C}(\gamma):=\begin{cases}\sqrt{-\gamma^{-2}\Delta+\gamma^{-4}}-\gamma^{-2}-1/|x|&\quad\text{if}\
\gamma\in(0,2/\pi],\\\ -\frac{1}{2}\Delta-1/|x|&\quad\text{if}\
\gamma=0\end{cases}$ (4.94)

is continuous, monotone decreasing, and satisfies $\mathcal{S}_{C}(0)=1/4$.

  2. (2)

As $|\underline{Z}|=\sum_{\kappa=1}^{K}Z_{\kappa}\to\infty$ and $c\to\infty$
with $\max_{\kappa}\\{Z_{\kappa}/c\\}\leq 2/\pi$, one has

$\displaystyle E_{c,C}(|\underline{Z}|,\underline{Z},\underline{R})=E^{\rm
TF}(\underline{z},\underline{r})|\underline{Z}|^{7/3}+q\sum_{\kappa=1}^{K}Z_{\kappa}^{2}\cdot\mathcal{S}_{C}\left(\frac{Z_{\kappa}}{c}\right)+\mathcal{O}(|\underline{Z}|^{2-1/30}).$
(4.95)

The error term $\mathcal{O}(|\underline{Z}|^{2-1/30})$ means that
$|\mathcal{O}(|\underline{Z}|^{2-1/30})|\lesssim|\underline{Z}|^{2-1/30}$,
where the implicit constant depends only on $r_{0}$ and $K$.

######  Remark 4.20.

The convergence of the one-particle ground state density on the Thomas–Fermi
scale in Coulomb norm can be proved using the argument in the proof of Theorem
4.8 together with Theorem 4.19. The convergence of the density on the Scott
scale has not been worked out so far.

###  4.5. Molecules with self-generated magnetic fields

Suppose that the kinetic energy of the electrons is described by the
Chandrasekhar in presence of a magnetic vector potential, i.e.,

$\displaystyle\mathcal{T}^{(c)}(\underline{A}):=\sqrt{c^{2}T(\underline{A})+c^{4}}-c^{2},$
(4.96)

where $T(\underline{A})$ is either the magnetic Schrödinger or Pauli operator
as in (3.44). We consider $\underline{A}$-fields in

$\displaystyle\tilde{\mathfrak{A}}:=\\{\underline{A}\in
L^{6}(\mathbb{R}^{3}:\mathbb{R}^{3}):\,{\rm
div}(\underline{A})=0,\,|\nabla\otimes\underline{A}|\in
L^{2}(\mathbb{R}^{3})\\}.$ (4.97)

Let

$\displaystyle
C_{N,V,\underline{A}}=\sum_{\nu=1}^{N}\left(\mathcal{T}_{\nu}^{(c)}(\underline{A})-V(x_{\nu})\right)+\sum_{1\leq\nu<\mu\leq
N}\frac{1}{|x_{\nu}-x_{\mu}|}+U\quad\text{in}\
\bigwedge_{\nu=1}^{N}L^{2}(\mathbb{R}^{3}:\mathbb{C}^{2})$ (4.98)

be a Hamilton operator for a relativistic molecule with given vector potential
$\underline{A}$, and $V$ and $U$ as in (1.2) and (1.3), respectively.
(Technically, $C_{N,V,\underline{A}}$ is defined as the Friedrichs extension
of the corresponding quadratic form with form domain
$\mathcal{S}(\mathbb{R}^{3}:\mathbb{C}^{2})$, whenever
$\max_{\kappa}Z_{\kappa}/c\leq\gamma_{C}$.)

For admissible vector potentials $\underline{A}\in\tilde{\mathfrak{A}}$, given
nuclear positions $\underline{R}$, and $N=|\underline{Z}|$ the ground state
energy is

$\displaystyle E_{c,C,{\rm
mag}}(\underline{Z},\underline{R},\underline{A}):=\inf\operatorname{spec}(C_{N,V,\underline{A}}),$
(4.99)

and the total energy including the magnetic field energy is

$\displaystyle E_{c,C,{\rm
mag}}(\underline{Z},\underline{R}):=\inf_{\underline{A}\in\tilde{\mathfrak{A}}}\left(E_{c,C,{\rm
mag}}(\underline{Z},\underline{R},\underline{A})+\frac{c^{2}}{8\pi^{2}}\int_{\mathbb{R}^{3}}|\nabla\times\underline{A}|^{2}\right).$
(4.100)

The following energetic result for the Scott scale was proved by Erdős et al.
[49].

