# On the discrete Dirac spectrum of general-relativistic hydrogenic ions with
anomalous magnetic moment

E. B. Kapengut  Department of Mathematics, Rutgers,The State University of New
Jersey, Piscataway, NJ  M. K.-H. Kiessling  Department of Mathematics,
Rutgers,The State University of New Jersey, Piscataway, NJ  E. Ling
Copenhagen Centre for Geometry and Topology (GeoTop), Department of
Mathematical Sciences, University of Copenhagen, Denmark  A. S. Tahvildar-
Zadeh  Department of Mathematics, Rutgers,The State University of New Jersey,
Piscataway, NJ

###### Abstract

The Reissner–Weyl–Nordström (RWN) spacetime of a point nucleus features a
naked singularity for the empirically known nuclear charges $Ze$ and masses
$M=A(Z,N)m_{\mathrm{p}}$, where $m_{\mathrm{p}}$ is the proton mass and
$A(Z,N)\approx Z+N$ the atomic mass number, with $Z$ the number of protons and
$N$ the number of neutrons in the nucleus. The Dirac hamiltonian for a test
electron with mass $m_{\mathrm{e}}$, charge $-e$, and anomalous magnetic
moment $\mu_{a}(\approx-\frac{1}{4\pi}\frac{e^{3}}{m_{\mathrm{e}}c^{2}})$ in
the electrostatic RWN spacetime of such a ‘naked point nucleus’ is known to be
essentially self-adjoint, with a spectrum that consists of the union of the
essential spectrum
$(-\infty,-m_{\mathrm{e}}c^{2}]\cup[m_{\mathrm{e}}c^{2},\infty)$ and a
discrete spectrum of infinitely many eigenvalues in the gap
$(-m_{\mathrm{e}}c^{2},m_{\mathrm{e}}c^{2})$, having $m_{\mathrm{e}}c^{2}$ as
accumulation point. In this paper the discrete spectrum is characterized in
detail for the first time, for all $Z\leq 45$ and $A$ that cover all known
isotopes. The eigenvalues are mapped one-to-one to those of the traditional
Dirac Hydrogen spectrum. Numerical evaluations that go beyond $Z=45$ into the
realm of not-yet-produced hydrogenic ions are presented, too. A list of
challenging open problems concludes this publication.

©The authors. Reproduction, in its entirety, is permitted for non-commercial
purposes.

###### Contents

  1. 1 Introduction
    1. 1.1 The non-relativistic problem
    2. 1.2 The special-relativistic problem
    3. 1.3 The general-relativistic problem
      1. 1.3.1 Informal statement of our main result
    4. 1.4 Outline of the ensuing sections
  2. 2 The Dirac equation of an electron with anomalous magnetic moment in the Reissner–Weyl–Nordström spacetime
    1. 2.1 The state of affairs
    2. 2.2 Dimensionless variables and parameters
    3. 2.3 The Prüfer transformation
    4. 2.4 Conversion to a dynamical system on a compact cylinder
    5. 2.5 Proof of the main result
  3. 3 A Numerical Challenge
  4. 4 Summary and Outlook
    1. 4.1 Summary
    2. 4.2 Outlook
  5. A The reduced Dirac hamiltonian as a linear matrix operator

##  1 Introduction

One of the fundamental axioms of quantum theory stipulates that the quantum
state of an isolated system evolves unitarily. By Stone’s theorem a unitary
time evolution is generated by a unique self-adjoint operator. In quantum
mechanics this operator is known as the hamiltonian of the system. The formal
expressions for such Hamilton operators as suggested by some basic physical
reasoning are rarely a priori self-adjoint, though. Instead they are only
symmetric on some convenient but often too small a domain. In the absence of a
universally accepted principle that would dictate which self-adjoint extension
would capture the physics correctly, physicists face the conceptual conundrum
of having to decide this question case by case — assuming here that a self-
adjoint extension exists in each case.

In the following we discuss this fundamental conceptual issue by focusing on
hydrogenic ions. Since the theory of hydrogenic ions has been such an
important stepping stone in the development of quantum mechanics, it may be
surprising to learn that the problem of choosing the physically correct self-
adjoint hamiltonian is far from being completely settled and understood. Our
main novel mathematical contribution in this paper, as summarized in the
abstract, is meant as one small further step on the way to settle the
hydogenic problem for good. Those readers familiar with this intriguing
background story may now want to skip directly to section 1.3.1 for the
informal statement of our main result, and then move on to the technical
sections of our paper. Everyone else is invited to read the rest of this
introduction and be intrigued by the unfolding story.

Throughout we will be concerned exclusively with the quantum-mechanical
problem of a single point electron in the electrostatic spacetime — which can
be flat or curved — of a fixed point nucleus. Thus we only treat the electron
degrees of freedom quantum-mechanically. The nucleus has charge $Q=Ze$, where
$e>0$ denotes the elementary charge (in Gaussian units), and $Z\in\mathbb{N}$
counts the number of elementary charges carried by the nucleus. The nuclear
ADM mass $M_{\mbox{\tiny{ADM}}}=M>0$; in particular, the proton mass is
denoted $m_{\mathrm{p}}$. Moreover, $m_{\mathrm{e}}$ is the empirical mass of
the electron, and $-e$ its charge; we shall also consider the anti-electron
(positron), having charge $+e$ and mass $m_{\mathrm{e}}$. Beside these
particle parameters, Newton’s constant of universal gravitation $G$, the speed
of light in vacuum $c$, and $\hbar$, Planck’s constant divided by $2\pi$, will
appear in our formulas.

###  1.1 The non-relativistic problem

Consider first the non-relativistic problem, with the Schrödinger hamiltonian

$H=-\frac{\hbar^{2}}{2m_{\mathrm{e}}}\Delta-(Ze^{2}+GMm_{\mathrm{e}})\frac{1}{r},$
(1.1)

acting classically and automorphically on complex-valued functions $\psi\in
D_{\rm min}(H)=C^{\infty}_{c}(\mathbb{R}^{3}\backslash\\{0\\})\subset
L^{2}(\mathbb{R}^{3})$. After a decomposition of the operator (1.1) into its
partial wave operators that act on the invariant subspaces indexed by the
angular momentum eigenvalues $\ell\in\\{0,1,2,...\\}$, an application of
Weyl’s limit point / limit circle criterion (see [39]) reveals that the radial
$\ell=0$ operators are not essentially self-adjoint on their minimal domain
$C^{\infty}_{c}(\mathbb{R}_{+}\backslash\\{0\\})$, but that there exists a
continuously parameterized family of self-adjoint extensions; see Thm.X.11 in
[31]. Their spectra [2], [1] generally differ from the textbook Bohr spectrum,
except for one special parameter value when the self-adjoint extension agrees
with the Friedrichs extension that is used by physicists. Without the
availability of empirical Hydrogen spectral data, the non-relativistic
Schrödinger operator on $C^{\infty}_{c}(\mathbb{R}^{3}\backslash\\{0\\})$
would not in itself be able to suggest the self-adjoint extension that would
capture the physical empiricism correctly.

The continuous family of self-adjoint extensions of (1.1) invokes some
“boundary condition” on the Schrödinger wave function $\psi$ at $r=0$. One
might want to argue that the theory that reproduces the correct physical
spectral data for hydrogenic ions should not depend on such fine details like
boundary conditions at the point nucleus. After all, the fiction of a point
nucleus has been chosen for mathematical convenience precisely for the same
reason: the typical distance between proton and electron in the Hydrogen
ground state is about $10^{5}$ times larger than the empirical size of the
proton, so that the details of the structure of the proton — and, by analogy,
of the nucleus of a hydrogenic ion (at least if not too massive) — should be
irrelevant for the determination of the main spectral structure. In essence
this seems to have been the line of reasoning of Tosio Kato, who in the late
1940s showed [22] that the Hamilton operator (1.1) is essentially self-adjoint
when defined on the dense domain $C^{\infty}_{c}(\mathbb{R}^{3})\subset
L^{2}(\mathbb{R}^{3})$. Its unique self-adjoint extension agrees with the
Friedrichs extension that corresponds to the energy functional conventionally
employed by physicists, and its discrete spectrum is identical to the Bohr
spectrum, as is well known.

Incidentally, everything we wrote about essential self-adjointness for (1.1)
applies equally well when both Coulomb electricity and Newton gravity are
taken into account, as featured explicitly in (1.1), or when only one but not
the other of these interactions is considered. The spectrum of the Friedrichs
extension of (1.1) can be explicitly computed (as done first by Schrödinger
himself [34]), revealing that the discrete spectrum consists of the
eigenvalues

$E_{n}^{\rm
Bohr}=-\frac{1}{2}\frac{(Ze^{2}+GMm_{\mathrm{e}})^{2}m_{\mathrm{e}}}{\hbar^{2}}\frac{1}{n^{2}}$
(1.2)

with $n\in\mathbb{N}$. Now recall that $\frac{GMm_{\mathrm{e}}}{Ze^{2}}\approx
10^{-40}$. This quantitative estimate is usually invoked in elementary atomic
physics textbooks in arguments for why the influence of atomic gravity on the
atomic spectra is immeasurably small for all practical purposes, and thus
negligible.

The practical success of Kato’s reasoning would seem to suggest that the
principle that guides the quest for the physically correct hamiltonian is to
choose the largest possible one of all the self-adjoint extensions. While one
cannot argue with success, as the saying goes, one has to swallow this
principle of the maximal domain of self-adjointness with a grain of salt.
Indeed, while it is plausible, it is not compelling, and is vindicated only by
its practical success. For those physicists who adhere too literally to the
characterization of physics as an experimental science, this empirical success
may be all that matters. Our paper is motivated by the conviction that
mathematically logical coherence of a physical theory is an equally important
ingredient to assess the merits of a physical theory.

In this vein, a manifestly more conceptually compelling principle is the
following.

###  1.2 The special-relativistic problem

Schrödinger’s operator (1.1) is only thought to be a non-relativistic
approximation to a putatively more fundamental relativistic operator.
Considering first only special relativity, the operator in question is the
Dirac operator of hydrogenic ions,

$H=-i\hbar
c\vec{\boldsymbol{\alpha}}\cdot\nabla+m_{\mathrm{e}}c^{2}\boldsymbol{\beta}-Ze^{2}\frac{1}{r}\boldsymbol{1}$
(1.3)

acting on four-component complex bi-spinor wave functions $\Psi$ on the
minimal domain
$\big{(}C^{\infty}_{c}(\mathbb{R}^{3}\backslash\\{0\\})\big{)}^{4}\subset\big{(}L^{2}(\mathbb{R}^{3})\big{)}^{4}$.
As shown first by Weidmann [39], this special-relativistic Dirac hamiltonian
for hydrogenic ions with purely electrical Coulomb interactions is essentially
self-adjoint on
$\big{(}C^{\infty}_{c}(\mathbb{R}^{3}\backslash\\{0\\})\big{)}^{4}$ for $Z\leq
118$; cf. [35]. This covers all empirically known nuclei, which includes not
only those that occur in nature ($Z\leq 92$) but also all those produced
artificially in nuclear reactors or heavy ion collisions.

Thus, the Dirac hamiltonian (1.3), defined on its minimal domain, has a unique
self-adjoint extension when $Z\leq 118$. Moreover, as shown first by Darwin
[10] and Gordon [16], and reproduced many times since with various techniques
(cf. [36], [17]), its spectrum can be computed in closed form. In particular,
the discrete spectrum reproduces the Sommerfeld fine structure formula, i.e.
for $n\in\mathbb{N}$ and $k\in\\{-n,\dots,-1,1,\dots,n-1\\}$ one has

$E_{n,k}^{\rm
Dirac}=\frac{m_{\mathrm{e}}c^{2}}{\sqrt{1+\frac{Z^{2}\alpha_{\mbox{\tiny{S}}}^{2}}{\left(n-|k|+\sqrt{k^{2}-Z^{2}\alpha_{\mbox{\tiny{S}}}^{2}}\right)^{2}}}},$
(1.4)

where

$\alpha_{\mbox{\tiny{S}}}\,=\,\frac{e^{2}}{\hbar c}\approx\frac{1}{137.036}$
(1.5)

is Sommerfeld’s fine structure constant. Armed with these explicit formulas it
is straightforward to take the non-relativistic limit (formally letting the
speed of light $c\to\infty$; for a technically careful procedure see [36]).
One finds that the discrete spectrum (after first subtracting
$m_{\mathrm{e}}c^{2}$) converges to the Bohr spectrum of the Friedrichs
extension of the hydrogenic Schrödinger Hamilton operator. The suggestive
conclusion is that relativity theory takes care of the theoretical self-
adjointness ambiguity that exists at the non-relativistic level of Schrödinger
theory. Indeed, it stands to reason that the non-relativistic limit of an
essentially self-adjoint hydrogenic Dirac hamiltonian with $Z\leq 118$ yields
a more compelling selection principle for the hydrogenic hamiltonian at the
non-relativistic level than the practical success of Kato’s maximal self-
adjoint extension principle.

But what about the regime $Z>118$? And what about gravity? We now address
these two issues one by one, beginning with the special-relativistic large $Z$
regime.

Since empirically $Z\leq 118$ so far, one may not be too alarmed about the
lack of essential self-adjointness of the special-relativistic Dirac operator
for hydrogenic ions with $Z>118$. Be that as it may, physicists for quite some
time already have been trying to create super-heavy ions with $Z>118$ using
heavy ion collisions, in the expectation of finding some novel states of
nuclear matter, and to gain some theoretical insights they have continued to
use the special-relativistic Dirac operator for hydrogenic ions with point
nuclei; cf. [17]. Precisely because there are no empirical data yet that could
guide one’s choice, this regime is perfectly suited to illustrate the dilemma
one faces when essential self-adjointness fails.

Thus, when $119\leq Z\leq 137$, the Dirac operator (1.3) defined on its
minimal domain does have a _distinguished_ self-adjoint extension, defined by
allowing $Z\in\mathbb{C}$ and demanding analyticity in $Z$ in the neighborhood
of $Z=0$ for its self-adjoint extension, as shown by Narnhofer [26]. The Dirac
operator (1.3) defined on its minimal domain is then essentially self-adjoint
for $Z\leq\sqrt{3}/2\alpha_{\mbox{\tiny{S}}}(\approx 118.677)$, and has a
self-adjoint extension that depends on $Z$ analytically for all
$Z\in(0,1/\alpha_{\mbox{\tiny{S}}})$, with $1/\alpha_{\mbox{\tiny{S}}}\approx
137.036$. Yet, is analyticity a compelling distinction? Recall that in
thermodynamics the important phenomenon of a phase transition is associated
with precisely such a point of non-analyticity for the pertinent thermodynamic
potential. Thus, is $Z=118$, or more accurately $Z=\frac{1}{2}\sqrt{3}$,
perhaps the hydrogenic spectral analog of a point of phase transition beyond
which analyticity fails to predict the correct self-adjoint extension? We
believe it’s fair to suspect that nobody knows.

As pointed out in [17], this analytic extension is also the only one for which
all bound states do have finite quantum expected values separately for kinetic
and for potential energy. This is an interesting observation, through which
the analytic self-adjoint extension is doubly distinguished, but is this a
compelling principle to select this self-adjoint extension? And even if one
supposes that for the theory of atomic spectra the principle of analyticity
rules supreme, or the principle that kinetic and potential energy expected
values have to be finite separately, it only gets one to $Z=137$.

A compelling way out of the dilemma was noted first by Behncke [6], and
amplified subsequently by Arnold–Kalf–Schneider [3] and by
Gesztezy–Simon–Thaller [15] (see also [36]). These authors pointed out that
the problem of the lack of essential self-adjointness of (1.3) for when
$Z>118$ goes away if one takes the anomalous magnetic moment $\mu_{a}$ of the
electron into account. This means to replace (1.3) by

$H=-i\hbar
c\vec{\boldsymbol{\alpha}}\cdot\nabla+m_{\mathrm{e}}c^{2}\boldsymbol{\beta}-Ze^{2}\frac{1}{r}\boldsymbol{1}+i\mu_{a}Ze\boldsymbol{\beta}\vec{\boldsymbol{\alpha}}\cdot\nabla\frac{1}{r},$
(1.6)

with $\mu_{a}<0$ [14]. The operator (1.6) on its minimal domain is essentially
self-adjoint for all $Z>0$. This would seem to be a weighty argument for its
physical significance, except for the following caveat: The anomalous magnetic
moment of the electron is conventionally explained as a quantum-
electrodynamical effect, and it is somewhat unsatisfactory if a non-rigorous
perturbative quantum field-theoretical calculation [9] is invoked to rescue
quantum mechanics from one of its conceptual dilemmas.

But now consider this: In magnitude, the leading order contribution to the
anomalous magnetic moment of the electron, according to QED’s expansion in
powers of Sommerfeld’s $\alpha_{\mbox{\tiny{S}}}$ and its logarithm, is the
product of the Bohr magneton
$\mu_{\mbox{\tiny{B}}}=\frac{1}{4\pi}\frac{he}{m_{\mathrm{e}}c}$ and the
factor $\frac{1}{2\pi}\alpha_{\mbox{\tiny{S}}}$, with
$\alpha_{\mbox{\tiny{S}}}$ given in (1.5). More to the point, the anomalous
magnetic moment in highly accurate approximation reads
$\mu_{a}\approx-\mu_{\rm cl}$, with

$\mu_{\rm cl}:=\frac{1}{4\pi}\frac{e^{3}}{m_{\mathrm{e}}c^{2}},$ (1.7)

which is independent of $\hbar$, and in this sense is a classical physics
quantity, named the _classical magnetic moment of the electron_ in [25].
Therefore the Dirac operator (1.6) can be viewed as obtained by quantizing a
classical special-relativistic theory of an electron with charge $-e$ and
anomalous magnetic moment of magnitude $\mu_{\rm cl}$, as expected for a
properly quantum-mechanical theory of hydrogenic ions. This removes the caveat
expressed above.

Having convincingly overcome the self-adjointness problem for this special-
relativistic model of hydrogentic ions with point nuclei, the next question is
whether the spectrum of (1.6) is compatible with empirical data. Since so far
we do not have any data for $Z>118$, for now this viability question can only
be decided by comparing the spectrum of (1.6) with data for when $Z\leq 118$.
We don’t attempt to do this here, but emphasize that in principle it is a
well-defined problem.

On the theoretical side the following contributions towards answering the
above viability question have so far been made:

An important analytical result has been contributed by Kalf and Schmidt [21],
and completed by Schmidt [33]. They noted that when $Z\leq 118$ (or rather
$Z\leq\frac{\sqrt{3}}{2}\frac{1}{\alpha_{\mbox{\tiny{S}}}}$), then (1.6)
converges in strong resolvent sense to (1.3) in the limit $\mu_{a}\to 0$. This
alone does not imply convergence of the point spectra. They then went on to
prove that when $Z\leq 137$ (or rather
$Z<\frac{1}{\alpha_{\mbox{\tiny{S}}}}$), then the eigenvalues of the unique
self-adjoint operator with anomalous magnetic moment converge to those of the
distinguished self-adjoint extension of the one without. Thus, if $|\mu_{a}|$
is small enough, then the spectral results will differ only ever so slightly.

But how small is “small enough”? In particular, what about spectral results
for the physical $\mu_{a}$ given in (1.7)? In the physics literature one
traditionally finds non-rigorous perturbative treatments; cf. [9]. A non-
perturbative numerical evaluation of a handful of low-lying eigenvalues has
been carried out by the Thallers, see [36] (section 7, and the dedication) and
[37], though unfortunately without revealing the details of how the numerical
evaluation has been accomplished. Numerically they compared the electron’s
lowest eigenvalue of (1.3) with that of (1.6). Their eigenvalue plot (see Fig.
7.1 in [36]) suggests that these lowest eigenvalues must be barely
distinguishable for all naturally occurring $Z$, while significant
discrepancies become visible when $Z>118$; the lowest eigenvalue of (1.3)
terminates when $Z\nearrow 137.036$, while its counterpart for (1.6) continues
to exist until it meets the previously second-lowest eigenvalue at about
$Z\approx 145$ with value $E\approx 0$. There a cross-over happens, and both
eigenvalues continue to decrease until they meet the negative continuum
roughly when $Z\approx 147$, resp. $Z\approx 160$. We have not seen any
confirmation of these numerical results by other authors. This is one of the
open issues that we address in our paper, as the special-relativistic limit of
our main inquiry.

###  1.3 The general-relativistic problem

While one may be inclined to dismiss the $Z>118$ problem as of rather
academical interest, gravity certainly acts between the electron and the
nucleus in all known hydrogenic ions, and although the non-relativistic
estimates suggest that gravity makes practically immeasurably small
contributions only, conceptually it is mandatory to include this force of
nature in the model. Its absence from the special-relativistic Dirac operator
for hydrogenic ions means that the essential self-adjointness of the latter
for $Z\leq 118$ (in the absence of the $\mu_{a}$ term) vindicates only the
Friedrichs extension of the Schrödinger operator (1.1) (that does not feature
$\mu_{a}$) with electrical Coulomb but no gravitational Newton interactions
(for $Z\leq 118$, although for $Z$ that large the Schrödinger operator is not
capturing the physics accurately any longer). To address this deficiency in a
conceptually principled manner one has to start from the general-relativistic
Dirac operator of hydrogenic ions.

The direct general-relativistic counterpart to both, Schrödinger’s (1.1) and
Dirac’s (1.3), is the Dirac hamiltonian extracted from Dirac’s equation for a
point electron in the Reissner–Weyl–Nordström (RWN) spacetime of a point
nucleus.

The RWN spacetime [41], [32], [27] is a static, spherically symmetric
Lorentzian manifold with two real parameters: the ADM mass $M\geq 0$ and
charge $Q$; for hydrogenic ions, $Q=Ze>0$. Using the $(+,-,-,-)$ signature,
the line element corresponding to the Lorentzian metric is

$ds^{2}\,=\,f(r)^{2}c^{2}dt^{2}-\frac{1}{f(r)^{2}}dr^{2}-r^{2}d\Omega^{2},$
(1.8)

where $d\Omega^{2}$ is the usual round metric on $\mathbb{S}^{2}$ and

$f(r)^{2}\,=\,{1-\frac{G}{c^{2}}\,\frac{2M}{r}+\frac{G}{c^{4}}\,\frac{Z^{2}e^{2}}{r^{2}}}\,.$
(1.9)

All known nuclei are in the naked singularity sector, $GM^{2}<Z^{2}e^{2}$; in
fact, we have $GM^{2}\ll Z^{2}e^{2}$ by more than thirty orders of magnitude.
In this case, $f(r)^{2}$ is strictly positive for all $r>0$, and we stipulate
$f(r)$ to be its positive square root. There is a curvature singularity at
$r=0$; consequently, the maximally extended analytic manifold is
${{\mathcal{M}}}=\mathbb{R}\times(0,\infty)\times\mathbb{S}^{2}$.