######  Theorem 4.21.

Let the notations and assumptions be as in Theorem 4.19. In particular, fix
$K$, $\underline{z}\in(0,1)^{K}$, and $\underline{r}\in\mathbb{R}^{3K}$.
Assume furthermore that there is $\gamma_{0}<2/\pi$ such that
$\max_{\kappa}Z_{\kappa}/c<\gamma_{0}$. Then

$\displaystyle E_{c,C,{\rm mag}}(\underline{Z},\underline{R})=E^{\rm
TF}(\underline{z},\underline{r})|\underline{Z}|^{7/3}+\sum_{\kappa=1}^{K}Z_{\kappa}^{2}\left(\frac{1}{2}-s_{C}(\gamma)\right)+o(|\underline{Z}|^{2})$
(4.101)

in the limits $|\underline{Z}|,c\to\infty$.

######  Remarks 4.22.

(1) Since $c^{-1}|\underline{Z}|$ is bounded, the prefactor $c^{2}/(8\pi^{2})$
of the magnetic energy in the relativistic case is of order
$|\underline{Z}|^{2}$ (at least if we assume additionally that
$|\underline{Z}|/c$ is kept constant when $|\underline{Z}|,c\to\infty$). This
is much larger than in the non-relativistic case (Theorem 3.18), where
$c^{-2}|\underline{Z}|$ was bounded. Thus, the self-generated magnetic field
has to be much smaller in the relativistic case, which explains why it does
not alter the Scott coefficient (contrary to the non-relativistic situation).
In fact, the prefactor $8\pi$ in front of the field energy is irrelevant and
can be replaced by any other fixed constant in the relativistic situation.

(2) The convergence of the density on the Scott scale has not been worked out
so far.

##  5\. Open questions

We collect some questions that – at least from our perspective – are
interesting both from physical and mathematical points of view.

  1. (1)

For $N\in\mathbb{N}$ and $\underline{Z}\in(0,\infty)^{K}$, we set

$\mathfrak{E}_{S}(N,\underline{Z}):=\inf\\{E_{S}(N,\underline{Z},\underline{R}):\underline{R}\in\mathbb{R}^{3K}\\}.$
(5.1)

This is the ground state energy of a non-relativistic molecule in static
approximation, provided there is a state $\psi$ in the form domain of
$H_{N,V}$ and nuclear positions $R\in\mathbb{R}^{3K}$ such that
$\langle\psi,H_{N,V}\psi\rangle=\mathfrak{E}_{S}(N,\underline{Z})$.

Question: Can one prove the Scott conjecture for
$\mathfrak{E}_{S}(N,\underline{Z})$, i.e., when one minimizes over the nuclear
positions? We expect additivity of the energy up to
$o(|\underline{Z}|^{\frac{5}{3}})$, i.e.,

$\mathfrak{E}_{S}(N,\underline{Z})=\sum_{\kappa=1}^{K}\left(E^{\mathrm{TF}}(Z_{\kappa})+\frac{q}{4}Z_{\kappa}^{2}-C_{\mathrm{DS}}Z^{\frac{5}{3}}_{\kappa}\right)+o(|\underline{Z}|^{\frac{5}{3}}).$
(5.2)

Even more, one might ask whether the strong Scott conjecture would hold around
each of the nuclei if (5.2) is true. A step in this direction was taken by
Iantchenko et al. [117, Theorem 3] under the additional hypothesis that the
minimal nuclear distance is bounded from below, e.g., by $\mathrm{const}\
|\underline{Z}|^{-\frac{1}{4}}$.

The analogue of $\mathfrak{E}_{S}$ can be defined for other many-particle
models discussed in this review, e.g., those in Section 4. As far as we know,
the Scott conjecture is also open for these problems. Again, we expect
additivity of the energy up to order $o(|\underline{Z}|^{5/3})$.

One could also consider a variant of this problem, where the kinetic energy of
the nuclei is taken into account.

  2. (2)

It is folklore in quantum chemistry that chemical accuracy is achieved without
taking a self-generated magnetic field into account. Therefore, we ask whether
in physical models at least the Scott conjecture does not depend on the self-
generated magnetic field.

  3. (3)

The Scott conjectures for the energy and the density are open for no-pair
operators where a self-consistent mean field is taken into account. In this
case the Hilbert space of admissible one-particle states (in the $x\mapsto
x/c$ rescaled picture) is
$\Lambda_{\chi}(L^{2}(\mathbb{R}^{3}:\mathbb{C}^{4}))$ with
$\Lambda_{\chi}:=\mathbf{1}_{(0,\infty)}(D_{\gamma}^{H}+\chi)$ with a (not
necessary local) mean field $\chi$. The resulting no-pair operator is
sometimes called Fuzzy operator. Possible choices for $\chi$ are the
Thomas–Fermi potential (the right sides of (2.11)-(2.12)), or the Hartree–Fock
potential

$\sum_{i}\left(|\varphi_{i}|^{2}\ast\frac{1}{|\cdot|}(x)-\frac{\varphi_{i}(x)\overline{\varphi_{i}(\cdot)}}{|x-\cdot|}\right)$

generated by a set of appropriately chosen orbitals $\\{\varphi_{i}\\}_{i}$.
The latter choice is especially popular in quantum chemistry, see, e.g., [125,
126, 194, 127]. It turns out, though, that numerically computed values of the
ground state energy for Coulomb systems with $Z\gtrsim 90$ in the Fuzzy
picture are quite close to those in the Furry picture; see also [171, 192,
195, 100].