The Dirac equation for a point electron in the RWN spacetime reads

$\tilde{\gamma}^{\mu}(i\hbar c\nabla_{\mu}-eA_{\mu})\Psi-
m_{\mathrm{e}}c^{2}\Psi\,=\,0.$ (1.10)

Here, $\Psi\colon{{\mathcal{M}}}\to\mathbb{C}^{4}$ is a bi-spinor field on the
spacetime manifold, $\nabla_{\mu}$ is the covariant derivative acting on bi-
spinors, and $\tilde{\gamma}^{\mu}$ are the Dirac matrices that satisfy the
anti-commutation relations with respect to the Lorentzian metric (1.8), (1.9).
The Dirac matrices generate the complexified Clifford algebra
$\text{Cl}_{1,3}(\mathbb{R})_{\mathbb{C}}$ over each tangent space
$T_{p}{{\mathcal{M}}}$ on the manifold. Finally, $A=-\frac{Q}{r}cdt$ is the
one-form of the electromagnetic potential of the RWN spacetime, which is
purely electrostatic. In the $(x^{0}=ct,r,\theta,\phi)$ coordinates, the only
nonzero component of $A$ is $A_{0}=-\frac{Q}{r}$. The quantum-mechanical Dirac
operator $H$ is now extracted from (1.10) by rewriting this Dirac equation
into the abstract Schrödinger format $i\hbar\partial_{t}\Psi=H\Psi$.

It came as a big surprise when in 1982 Cohen and Powers [8] proved that this
general-relativistic Dirac operator for hydrogenic ions is not essentially
self-adjoint on its minimal domain but has a continuously parameterized family
of self-adjoint extensions, each one (presumably) with its own discrete
spectrum. It seems that general relativity throws us back to the same self-
adjointness ambiguity that plagues the non-relativistic Schrödinger problem
(1.1). Worse, since the Dirac operator is unbounded above and below, one now
cannot invoke a Friedrichs extension as a putatively distinguished self-
adjoint extension. Einstein’s gravity clearly is not a weak perturbation of
Coulomb electricity in the sense that Newton’s gravity is — it has devastating
effects on the quantum-mechanical theory of all the hydrogenic spectra,
Hydrogen itself included.

About 20 years later, Belgiorno, Martinelli, and Baldicchi [7] showed that the
replacement of (1.10) with the Dirac equation that includes a term for the
electron’s anomalous magnetic moment coupled to the gradient of the Coulomb
field (see next section) restores essential self-adjointness — though only if
the magnitude of the anomalous magnetic moment is larger than the critical
value $\mu_{\rm crit}$ given by

$\mu_{\rm crit}:=\frac{3}{2}\frac{\hbar\sqrt{G}}{c}.$ (1.11)

Put differently, the essential self-adjointness of the special-relativistic
Dirac operator for hydrogenic ions with any $Z>0$ when the electron has both
charge and anomalous magnetic moment is not structurally stable against the
switching on of Einstein’s gravity, unless the anomalous magnetic moment is
larger than the critical value (1.11) in magnitude. Therefore, even if the
anomalous magnetic moment of the electron is taken into account, technically
speaking Einstein’s gravity is not generally a weak perturbation of the
special-relativistic hydrogenic ion problem but can have devastating effects
on the spectra, Hydrogen included. Whether it does or does not have such
devastating effects on the problem with physical parameter values depends on
whether the empirical anomalous magnetic moment is larger than the critical
value or not.

Comparison of (1.7) with (1.11) reveals that the empirical value of the
anomalous magnetic moment is about $10^{18}$ times larger than the theoretical
critical value. Therefore, essential self-adjointness of this general-
relativistic Dirac operator for a physically relevant model of hydrogenic ions
with empirically accurate anomalous magnetic moment of the electron has been
established. This holds for all $Z>0$; though, for $Z>118$ we leave the
empirically confirmed realm of nuclei, as noted earlier.

####  1.3.1 Informal statement of our main result

For this essentially self-adjoint general-relativistic Dirac operator, the
authors in [7] conclude that the discrete spectrum is infinite. The main
purpose of the present paper is to supply a complete characterization of the
discrete spectrum of the Dirac hamiltonian for a point electron with larger-
than-critical anomalous magnetic moment in the RWN spacetime of a point
nucleus. Our main result, Theorem 2.23, states that the discrete spectrum is
indexed by two integers $k$ and $N$, where $k$ is the eigenvalue of the spin-
orbit operator, and $N$ can be identified with the winding number of
heteroclinic orbits of a certain dynamical system on a compact cylinder that
connect a saddle point on one cylinder end to a saddle point on the other.
This identification allows us to find a one-to-one correspondence between the
eigenfunctions of our hamiltonian and the orbitals of Hydrogen found in
textbook quantum mechanics. The correspondence allows us to relate $N$ and $k$
to the principal quantum number $n$ by defining $n:=N+|k|$. Section 4 of [23]
supplies details on this correspondence.

###  1.4 Outline of the ensuing sections

Section 2 is our main technical section. After a recap of the state of affairs
concerning the Dirac equation of a point electron with anomalous magnetic
moment in the Reissner–Weyl–Nordström spacetime in section 2.1, we pave the
ground for our approach by introducing suitable dimensionless variables and
parameters in section 2.2, then invoke the Prüfer transform in section 2.3 to
map the reduced Dirac eigenvalue problem into a dynamical system on a half-
infinite cylinder. In section 2.4 we compactify the dynamical system by
mapping it to a finite cylinder. Section 2.5 is devoted to the discussion of
the compactified dynamical system, establishing a detailed characterization of
its orbits, culminating in our main Theorem 2.23. Section 3 features
numerically computed eigenvalue branches as functions of $Z$, suggesting that
the spectrum of an electron with anomalous magnetic moment in the
Reissner–Weyl–Nordström spacetime is practically indistinguishable from its
counterpart in Minkowski spacetime equipped with a point nucleus. We also
emphasize the numerical challenge to resolve the differences in a satisfactory
manner. Our appendix A contains pertinent supplementary material. We conclude
with a summary and outlook in section 4.

##  2 The Dirac equation of an electron with anomalous magnetic moment in the
Reissner–Weyl–Nordström spacetime

###  2.1 The state of affairs

The RWN spacetime, with metric given by (1.8) and (1.9), is a solution to the
Einstein–Maxwell(–Maxwell)111The “second Maxwell” refers to his law of the
electromagnetic vacuum, while the “first Maxwell” stands for the pre-metric
Maxwell equations. In our last section we will comment on other
electromagnetic vacua, with the “second Maxwell” replaced accordingly.
equations with electromagnetic Faraday tensor $F=dA$, where $A$ is the one-
form $A=-\frac{Q}{r}cdt$. Therefore, in the conventional
$(x^{0}=ct,r,\theta,\phi)$ coordinates, the only nonzero component of $A$ is
$A_{0}=-\frac{Q}{r}$. Hence, the only nonzero components of $F$ are
$F_{0r}=-F_{r0}=\frac{Q}{r^{2}}$.

In an electromagnetic spacetime, the Dirac equation for a point electron with
mass $m_{\mathrm{e}}$, charge $-e$, and anomalous magnetic moment $\mu_{a}$ is
given by (note that each term acting on $\Psi$ has units of energy)

$\tilde{\gamma}^{\mu}(i\hbar
c\nabla_{\mu}-eA_{\mu})\Psi+\tfrac{1}{2}\mu_{a}\tilde{\sigma}^{\mu\nu}F_{\mu\nu}\Psi-
m_{\mathrm{e}}c^{2}\Psi\,=\,0.$ (2.12)

The anomalous magnetic moment term,
$\frac{1}{2}\mu_{a}\tilde{\sigma}^{\mu\nu}F_{\mu\nu}$, is the covariant
generalization of the corresponding term in flat spacetime, see section 4.2.3
in [36]. This is accomplished by setting
$\tilde{\sigma}^{\mu\nu}=i\tilde{\gamma}^{\mu}\tilde{\gamma}^{\nu}$. In the
following, we leave $\mu_{a}$ as a parameter. Physically, the anomalous
magnetic moment is measured in multiples of the Bohr magneton:
$\mu_{a}=-a\frac{e\hbar}{2m_{\mathrm{e}}c}$, where $a$ is dimensionless. The
first perturbative term in QED in flat spacetime gives
$a=\frac{1}{2\pi}\alpha_{\mbox{\tiny{S}}}\approx 0.00116$; cf. our discussion
in the introduction.

By multiplying the Dirac equation (2.12) by $\gamma^{0}$, we can recast it
into a hamiltonian form

$i\hbar\partial_{t}\Psi\,=\,H\Psi.$ (2.13)

Due to the static nature of the spacetime, the operator $H$ does not depend on
time.

For each time $t$, let $\S_{t}$ denote the hypersurface of constant $t$ within
the $(t,r,\theta,\phi)$ coordinates. We define an inner product on $\S_{t}$
via

$(\Psi,\Phi)_{\Sigma_{t}}\,=\,\int_{\Sigma_{t}}\overline{\Psi}\tilde{\gamma}^{\mu}\Psi
n_{\mu}dV_{\Sigma_{t}}.$ (2.14)

Here $\overline{\Psi}$ is the conjugate bispinor defined as
$\overline{\Psi}:=\Psi^{\dagger}\gamma^{0}$, $n$ is the unit future directed
timelike normal to $\Sigma_{t}$, and $dV_{{\Sigma}_{t}}$ is the volume form on
$\Sigma_{t}$ induced from its Riemannian metric. This choice of inner product
can be motivated from an action principle. It can also be motivated from the
conservation of current density, i.e. $\nabla_{\mu}j^{\mu}=0$ where
$j^{\mu}=\overline{\Psi}\tilde{\gamma}^{\mu}\Psi$.

Thus, for each $t$, we have a Hilbert space

$\mathcal{H}_{t}\,=\,\\{\Psi\colon
M\to\mathbb{C}^{4}\,\mid\,(\Psi,\Phi)_{\Sigma_{t}}<\infty\\}.$ (2.15)

The operator $H$ in (2.13) on the Hilbert space $\mathcal{H}_{t}$ is called
the _Dirac hamiltonian_ on $\S_{t}$.

The spherical symmetry present within the Lorentzian metric and the
electromagnetic field allows us to perform a separation of variables and
decompose the Hilbert space $\mathcal{H}_{t}$ into a direct sum of partial
wave subspaces [8], see also [36, Sec. 4.6]. The partial wave subspaces are
indexed by a triplet $(j,m_{j},k_{j})$ where

$j\,=\,\frac{1}{2},\frac{3}{2},\frac{5}{2}\dotsc;\quad
m_{j}\,=\,-j,-j+1,\dotsc,j-1,j;\quad k_{j}\,=\,\pm\left(j+\frac{1}{2}\right).$
(2.16)

These correspond to certain eigenvalues of well-known operators on
$L^{2}(\mathbb{S}^{2})^{4}$, see [36, p. 126]. The Dirac hamiltonian $H$
acting on each partial wave subspace is given by a two-dimensional _reduced
hamiltonian_ (compare with [36, eq. (5.46)]):

$H_{k_{j}}\,=\,\begin{pmatrix}m_{\mathrm{e}}c^{2}f(r)-\frac{eQ}{r}&-\hbar
cf^{2}(r)\partial_{r}+\hbar cf(r)\frac{k_{j}}{r}+\mu_{a}f(r)\frac{Q}{r^{2}}\\\
\hbar cf^{2}(r)\partial_{r}+\hbar
cf(r)\frac{k_{j}}{r}+\mu_{a}f(r)\frac{Q}{r^{2}}&-m_{\mathrm{e}}c^{2}f(r)-\frac{eQ}{r}\end{pmatrix}$
(2.17)

which acts on the Hilbert space

$L^{2}\big{(}(0,\infty),f(r)^{-2}dr\big{)}^{2}.$ (2.18)

For a derivation of this form of the reduced hamiltonian, see [8, p. 76] (see
also [4]). Although each partial wave subspace depends on the triplet
$(j,m_{j},k_{j})$, the reduced hamiltonian $H_{k_{j}}$ only depends on the
integer $k_{j}$. (The independence of the reduced hamiltonian on $m_{j}$ is
the reason for the degeneracy of the different spin states within the Hydrogen
atom; it’s a consequence of spherical symmetry.)

The Dirac hamiltonian $H$ on $\Sigma_{t}$ is essentially self-adjoint on the
domain of $C^{\infty}$ bispinors compactly supported away from the origin
$r=0$ if and only if each reduced hamiltonian $H_{k_{j}}$ is essentially self-
adjoint on $C^{\infty}_{c}(0,\infty)^{2}$, in which case, the spectrum of the
corresponding self-adjoint operator (still denoted by $H$) is the union of the
spectra of the self-adjoint operators for the reduced hamiltonians (still
denoted by $H_{m_{j},k_{j}}$). For a proof, see [36, Lem. 4.15] which
generalizes to our setting. This reduces the problem to finding the spectrum
for each reduced hamiltonian (hence the name).

The following theorem was proved in [7]. It says that essential self-
adjointness is guaranteed for each reduced hamiltonian $H_{k_{j}}$ provided
the anomalous magnetic moment is not too small.

######  Theorem 2.1 (Belgiorno–Martellini–Baldicchi[7]).

Each reduced hamiltonian $H_{k_{j}}$ is essentially self-adjoint on
$C^{\infty}_{c}(0,\infty)^{2}$ if and only if
$|\mu_{a}|\geq\frac{3}{2}\frac{\sqrt{G}\hbar}{c}$. In which case, the
following hold for the self-adjoint Dirac hamiltonian $H$:

  * _(a)_

The essential spectrum of $H$ is
$(-\infty,-m_{\mathrm{e}}c^{2}]\cup[m_{\mathrm{e}}c^{2},\infty)$;

  * _(b)_

the purely absolutely continuous spectrum of $H$ is
$(-\infty,-m_{\mathrm{e}}c^{2})\cup(m_{\mathrm{e}}c^{2},\infty)$;

  * _(c)_

the singular continuous spectrum of $H$ is empty;

  * _(d)_

$H$ has infinitely many eigenvalues in the gap
$(-m_{\mathrm{e}}c^{2},m_{\mathrm{e}}c^{2})$ of its essential spectrum.

######  Remark 2.2.

_Recall that the empirical anomalous magnetic moment is
$\mu_{a}=-a\frac{e\hbar}{2m_{\mathrm{e}}c}$, where $a$ is dimensionless, see
[14]. In first perturbative order, QED in flat spacetime gives
$a=\frac{1}{2\pi}\alpha_{\mbox{\tiny{S}}}\approx 0.00116$. With this
_classical value_ for the anomalous magnetic moment of the electron, the
hurdle for essential self-adjointness is overwhelmingly cleared:_

$\hskip 28.45274pt\frac{3}{2}\frac{\sqrt{G}\hbar}{c}\frac{1}{|\mu_{\rm
cl}|}\,=\,6\pi\frac{\sqrt{G}m_{\mathrm{e}}}{\alpha_{\mbox{\tiny{S}}}e}\,\approx\,1.27\cdot
10^{-18}\,\ll\,1.\hskip 28.45274pt\square$

Theorem 2.1(d) implies that the discrete spectrum of the Dirac hamiltonian on
the RWN spacetime has infinitely many eigenvalues. In this paper, we
rigorously investigate the discrete spectrum and classify it. We also wish to
compare it to the the discrete spectrum of the corresponding special
relativistic Hydrogen ion problem. A naturally tempting conjecture is that the
discrete spectrum of the Dirac equation for an electron with anomalous
magnetic moment in the RWN spacetime will converge to the discrete spectrum of
the corresponding special-relativistic problem in the limit when Newton’s
$G\to 0$, given realistic values for the electron parameters. While we will
not attempt to prove this conjecture in the present paper, we have carried out
a numerical study of the eigenvalue problem which supports this conjecture.

###  2.2 Dimensionless variables and parameters

The conventional Gaussian units (or, for that matter, also the modern SI
units) come equipped with numerical values that obscure the understanding of
the Dirac equation for hydrogenic ions more than they illuminate it. It is
advisable to switch to dimensionless quantities by choosing reference units
that are more suitable to the atomic realm.

Some choices of reference quantities are canonical.

In particular, it is self-evident that the elementary charge $e$ should serve
as the unit of charge (and not the unit _Coulomb_ or the older _electrostatic
unit_). The ratio $\frac{Q}{e}=:Z$ then becomes the dimensionless charge of
the nuclei in atoms or ions. Thus, $Z\in\mathbb{N}$ for the physical nuclei
($Z$ in a subset of $\mathbb{N}$, to be accurate), yet sometimes we allow a
real $Z>0$ for mathematical convenience.

Similarly, the mass of the electron, $m_{\mathrm{e}}$, as the smallest mass of
the participating particles in the problem is a canonical choice for the unit
of mass (and not the _kilogram_ or the _gram_). The dimensionless nuclear mass
number $A(Z,N)$, and the dimensionless ratio of proton mass $m_{\mathrm{p}}$
to the electron mass $m_{\mathrm{e}}$, jointly give
$\frac{M}{m_{\mathrm{e}}}=A(Z,N)\frac{m_{\mathrm{p}}}{m_{\mathrm{e}}}$; here,
$N$ is the number of neutrons in the nucleus. Empirically,
$\frac{m_{\mathrm{p}}}{m_{\mathrm{e}}}\approx 1836$, while $A(Z,N)$ does not
seem to have a simple explicit formula; note, though, that for empirically
confirmed nuclei one has the bounds $Z\leq A(Z,N)<Z+N<3Z$. For mathematical
convenience, we may treat $\frac{M}{m_{\mathrm{e}}}$ as a positive real
parameter, though.

Since relativity theory says that each mass $m$ is associated with an energy
$mc^{2}$, it is now a natural choice to use $m_{\mathrm{e}}c^{2}$ as reference
unit for energy $E$. We set $\frac{E}{m_{\mathrm{e}}c^{2}}=:\varepsilon$.

In a relativistic theory it is also natural to use $ct$ as timelike basic
variable rather than $t$, and to use the same reference length as unit for the
timelike and for the space variables. There seems no canonical choice, though,
for this reference length, not even for a theory of the atomic realm. For
instance, in the atomic physics and chemistry literature one often finds the
choice of the Bohr radius of hydrogen, $a_{\rm
Bohr}:=\frac{1}{\alpha_{\mbox{\tiny{S}}}}\frac{\hbar}{m_{\mathrm{e}}c}$, as
reference unit. Since chemistry and atomic physics is historically concerned
with the nature of, well, atoms and molecules, and at most moderately charged
ions, this is a convenient choice of reference length. Yet, in those atomic
units the critical size of a hydrogenic ion where the analytical self-adjoint
extension of the special-relativistic Dirac operator (without anomalous
magnetic moment term) ends is a factor $\frac{1}{137}$ smaller than the
reference length (the radius of Hydrogen in such atomic units). For an
investigation like ours where precisely such critical situations take center
stage, it is more meaningful to have this critical size as reference length.
More explicitly, the reference length should be the reduced Compton length of
the electron, $\frac{\hbar}{m_{\mathrm{e}}c}$, and the unit of time $t$ is the
time it takes light to cover the reference length, viz.
$\frac{\hbar}{m_{\mathrm{e}}c^{2}}$. In such units the narrative then becomes
that in Hydrogen the separation of the electron from the nucleus is much
larger than the critical distance, and therefore on the safe side; the fact
that “much larger” means “137 times larger” is then only a matter of detail.

Finally, there are some dimensionless parameters to agree upon. It is self-
evident that Sommerfeld’s fine structure constant
$\alpha_{\mbox{\tiny{S}}}=\frac{e^{2}}{\hbar c}$ is a canonical dimensionless
parameter to work with. In atomic physics it plays the role of a dimensionless
electrical coupling strength between two particles that carry one elementary
charge each, for instance: an electron and a proton, or two electrons.
Similarly, there is an analogous dimensionless quantity that may be called the
gravitational fine structure constant of the electron,
$\frac{Gm_{\mathrm{e}}^{2}}{\hbar c}=:\alpha_{G}$, that plays the role of a
basic gravitational coupling strength between two electrons; although
hydrogenic ions posses only a single electron, any other atom or ion has at
least two electrons, so this is a physically meaningful quantity. With the
help of the dimensionless mass ratios $\frac{M}{m_{\mathrm{e}}}$ this
gravitational reference coupling strength can be converted into the
dimensionless gravitational coupling strengths of an electron and a nucleus.

Of importance is also the ratio of gravitational to electrical coupling
constants, which is the incredibly small dimensionless number

$\mathcal{G}\,:=\,\frac{Gm_{\mathrm{e}}^{2}}{e^{2}}\,\approx\,2.40\cdot
10^{-43}.$ (2.19)

When we will speak of the limit of vanishing gravity, we technically mean
sending the parameter $\mathcal{G}\searrow 0$.

After these preparations, we can convert the reduced hamiltonian (2.17) into a
dimensionless format as follows.

  1. 1.

We make the replacement $r\mapsto\frac{\hbar}{m_{\mathrm{e}}c}r$. Thus $r$ is
rendered dimensionless and measures distance as multiple of the electron’s
Compton wavelength. As a result, $f(r)$ now reads

$f(r)\,=\,\sqrt{1-2\frac{GMm_{\mathrm{e}}}{\hbar
c}\,\frac{1}{r}+\frac{Gm_{\mathrm{e}}^{2}Q^{2}}{(\hbar
c)^{2}}\,\frac{1}{r^{2}}}.$ (2.20)

  2. 2.

Recalling that $Q=Ze$ and $M=A(Z,N)m_{\mathrm{p}}$, we define the rescaled
dimensionless quantities $A_{*}\geq 0$ and $Z_{*}>0$ via

$\displaystyle A_{*}\,$ $\displaystyle:=\,\frac{GMm_{\mathrm{e}}}{\hbar
c}\,=\,\frac{Gm_{\mathrm{e}}^{2}}{e^{2}}\left(\frac{e^{2}}{\hbar
c}\right)\frac{m_{\mathrm{p}}}{m_{\mathrm{e}}}A\,=\,\mathcal{G}\alpha_{\mbox{\tiny{S}}}\frac{m_{\mathrm{p}}}{m_{\mathrm{e}}}A,$
(2.21) $\displaystyle Z_{*}\,$
$\displaystyle:=\,\frac{\sqrt{G}m_{\mathrm{e}}Q}{\hbar
c}\,=\,\sqrt{\frac{Gm_{\mathrm{e}}^{2}}{e^{2}}}\left(\frac{e^{2}}{\hbar
c}\right)Z\,=\,\sqrt{\mathcal{G}}\alpha_{\mbox{\tiny{S}}}Z.$ (2.22)

######  Remark 2.3.

_Our quantity $A_{*}$ has been called the gravitational fine structure
constant for the interaction of a particle of mass $M$ with one of mass
$m_{\mathrm{e}}$, see [20], but we prefer to reserve the name gravitational
fine structure constant for that quantity when $M=m_{\mathrm{e}}$. $\square$_

Using $A_{*}$ and $Z_{*}$, $f(r)$ is simply

$f(r)\,=\,\sqrt{1-\frac{2A_{*}}{r}+\frac{Z_{*}^{2}}{r^{2}}}\,,$ (2.23)

and the condition for a naked singularity becomes $A_{*}<Z_{*}$.