Nevertheless, from a mathematical point of view it is interesting to
investigate the precise value of the Scott correction for the Fuzzy operator.
The following tasks seem natural.

    1. (a)

Show that for any (reasonable?) choice of the mean field $\chi$, the leading
order of the ground state energy is still the Thomas–Fermi energy.

    2. (b)

Show the Scott conjecture for the Fuzzy model defined in the spirit of
Mittleman (minimization of the electronic degrees followed by a maximization
of the splitting).

    3. (c)

Show that, to within the order of accuracy of Scott’s correction, the
maximization is attained in the Furry picture. (Recall that the
$Z^{2}$-correction in the Furry picture is exclusively generated by the
effective one-particle problem involving the hydrogenic operator
$D_{\gamma}^{H}$. Thus, our intuition is supported by the variational
principle for operators with spectral gaps [99, 173], which leads to the
largest eigenvalues of $\Lambda_{\chi}D_{\gamma}^{H}\Lambda_{\chi}$, when one
chooses $\Lambda_{\chi}$ to be the Furry projection. This is also the
underlying spirit of Schwinger’s derivation [198] of the relativistic Scott
correction.)

  4. (4)

Can one show first and second order asymptotics for the Lieb–Loss model [140]?
See Bach and Hach [6] for the first order in the non-relativistic setting.

  5. (5)

Let us formulate some questions regarding the hydrogenic densities
$\rho_{\\#}^{H}$ with $\\#\in\\{C,B,F\\}$. For $\\#=F$ one might be able to
exploit the explicitly known eigenfunctions of the Coulomb–Dirac operator (cf.
[96, 32, 185, 17, 231]) as in Theorem 3.11 by Heilmann and Lieb.

    1. (a)

Inspired by Theorems 4.1 and 4.4, we ask whether the densities
$\rho_{\\#}^{H}$ satisfy a power law at the origin, i.e., whether there are
model-dependent constants $a_{\gamma,\\#},b_{\gamma,\\#}>0$ such that

$\displaystyle\lim_{|x|\to
0}\rho_{\\#}^{H}(x)|x|^{2a_{\gamma,\\#}}=b_{\gamma,\\#}.$ (5.3)

In view of Theorems 4.1 and 4.4 and the fact $|\psi_{n=0,\kappa=\pm
1,m}^{D}(x)|\sim|x|^{-\Sigma_{\gamma}}$ for $m=-j_{\kappa},...,j_{\kappa}$ and
$|x|\ll 1$ (cf. [17, p. 316] or [232, p. 427]), it seems natural to believe
that $a_{\sigma,C}=\sigma_{\gamma}$ (cf. (4.16)) and
$a_{\sigma,F}=\Sigma_{\gamma}$ (cf. (4.40)).

    2. (b)

Electrons far away from the nucleus are expected to behave non-
relativistically. Can one show

$\displaystyle\lim_{|x|\to\infty}|x|^{3/2}\rho_{\\#}^{H}(x)=\lim_{|x|\to
0}|x|^{3/2}\rho_{z=1}^{\rm TF}(x)$ (5.4)

as in Theorem 3.11? (Here $\rho_{z=1}^{\rm TF}$ is the hydrogenic TF density
with $q=2$.) In case $\\#=F$ one might be able to derive an asymptotic
expansion as in (3.30) using the explicitly known eigenfunctions.

    3. (c)

The non-relativistic density $\rho_{S}^{H}(x)$ decreases monotonically in
$|x|>0$. Is this also true for the relativistic hydrogenic densities?

  6. (6)

As we have seen in Theorem 2.10 and the ensuing discussion, Weizsäcker’s
parameter $A$ in the TFW functional (2.23) can be tuned to achieve _either_
energy agreement, _or_ agreement of the TFW density near the origin. This
leads to the following (vague) question of whether one can construct a
modification of Weizsäcker’s gradient term that gives _simultaneously_ the
energy and the density at the origin correctly.

## Acknowledgments

Partial support through U.S. National Science Foundation grant DMS-1954995
(R.L.F.) and by the Deutsche Forschungsgemeinschaft through Germany’s
Excellence Strategy, grant EXC-2111-390814868 (R.L.F.& H.S) is acknowledged.

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