The asymptotic behavior of $1/f^{2}$ will be useful, so we record it here:

$\frac{1}{f^{2}(r)}\,=\,\left\\{\begin{array}[]{ll}Z_{*}^{-2}r^{2}+O(r^{3})&\text{
as }r\to 0\\\ &\\\ 1+O(r^{-1})&\text{ as }r\to\infty.\end{array}\right.$
(2.24)

  3. 3.

We relabel $k_{j}$ as $k$ and define the operator
$K_{k}:=\frac{1}{m_{\mathrm{e}}c^{2}}H_{k_{j}}$.

  4. 4.

In order to facilitate the comparison of our results with those in [36], we
also introduce $\gamma:=-Z\frac{e^{2}}{\hbar c}=-Z\alpha_{\mbox{\tiny{S}}}$.
Note that $Z_{*}=-\sqrt{\mathcal{G}}\gamma$.

  5. 5.

Recalling that $\mu_{a}=-a\frac{e\hbar}{2m_{\mathrm{e}}c}$, with
$a=\frac{\alpha_{\mbox{\tiny{S}}}}{2\pi}$ to first order in perturbative flat
spacetime QED, where $\alpha_{\mbox{\tiny{S}}}$ is Sommerfeld’s fine structure
constant, we now define $\lambda:=-\frac{1}{2}a\gamma$. With our choice of $a$
we have $\lambda=\frac{1}{4\pi}Z\alpha_{\mbox{\tiny{S}}}^{2}$. Yet, for
mathematical convenience, we simply assume $\lambda>0$ for the rest of this
paper.

  6. 6.

For $E\in(-m_{\mathrm{e}}c^{2},m_{\mathrm{e}}c^{2})$, define
$\varepsilon:=\frac{E}{m_{\mathrm{e}}c^{2}}$. Hence $\varepsilon\in(-1,1)$ is
dimensionless. In some of the following proofs, $\varepsilon=\pm 1$ will be
considered even though it is not part of the discrete spectrum by Theorem 2.1.

With this new notation, the operator $K_{k}$ becomes

$K_{k}\,=\,\begin{pmatrix}f(r)+\frac{\gamma}{r}&-f^{2}(r)\partial_{r}+f(r)\frac{k}{r}-f(r)\frac{\lambda}{r^{2}}\\\
f^{2}(r)\partial_{r}+f(r)\frac{k}{r}-f(r)\frac{\lambda}{r^{2}}&-f(r)+\frac{\gamma}{r}\end{pmatrix}.$
(2.25)

######  Remark 2.4.

_Note that $f(r)$, given in (2.23), is the only place in $K_{k}$ where
Newton’s gravitational constant $G$, or for that matter its dimensionless
version $\mathcal{G}$, enters, through $A_{*}$ and $Z_{*}$. Formally, the
$\mathcal{G}\to 0$ limit yields $f(r)\to 1$ for $r>0$, and the reduced
hamiltonian (2.25) defined on its minimal domain
$C^{\infty}_{c}(0,\infty)^{2}$ becomes the reduced hamiltonian in the
corresponding special-relativistic problem, see [36, eq. (7.167))]. $\square$
_

###  2.3 The Prüfer transformation

The eigenvalue problem for the reduced hamiltonian yields a coupled system of
differential equation. In this section, we show how to uncouple them via a
Prüfer transformation. This technique has been recognized in the past to help
numerically solve eigenvalue problems for Dirac operators [42].

Note that $E$ is an eigenvalue for $H_{m_{j},k_{j}}$ if and only if
$\varepsilon$ is an eigenvalue for $K_{k}$, where $K_{k}$ is given by (2.25).
The eigenvalue problem

$K_{k}\begin{pmatrix}u\\\ v\end{pmatrix}\,=\,\varepsilon\begin{pmatrix}u\\\
v\end{pmatrix}$ (2.26)

yields the following coupled system of first order linear differential
equations

$\displaystyle
u^{\prime}(r)+\frac{1}{f}\left(\frac{k}{r}-\frac{\lambda}{r^{2}}\right)u(r)-\frac{1}{f^{2}}\left(f-\frac{\gamma}{r}+\varepsilon\right)v(r)\,=\,0$
(2.27) $\displaystyle
v^{\prime}(r)-\frac{1}{f}\left(\frac{k}{r}-\frac{\lambda}{r^{2}}\right)v(r)-\frac{1}{f^{2}}\left(f+\frac{\gamma}{r}-\varepsilon\right)u(r)\,=\,0.$
(2.28)

To uncouple this system, we introduce the variables $R(r)$ and $\Omega(r)$ via
a Prüfer transform:

$R(r)\,:=\,\sqrt{u^{2}(r)+v^{2}(r)}\quad\text{ and
}\quad\Omega(r)\,:=\,2\arctan\left(\frac{u(r)}{v(r)}\right),$ (2.29)

where $R\in[0,\infty)$ and $\Omega\in(-\pi,\pi]$. Then

$u(r)\,=\ R\cos\left(\frac{1}{2}\Omega\right)\quad\text{ and }\quad
v(r)\,=\,R\sin\left(\frac{1}{2}\Omega\right).$ (2.30)

Summing $\eqref{eq: du/dr}\cdot u$ and $\eqref{eq: dv/dr}\cdot v$ gives

$\frac{R^{\prime}}{R}\,=\,\frac{1}{f}\sin\Omega+\frac{1}{f}\left(-\frac{k}{r}+\frac{\lambda}{r^{2}}\right)\cos\Omega.$
(2.31)

Likewise, summing $\eqref{eq: dv/dr}\cdot u$ and $-\eqref{eq: du/dr}\cdot v$
gives

$\Omega^{\prime}\,=\,\frac{2}{f}\cos\Omega+\frac{2}{f}\left(\frac{k}{r}-\frac{\lambda}{r^{2}}\right)\sin\Omega+\frac{2}{f^{2}}\left(\frac{\gamma}{r}-\varepsilon\right).$
(2.32)

The system (2.31) and (2.32) is now decoupled. Although (2.27) and (2.28) is a
linear system and (2.31) and (2.32) is nonlinear, the advantage we gain is
that we only need to study the differential equation (2.32) for $\Omega$ and
then $R$ can be solved by integrating (2.31).

Once we solve for $\Omega(r)$ and $R(r)$, we can define $u(r)$ and $v(r)$ via
(2.30). If $u$ and $v$ belong to the Hilbert space
$L^{2}\big{(}(0,\infty),f(r)^{-2}dr^{2}\big{)}$, which holds if and only if
$R/f\in L^{2}\big{(}(0,\infty),dr\big{)}$, then the corresponding wave
function $\Psi$ represents an element of the Hilbert space $\mathcal{H}_{t}.$

###  2.4 Conversion to a dynamical system on a compact cylinder

The study of (2.32) is facilitated by converting the differential equation
into a dynamical system on a compact cylinder. First, we make the system
autonomous by introducing a new differential equation, $r^{\prime}=1$, so that
$r$ is now an independent variable. However, the autonomous two-dimensional
system, $r^{\prime}=1$ and (2.32), is not compact and is singular at the
origin $r=0$. We rectify both of these concerns. We introduce transformations
that will make the system compact and remove the singularity at the origin
$r=0$.

We first compactify the system by bringing $r=\infty$ into a finite value.
Introduce a new independent variable $\eta$ via

$\eta(r)\,:=\,\frac{r}{1+r}\quad\text{ with inverse }\quad
r(\eta)\,=\,\frac{\eta}{1-\eta}.$ (2.33)

$\eta$ maps $(0,\infty)$ diffeomorphically onto $(0,1)$. This transformation
has an effect on $f(r)$; from (2.23), we have

$\displaystyle f\big{(}r(\eta)\big{)}\,$
$\displaystyle=\,\sqrt{1+2A_{*}+Z_{*}^{2}-\frac{2}{\eta}(A_{*}+Z_{*}^{2})+\frac{Z_{*}^{2}}{\eta^{2}}}$
(2.34) $\displaystyle=\,\frac{1}{\eta}g(\eta),$ (2.35)

where

$g(\eta)\,:=\,\sqrt{(1+2A_{*}+Z_{*}^{2})\eta^{2}-2(A_{*}+Z_{*}^{2})\eta+Z_{*}^{2}}.$
(2.36)

Note that $g(\eta)$ is always positive since the roots of the quadratic inside
the radical are complex; this follows since we are assuming that we’re in the
naked singularity sector $A_{*}<Z_{*}$.

Similar to (2.24), we have

$\frac{1}{f^{2}\big{(}r(\eta)\big{)}}\,=\,\frac{\eta^{2}}{g^{2}(\eta)}\,=\,\left\\{\begin{array}[]{ll}\frac{\eta^{2}}{Z_{*}^{2}}+O(\eta^{3})&\text{
as }\eta\to 0\\\ &\\\ 1+O(\frac{1-\eta}{\eta})&\text{ as }\eta\to
1.\end{array}\right.$ (2.37)

The system, $r^{\prime}=1$ and (2.32), becomes

$\left\\{\begin{array}[]{ll}\eta^{\prime}&=\,(1-\eta)^{2}\\\
\Omega^{\prime}&=\,\frac{2\eta}{g}\cos\Omega+\frac{2}{g}\left(k(1-\eta)-\lambda\frac{(1-\eta)^{2}}{\eta}\right)\sin\Omega+\frac{2\eta}{g^{2}}\big{(}\gamma(1-\eta)-\varepsilon\eta\big{)}.\end{array}\right.$
(2.38)

The system (2.38) is now precompact. It’s not compact since there is still a
singularity at $r=0$ – the coefficient for the anomalous magnetic moment
$\lambda$ behaves like $1/\eta$ as $\eta\to 0$. To rectify this, we introduce
a new independent variable $\tau$ via $\tau:=r+\log(r)$, so that
$\frac{dr}{d\tau}=\eta$. Multiplying $\eta^{\prime}$ and $\Omega^{\prime}$ by
$\eta$ in (2.38) and introducing the notation
$\dot{\Omega}=\frac{d\Omega}{d\tau}$ and $\dot{\eta}=\frac{d\eta}{d\tau}$, the
system becomes

$\left\\{\begin{array}[]{ll}\dot{\eta}&=\,\eta(1-\eta)^{2}\\\
\dot{\Omega}&=\,\frac{2\eta^{2}}{g}\cos\Omega+\frac{2}{g}\big{(}k\eta(1-\eta)-\lambda(1-\eta)^{2}\big{)}\sin\Omega+\frac{2\eta^{2}}{g^{2}}\big{(}\gamma(1-\eta)-\varepsilon\eta\big{)}.\end{array}\right.$
(2.39)

The system (2.39) is now compact and free of singularities. Understanding the
solutions to this system will be the main objective for the remainder of the
paper.

######  Remark 2.5.

_In the case of Minkowski spacetime, i.e. $f\equiv 1$ (and hence
$g(\eta)=\eta$), one would have to multiply $\eta^{\prime}$ and
$\Omega^{\prime}$ by $\eta^{2}$ instead of $\eta$. In this case, the
independent variable, call it $\tilde{\tau}$, would have to satisfy
$\frac{dr}{d\tilde{\tau}}=\eta^{2}$. $\square$ _

(2.39) now forms a dynamical system on the compact cylinder
$\mathcal{C}=[0,1]\times\mathbb{S}^{1}$. There are only four equilibrium
points. If we identify $\mathcal{C}$ with the fundamental domain
$\mathcal{C}_{*}=[0,1]\times[-\pi,\pi)$, these four points are:

$S^{-}\,:=\,(0,0),\quad N^{-}\,:=\,(0,-\pi),\quad
S^{+}_{\varepsilon}\,:=\,(1,-\arccos\varepsilon),\quad
N^{+}_{\varepsilon}\,:=\,(1,\arccos\varepsilon).$ (2.40)

Note that $S^{-}$ and $N^{-}$ appear on the left boundary of the cylinder,
while $S^{+}_{\varepsilon}$ and $N^{+}_{\varepsilon}$ appear on the right
boundary of the cylinder. The jacobians are

$\displaystyle J(S^{-})\,=\,\begin{pmatrix}1&0\\\
0&-2\frac{\lambda}{Z_{*}}\end{pmatrix}\quad$ $\displaystyle\text{ and }\quad
J(N^{-})\,=\,\begin{pmatrix}1&0\\\ 0&2\frac{\lambda}{Z_{*}}\end{pmatrix},$
$\displaystyle J(S^{+}_{\varepsilon})\,=\,\begin{pmatrix}0&0\\\
c_{+}&\sqrt{1-\varepsilon^{2}}\end{pmatrix}\quad$ $\displaystyle\text{ and
}\quad J(N^{+}_{\varepsilon})\,=\,\begin{pmatrix}0&0\\\
c_{-}&-\sqrt{1-\varepsilon^{2}}\end{pmatrix},$

where

$c_{\pm}\,:=\,2(1-A_{*}\pm
k)\sqrt{1-\varepsilon^{2}}+2\varepsilon(2A_{*}-1)-2\gamma.$

$S^{-}$ and $N^{-}$ are hyperbolic equilibrium points; they correspond to a
saddle and node, respectively. From the stable manifold theorem [30], there is
a unique (up to translation by a constant) orbit emanating out of $S^{-}$ into
the cylinder, called the _unstable manifold_. $S^{+}_{\varepsilon}$ and
$N^{+}_{\varepsilon}$ are non-hyperbolic equilibrium points, and so the stable
manifold theorem does not apply. They correspond to saddle-nodes with the
saddle part of $S^{+}_{\varepsilon}$ pointing into the cylinder, and the
saddle part of $N^{+}_{\varepsilon}$ pointing out of the cylinder. See Theorem
1 in section 2.11 of [30]. From Theorem 2.19 in [12], it follows that there is
a unique (up to translation by a constant) orbit flowing into
$S^{+}_{\varepsilon}$ which will be referred to as its _stable manifold_. We
will let $\mathcal{W}^{-}_{\varepsilon}$ and $\mathcal{W}^{+}_{\varepsilon}$
denote the unstable and stable manifolds associated with $S^{-}$ and
$S^{+}_{\varepsilon}$, respectively.

A _heteroclinic orbit_ is an orbit of the dynamical system whose $\alpha$ and
$\omega$-limit sets are equilibrium points. For the system (2.39), a
heteroclinic orbit is one that begins and ends at the equilibrium points.
Since $\dot{\eta}>0$ for $\eta\in(0,1)$, a heteroclinic orbit that does not
lie on the boundary, $\eta=0$ or $\eta=1$, must connect one of the equilibrium
points $S^{-}$ or $N^{-}$ to either $S^{+}_{\varepsilon}$ or
$N^{+}_{\varepsilon}$; these will be referred to as _non-boundary heteroclinic
orbits._ A non-boundary heteroclinic orbit that begins at $S^{-}$ most likely
ends at $N^{+}_{\varepsilon}$. If, in the rare situation, a non-boundary
heteroclinic orbit joins $S^{-}$ to $S^{+}_{\varepsilon}$, then the orbit will
be called a _saddles connector._ If an orbit is a saddles connector, then
$\mathcal{W}^{-}_{\varepsilon}$ and $\mathcal{W}^{+}_{\varepsilon}$ coincide.

######  Theorem 2.6.

Suppose $\Psi$ is constructed from a non-boundary heteroclinic orbit of the
system (2.39). Assume $\lambda\geq\frac{3}{2}Z_{*}$. Then $\Psi$ is an element
of the Hilbert space if and only if the non-boundary heteroclinic orbit is a
saddles connector.

###### Proof.

$\Psi$ is an element of the Hilbert space if and only if $R/f\in
L^{2}\big{(}(0,\infty),dr\big{)}$. From (2.31), we have

$R(r)\,=\,\exp\left(\int_{r_{0}}^{r}\frac{\sin\Omega(\tilde{r})}{f(\tilde{r})}d\tilde{r}\right)\exp\left(\int_{r_{0}}^{r}\frac{\cos\Omega(\tilde{r})}{f(\tilde{r})}\left(-\frac{k}{\tilde{r}}+\frac{\lambda}{\tilde{r}^{2}}\right)d\tilde{r}\right)$

for some $r_{0}\in(0,\infty).$ We first examine the $L^{2}$ condition for left
end of the cylinder, $\eta=0$ (i.e. $r=0$). Using (2.24) in the above
expression for $R(r)$, we conclude that

  * (i)

If the orbit begins at $S^{-}$, then $R(r)\propto r^{\lambda/Z_{*}}+o(1)$ as
$r\to 0$.

  * (ii)

If the orbit begins at $N^{-}$, then $R(r)\propto r^{-\lambda/Z_{*}}+o(1)$ as
$r\to 0$.

Therefore $R^{2}(r)/f^{2}(r)\propto r^{2\pm 2\lambda/Z_{*}}+o(1)$ as $r\to 0$.
The $L^{2}$ condition is satisfied in a neighborhood of $r=0$ if and only if
$2\pm 2\lambda/Z_{*}>-1$. Therefore (i) implies the $L^{2}$ condition in a
neighborhood of $r=0$ while (ii) does not so long as
$\lambda\geq\frac{3}{2}Z_{*}$.

Now we examine the $L^{2}$ condition for the right end of the cylinder,
$\eta=1$ (i.e. $r=\infty$). Fix $\alpha>0$ such that
$\alpha<\sqrt{1-\varepsilon^{2}}$. Set
$\beta_{\pm}:=\sqrt{1-\varepsilon^{2}}\pm\alpha>0$. If the orbit ends at
$S^{+}_{\varepsilon}$, then $\sin\Omega/f=-\sqrt{1-\varepsilon^{2}}+o(1)$ as
$r\to\infty$. Therefore there is an $R_{\alpha}$ such that
$\sin\Omega(r)/f(r)<-\beta_{-}$ whenever $r>R_{\alpha}$. Using similar
estimates for $\cos\Omega/f$, we observe:

  * (iii)

If the orbit ends at $S^{+}_{\varepsilon}$, then for some constants $c_{1}$
and $c_{2}$, we have the following:

    * If $k>0$, then

$\displaystyle R(r)\,$
$\displaystyle\leq\,c_{1}e^{-r\beta_{-}}\frac{e^{-\lambda\beta_{+}/r}}{r^{k\beta_{-}}}$
$\displaystyle\leq\,c_{1}e^{-r\beta_{-}}\frac{1}{r^{k\beta_{-}}}$
$\displaystyle\leq\,c_{1}e^{-r\beta_{-}}\quad\quad\quad\quad\text{ for all
sufficiently large $r$.}$

    * If $k<0$, then

$\displaystyle R(r)\,$
$\displaystyle\leq\,c_{2}e^{-r\beta_{-}}\frac{e^{-\lambda\beta_{+}/r}}{r^{k\beta_{+}}}$
$\displaystyle\leq\,c_{2}e^{-r\beta_{-}}\frac{1}{r^{k\beta_{+}}}$
$\displaystyle\leq\,c_{2}e^{-r\beta_{-}/2}\quad\quad\quad\quad\text{ for all
sufficiently large $r$.}$

In either case, the $L^{2}$ condition is satisfied for $R^{2}/f^{2}$ in a
neighborhood of $r=\infty$.

  * (iv)

If the orbit ends at $N^{+}_{\varepsilon}$, then
$\sin\Omega/f=\sqrt{1-\varepsilon^{2}}+o(1)$ as $r\to\infty$. Hence $R(r)$
exhibits exponential growth as $r\to\infty$ and so the $L^{2}$ condition is
never satisfied. ∎

######  Remark 2.7.

_If we instead considered negative values of $\lambda$ in Theorem 2.6, then
$N^{-}$ and $S^{-}$ would have switched roles and we would have required
$|\lambda|\geq\frac{3}{2}Z_{*}$. The assumption
$|\lambda|\geq\frac{3}{2}Z_{*}$ is equivalent to the assumption
$|\mu_{a}|\geq\frac{3}{2}\frac{\sqrt{G}\hbar}{c}$, which appeared in Theorem
2.1. In fact Theorem 2.6 offers another proof of the fact that $H$ is
essentially self-adjoint if and only if
$|\mu_{a}|\geq\frac{3}{2}\frac{\sqrt{G}\hbar}{c}$, see [36, Thm. 4.16].
$\square$ _

###  2.5 Proof of the main result

In this section, we will always assume the following conditions on our
parameters

$Z_{*}\,>\,0,\quad\quad 0\leq
A_{*}<Z_{*}\,\quad\quad\lambda\geq\frac{3}{2}Z_{*}.$

Recall that the second assumption implies $g(\eta)>0$ (naked singularity
sector), and the last assumption guarantees essential self adjointness. Also,
the other parameter is $k$ which is a nonzero integer. Lastly, recall that
$\gamma$ and $Z_{*}$ are not independent parameters but are related via
$Z_{*}=-\sqrt{\mathcal{G}}\gamma$ where $\mathcal{G}$ is given by (2.19).

From Theorem 2.6, we wish to obtain saddles connectors for our system, which
is our main result, Theorem 2.23, at the end of this section. We will follow
the strategy first laid out in [24] and generalized in [23]; the latter will
be our primary reference. (2.39) is a compact dynamical system on the cylinder
$\mathcal{C}=[0,1]\times\mathbb{S}^{1}$. So long as $-1<\varepsilon<1$, the
dynamical system $\mathcal{C}$ satisfies assumptions (a) - (e) in section 3.1
of [23] with $x=\eta$, $y=\Omega$, parameter $\mu=\varepsilon$, and
fundamental domain $\mathcal{C}_{*}=[0,1]\times[-\pi,\pi)$.

Assumptions (a), (c), and (d) are trivial to verify. Assumption (b) follows
from the analysis of the equilibrium points given in section 2.4. Now we
briefly explain why assumption (e) holds. A quick calculation shows that
Proposition 3.16 from [23] holds for $\varepsilon\in(-1,1]$. (Note that
$\varepsilon$ is analogous to $E$ in [23].) A similar argument as used in
Lemma 3.17 from [23] also holds. This combined with Theorem 3.5 in [23] shows
that assumption (e) holds.

Therefore the consequences of assumptions (a) - (e) laid out in section 3.2 of
[23] hold for our system which we briefly discuss. Associated to the dynamical
system on $\mathcal{C}$ is a topological number called the _winding number_
$w_{\varepsilon}$ [23, Def. 3.2] which counts the number of times the unstable
manifold $\mathcal{W}^{-}_{\varepsilon}$ winds around the cylinder.
Specifically, let $\widetilde{\mathcal{W}}^{-}_{\varepsilon}$ denote the lift
of $\mathcal{W}^{-}_{\varepsilon}$ to the universal cover
$\widetilde{C}=[0,1]\times\mathbb{R}$ beginning at $(0,0)$. Then
$\widetilde{\mathcal{W}}^{-}_{\varepsilon}$ terminates at either a lift of the
nodal point $N^{+}_{\varepsilon}$ or a lift of the saddle point
$S^{+}_{\varepsilon}$; hence it terminates at $(1,-2\pi
N\pm\arccos\varepsilon)$ for some $N\in\mathbb{Z}$. This number $N$ is
precisely the winding number.

To establish existence and uniqueness of saddles connectors with different
winding numbers, we rely on the following proposition.

######  Proposition 2.8.

Let $N$ be an integer. Suppose there exist two parameter values
$\varepsilon^{\prime}<\varepsilon^{\prime\prime}$ such that

$w_{\varepsilon^{\prime}}\,\leq\,N\quad\text{ and }\quad
w_{\varepsilon^{\prime\prime}}\,\geq\,N+1.$

Then there exists a unique
$\varepsilon\in[\varepsilon^{\prime},\varepsilon^{\prime\prime})$ such that
$\mathcal{W}^{-}_{\varepsilon}$ is a saddles connector with winding number
$w_{\varepsilon}=N$.

###### Proof.

Existence of the saddles connector follows from Proposition 3.4 in [23]. To
prove uniqueness, suppose there were two values
$\varepsilon_{1}<\varepsilon_{2}$ which correspond to saddles connectors
$\mathcal{W}^{-}_{1}$ and $\mathcal{W}^{-}_{2}$, respectively, each with the
same winding number $N$. Let $\widetilde{\mathcal{W}}^{-}_{1}$ and
$\widetilde{\mathcal{W}}^{-}_{2}$ denote their lifts into the universal cover
$\widetilde{\mathcal{C}}=[0,1]\times\mathbb{R}$ whose $\alpha$-limit point is
(0,0). By Lemma 3.3 in [23], it follows that $\widetilde{\mathcal{W}}^{-}_{2}$
lies below $\widetilde{\mathcal{W}}^{-}_{1}$. However, the $\omega$-limit
points of $\widetilde{\mathcal{W}}^{-}_{1}$ and
$\widetilde{\mathcal{W}}^{-}_{2}$ occur at $\Omega_{1}=-2\pi
N-\arccos\varepsilon_{1}$ and $\Omega_{2}=-2\pi N-\arccos\varepsilon_{2}$,
respectively. Since $\Omega_{2}<\Omega_{1}$, it follows that
$\widetilde{\mathcal{W}}_{2}^{-}$ lies above $\widetilde{\mathcal{W}}_{1}^{-}$
sufficiently close to the right hand side of the boundary ($\eta=1$). This is
a contradiction. ∎

Proposition 2.8 suggests a strategy of using barriers to constrain the winding
numbers. For example, the proof of the next proposition shows that negative
winding numbers are impossible since the slope field is always negative along
the horizontal line $\Omega=\pi$ (in the universal cover).

######  Proposition 2.9.

For any $\varepsilon\in(-1,1)$, the winding number satisfies
$w_{\varepsilon}\geq 0$.

###### Proof.

We work in the universal cover $\widetilde{\mathcal{C}}$. From (2.39), the
slope field at $\Omega=\pi$ is

$\dot{\Omega}|_{\Omega=\pi}(\eta)\,=\,-\frac{2\eta^{2}}{g(\eta)}+\frac{2\eta^{2}}{g^{2}(\eta)}\big{(}\gamma(1-\eta)-\varepsilon\eta).$

We want to show the above expression is negative for all $\eta\in(0,1)$. It’s
sufficient to show it’s negative for $\varepsilon=-1$. Therefore it suffices
to show $-g(\eta)+\gamma(1-\eta)+\eta<0$. Since $g$ is positive, this
expression is clearly negative for $\eta\in(0,\eta_{0})$ where
$\eta_{0}=-\gamma/(1-\gamma)$. For $\eta\in[\eta_{0},1)$, it suffices to show
$\big{[}\gamma(1-\eta)+\eta\big{]}^{2}<g^{2}(\eta)$. Collecting terms in
$\eta$, this is equivalent to

$\eta^{2}\big{[}(\gamma^{2}-2\gamma)-(2A_{*}+Z_{*}^{2})\big{]}+\eta\big{[}2\gamma(1-\gamma)+2(A_{*}+Z_{*}^{2})\big{]}-Z_{*}^{2}\,<\,0.$

Assume for now that the factor multiplying $\eta^{2}$ is positive. Then this
quadratic in $\eta$ is concave up. At $\eta=0$, the quadratic evaluates to
$-Z_{*}^{2}<0$. At $\eta=1$, it evaluates to $-\gamma^{2}<0$. Therefore the
quadratic is negative between $0$ and $1.$

It suffices to show $\gamma^{2}-2\gamma>2A_{*}+Z_{*}^{2}$. Recall
$\gamma=-Z_{*}/\sqrt{\mathcal{G}}$ where $\mathcal{G}$ is given by (2.19).
Hence it suffices to show
$A_{*}<Z_{*}^{2}(1-\mathcal{G})/2\mathcal{G}+Z_{*}/\sqrt{\mathcal{G}}$. Indeed
this inequality follows from the naked singularity sector hypothesis,
$A_{*}<Z_{*}$. ∎

An open problem in [24] was to show the existence of winding numbers for all
non-negative integers. This problem was resolved in [23] where the key
argument is Proposition 3.22 in that paper. An analogue of that proposition
holds for our system (2.39) as well.

######  Proposition 2.10.

For any integer $N\geq 0$, there is an $\varepsilon^{\prime\prime}\in(-1,1)$
such that $w_{\varepsilon^{\prime\prime}}\geq N+1$.

###### Proof.

The proof is by contradiction and follows the same strategy as the proof of
Proposition 3.22 in [23]. Suppose there is an integer $N\geq 0$ such that for
all $\varepsilon\in(-1,1)$, the winding number $w_{\varepsilon}$ is no greater
than $N$. By Proposition 2.9, the lift of the unstable manifold
$\widetilde{\mathcal{W}}^{-}_{\varepsilon}$ emanating from $S^{-}$ is
contained in the compact set $[0,1]\times[-\pi-2\pi N,\pi]$ for all
$\varepsilon$. It follows that, as $\varepsilon\to 1$, the unstable manifolds
$\widetilde{\mathcal{W}}^{-}_{\varepsilon}$ converge to an orbit of the
dynamical system, see Proposition 3.24 in [23]. Call this limiting orbit
$c(\tau).$ Proposition 3.18 in [23] holds for our system (2.39) as well. This
fact together with the Poincaré–Bendixson theorem implies that $c(\tau)$ must
converge to an equilibrium point on the boundary $\eta=1$. For
$\varepsilon=1$, the equilibrium points $S^{+}_{1}$ and $N^{+}_{1}$ are both
equal to the point $(1,0)$ in the $(\eta,\Omega)$ plane. Therefore $c(\tau)$
must converge to some copy of this point in the universal cover.

However at $\varepsilon=1$, the equilibrium point $(1,0)$ is a nilpotent
singularity. We want to apply Theorem 3.5 in [12] to (2.39). To do this, we
shift our system $\eta_{1}=\eta-\frac{\pi}{2}$ so that the equilibrium point
is at the origin $(0,0)$ in the $(\eta_{1},\Omega)$ plane. We also perform the
compression $\eta_{2}=-2\gamma\eta_{1}$. Our system becomes

$\left\\{\begin{array}[]{ll}\dot{\eta}_{2}&=\,-\frac{1}{2\gamma}\eta^{2}_{2}\left(1-\frac{\eta_{2}}{2\gamma}\right)\\\
\dot{\Omega}&=\,\eta_{2}+A(\eta_{2},\Omega).\end{array}\right.$

$A(\eta_{2},\Omega)$ contains quadratic terms and higher. (The point of the
compression was to ensure that the constant in front of $\eta_{2}$ in the
expression for $\dot{\Omega}$ equals 1, so we can directly apply Theorem 3.5
in [12].) Let $\eta_{2}=f(\Omega)$ be the solution to
$\eta_{2}+A(\eta_{2},\Omega)=0.$ Using the implicit function theorem, one
calculates: $f^{\prime}(0)=0$ and $f^{\prime\prime}(0)=2$. Therefore
$f(\Omega)=\Omega^{2}+\text{h.o.t.}$ Define
$F(\Omega)=B\big{(}f(\Omega)\big{)}$ where $B=\dot{\eta}_{2}.$ Then
$F(\Omega)=-\frac{1}{2\gamma}\Omega^{4}+\text{h.o.t.}$ Define

$G(\Omega)\,=\,\left(\frac{\partial A}{\partial\Omega}+\frac{\partial
B}{\partial\eta_{2}}\right)\bigg{|}_{\big{(}f(\Omega),\Omega\big{)}}\,=\,-2\Omega+\text{h.o.t.}$

Applying Theorem 3.5(4) in [12] with $x=\Omega$ and $y=\eta_{2}$ shows that
$m=4$ and $n=1$; hence $(i2)$ is relevant. The phase portrait depicted in
Figure 3.21(i) in [12] shows that no orbit starting within the cylinder (i.e.
a non-boundary orbit) can reach the equilibrium point. ∎

Recall that $\gamma$ and $Z_{*}$ are not independent parameters but are
related via $Z_{*}=-\sqrt{\mathcal{G}}\gamma$ where $\mathcal{G}$ is given by
(2.19). An assumption on $\gamma$ implicitly defines an assumption on $Z_{*}.$
For example, $\gamma\in[-1,0)$ is equivalent to
$Z_{*}\in(0,\sqrt{\mathcal{G}}\,]$; this is relevant for the next proposition.

######  Proposition 2.11.

For $\varepsilon=0$, the winding number satisfies $w_{0}=0$ for all
$\gamma\in[-1,0)$, $A_{*}\in[0,\frac{1}{2}Z_{*}]$,
$\lambda\geq\frac{3}{2}Z_{*}$, and $k\leq-1$.

###### Proof.

By Proposition 2.9, it suffices to show $w_{0}\leq 0$. We show that
$\Omega=-\frac{\pi}{2}$ is a lower barrier. That is we will establish the
following inequality.

$\dot{\Omega}|_{\Omega=-\frac{\pi}{2},\,\varepsilon=0}(\eta)\,=\,\frac{2}{g(\eta)}\big{(}-k\eta(1-\eta)+\lambda(1-\eta)^{2}\big{)}+\frac{2\eta^{2}}{g^{2}(\eta)}\gamma(1-\eta)\,>\,0.$

Eliminating the positive factor $(1-\eta)$ and rationalizing the denominator,
we want to show

$g^{2}(\eta)\big{(}-k\eta+\lambda(1-\eta)\big{)}^{2}-\gamma^{2}\eta^{4}\,>\,0\quad\quad\text{
for all }\quad\quad\eta\in(0,1).$

The above expression decreases as $k$ increases, so it suffices to show that
the inequality holds for $k=-1$. Similarly, it suffices to show that it holds
for $\lambda=\frac{3}{2}Z_{*}$ and $A_{*}=\frac{1}{2}Z_{*}$. Therefore the
desired inequality is

$p(\eta)\,>\,0,$

where

$p(\eta)\,:=\,g^{2}(\eta)\big{|}_{A_{*}=\frac{Z_{*}}{2}}\left(\eta+\frac{3Z_{*}}{2}(1-\eta)\right)^{2}-\gamma^{2}\eta^{4}.$

We define

$p_{0}(\eta)\,:=\,p(\eta)\big{|}_{Z_{*}=0}\,=\,\eta^{4}(1-\gamma^{2})\quad\text{
and }\quad p_{1}(\eta)\,:=\,p(\eta)-p_{0}(\eta).$

Since $\gamma\in[-1,0)$, we have $p_{0}(\eta)\geq 0$ for $\eta\in[0,1]$.
Therefore it remains to show that $p_{1}(\eta)>0$ for $\eta\in(0,1)$. We find

$p_{1}(\eta)\,=\,\frac{Z_{*}(1-\eta)}{4}\big{(}a_{3}\eta^{3}+a_{2}\eta^{2}+a_{1}\eta+a_{0}\big{)},$

where

$\displaystyle a_{3}\,$ $\displaystyle=\,-9Z_{*}^{3}+3Z_{*}^{2}-Z_{*}+8$
$\displaystyle a_{2}\,$ $\displaystyle=\,Z_{*}(27Z_{*}^{2}-6Z_{*}+1)$
$\displaystyle a_{1}\,$ $\displaystyle=\,3Z_{*}^{2}(-9Z_{*}+1)$ $\displaystyle
a_{0}\,$ $\displaystyle=\,9Z_{*}^{3}.$

Recall that $\gamma\in[-1,0)$ is equivalent to
$Z_{*}\in(0,\sqrt{\mathcal{G}}\,]$. Since $\sqrt{\mathcal{G}}\approx 4.90\cdot
10^{-22}$, it’s not hard to estimate the maximum and minimum values of the
coefficients $a_{i}$ as $Z_{*}$ runs through $(0,\sqrt{\mathcal{G}}\,]$.
Indeed, it follows that for all $Z_{*}\in(0,\sqrt{\mathcal{G}}\,]$, the
coefficients $(a_{3},a_{2},a_{1},a_{0})$ have signs $(+,+,+,+)$. Therefore the
result follows by Descartes’ rule of signs. ∎

######  Remark 2.12.

_The condition $A_{*}\in[0,\frac{1}{2}Z_{*}]$ is not strict and can be
improved. Ideally, one would want Proposition 2.11 to hold for all
$A_{*}\in[0,Z_{*})$. However, $p_{1}(\eta)$ is negative near $Z_{*}$ in the
case $A_{*}=Z_{*}$. Physically, if $Z_{*}$ represents the charge of the proton
(i.e. $Q=e$), then $A_{*}=\frac{1}{2}Z_{*}$ is on the order of $10^{-7}$ grams
which is about $10^{17}$ proton masses. $\square$ _

######  Proposition 2.13.

Suppose $\gamma\in[-1,0)$, $A_{*}\in[0,\frac{1}{2}Z_{*}]$,
$\lambda\geq\frac{3}{2}Z_{*}$, and $k\leq-1$. Then for each integer $N\geq 0$,
there is a unique $\varepsilon_{N}\geq 0$ such that
$\mathcal{W}^{-}_{\varepsilon_{N}}$ is a saddles connector with winding number
$w_{\varepsilon_{N}}=N$. Moreover, they are monotonic:
$0\leq\varepsilon_{0}\leq\varepsilon_{1}\leq\dotsb$.

###### Proof.

Fix $N\geq 0$. By Proposition 2.10, there is an $\varepsilon^{\prime\prime}$
such that $w_{\varepsilon^{\prime\prime}}\geq N$. By Lemma 3.3 in [23], it
follows that $w_{\varepsilon}\geq N$ for all
$\varepsilon>\varepsilon^{\prime\prime}$. Therefore we can assume
$\varepsilon^{\prime\prime}>0$ without loss of generality. Using
$\varepsilon^{\prime}=0$ from Proposition 2.11, the existence and uniqueness
of $\varepsilon_{N}\geq 0$ follows from Proposition 2.8. The monotonicity
follows from another application of Lemma 3.3 in [23]. ∎

The previous proposition establishes saddles connectors for $k\leq-1$. Next we
establish them for $k\geq 1$.

######  Lemma 2.14.

Fix $\mathcal{A}\in(0,Z_{*}]$ and set
$\delta:=1+\frac{\mathcal{A}^{2}}{Z_{*}^{2}}$. Let $A_{*}\in[0,\mathcal{A})$,
$\gamma\in(-\frac{1}{\delta},0)$, and $\varepsilon=0$. Then for each integer
$k\geq 1$, we have

  * _(a)_

$\dot{\Omega}|_{\Omega=-\frac{\pi}{2}}(\eta)>0$ for all $\eta\in[0,a_{k}]$
where $a_{k}=\frac{\lambda}{k+\lambda-\delta\gamma}$,

  * _(b)_

$\dot{\Omega}|_{\Omega=-\frac{3\pi}{2}}(\eta)>0$ for all $\eta\in[b_{k},1]$
where $b_{k}=\frac{\lambda}{k+\lambda+\delta\gamma}$.

######  Remark 2.15.

_The assumption $\gamma\in(-\frac{1}{\delta},0)$ in the previous lemma is used
only to ensure that $b_{k}\in(0,1)$ for all $k$. $\square$ _

###### Proof.

We prove (a); the proof of (b) is analogous. At $\varepsilon=0$, we have

$\displaystyle\dot{\Omega}|_{\Omega=-\frac{\pi}{2}}(\eta)\,=\,-\frac{2}{g(\eta)}\big{(}k\eta(1-\eta)-\lambda(1-\eta)^{2}\big{)}+\frac{2\eta^{2}}{g^{2}(\eta)}\gamma(1-\eta).$

It suffices to show

$-k\eta+\lambda(1-\eta)+\frac{\eta^{2}\gamma}{g(\eta)}>0\quad\text{ for
}\quad\eta\in[0,a_{k}].$ (2.41)

From eq. (2.36), we see that the assumption $A_{*}<\mathcal{A}$ implies that
$g^{2}(\eta)>\frac{\eta^{2}}{\delta^{2}}$; this follows since the discriminant
of the quadratic $g^{2}(\eta)-\frac{\eta^{2}}{\delta^{2}}$ is negative for
$A_{*}<\mathcal{A}$:

$\displaystyle\text{discriminant}\left(g^{2}(\eta)-\frac{\eta^{2}}{\delta^{2}}\right)\,$
$\displaystyle=\,4\bigg{(}A^{2}_{*}+Z^{2}_{*}(1-\delta^{2})\bigg{)}$
$\displaystyle<\,4\bigg{(}\mathcal{A}^{2}+Z^{2}_{*}(1-\delta^{2})\bigg{)}$
$\displaystyle=\,0.$

Plugging in $g^{2}(\eta)>\frac{\eta^{2}}{\delta^{2}}$ into eq. (2.41) (recall
$\gamma<0$), we have

$\displaystyle-k\eta+\lambda(1-\eta)+\frac{\eta^{2}\gamma}{g(\eta)}\,$
$\displaystyle>\,-k\eta+\lambda(1-\eta)+\delta\gamma\eta$
$\displaystyle=\,-\eta(k+\lambda-\delta\gamma)+\lambda$ $\displaystyle\geq 0.$

The last step follows since $\eta\leq a_{k}$. ∎

Consider the line segment in the $(\eta,\Omega)$ plane connecting the points
$(a_{k},-\frac{\pi}{2})$ and $(b_{k},-\frac{3\pi}{2})$ where $a_{k}$ and
$b_{k}$ are the same as in Lemma 2.14. The next lemma shows that the slope
field $\frac{\dot{\Omega}}{\dot{\eta}}$ is larger than the slope of this line
(i.e. larger than $-\frac{\pi}{b_{k}-a_{k}}$). Therefore this line segment
joined together with the horizontal line segments in Lemma 2.14 produce a
lower barrier.

######  Lemma 2.16.

Set $\mathcal{A}=0.1Z_{*}$, so
$\delta=1+\frac{\mathcal{A}^{2}}{Z_{*}^{2}}=1.01$. Set $\nu=3$ and
$\Lambda=\frac{\alpha_{\mbox{\tiny{S}}}}{2\pi}\frac{1}{\nu\delta}\approx
0.000383$. Assume $A_{*}\in[0,\mathcal{A})$,
$\gamma\in(-\frac{1}{\nu\delta},0)$, $\lambda\in[\frac{3}{2}Z_{*},\Lambda]$,
and $\varepsilon=0$. Then for each integer $k\geq 1$, the slope field
satisfies $\frac{\dot{\Omega}}{\dot{\eta}}>-\frac{\pi}{b_{k}-a_{k}}$ for all
$\eta\in[a_{k},b_{k}]$ and all $\Omega\in[-\pi,\pi)$, where $a_{k}$ and
$b_{k}$ are the same as in Lemma _2.14_.

######  Remark 2.17.

_The largest value $\Lambda$ for the anomalous magnetic moment term $\lambda$
is twice what is experimentally observed, so our range for $\lambda$ agrees
with empirical data. $\square$ _

###### Proof.

We want to show $\dot{\Omega}+\dot{\eta}\frac{\pi}{b_{k}-a_{k}}>0$. For
$\varepsilon=0$, we have

$\dot{\Omega}+\dot{\eta}\frac{\pi}{b_{k}-a_{k}}\,=\,A\cos\Omega+B\sin\Omega+C,$

where

$\displaystyle A\,$ $\displaystyle:=\,\frac{2\eta^{2}}{g(\eta)}$
$\displaystyle B\,$
$\displaystyle:=\,\frac{2}{g(\eta)}\big{(}k\eta(1-\eta)-\lambda(1-\eta)^{2}\big{)}$
$\displaystyle C\,$
$\displaystyle:=\,\frac{2\eta^{2}}{g^{2}(\eta)}\gamma(1-\eta)+\eta(1-\eta)^{2}\frac{\pi}{b_{k}-a_{k}}.$

Compare $A\cos\Omega+B\sin\Omega+C$ with the line $Ax+By+C=0$ in the $(x,y)$
plane. The distance from the origin $(0,0)$ to the line is given by
$\frac{|C|}{\sqrt{A^{2}+B^{2}}}$. Therefore the line never intersects the unit
circle provided $C^{2}-A^{2}-B^{2}>0$. (Notice that the dependence on $\Omega$
has vanished; hence these results will be valid for all
$\Omega\in[-\pi,\pi)$.) If this holds, and we show for some $\Omega_{0}$ that
$A\cos\Omega_{0}+B\sin\Omega_{0}+C>0$ for all $\eta\in[a_{k},b_{k}]$, then the
unit circle lies in the half of the $(x,y)$ plane where $Ax+By+C>0$. For
simplicity, we choose $\Omega_{0}=0$. Therefore it suffices to show the
following:

  * (1)

$C^{2}-A^{2}-B^{2}>0$ for all $\eta\in[a_{k},b_{k}]$ and all integers $k\geq
1$.

  * (2)

$A+C>0$ for all $\eta\in[a_{k},b_{k}]$ and all integers $k\geq 1$.

We first show (2). Recall from the proof of Lemma 2.14 that
$g^{2}(\eta)>\frac{\eta^{2}}{\delta^{2}}$ since $A_{*}<\mathcal{A}$. Therefore

$\displaystyle\frac{A+C}{1-\eta}\,>\,\frac{C}{1-\eta}\,$
$\displaystyle>\,\frac{2}{\delta}\gamma+\eta(1-\eta)\frac{\pi}{b_{k}-a_{k}}$
$\displaystyle\text{since }g^{2}(\eta)>\frac{\eta^{2}}{\delta^{2}},$
$\displaystyle>\,\frac{2}{\delta}\gamma+a_{k}(1-b_{k})\frac{\pi}{b_{k}-a_{k}}$
$\displaystyle=\,\frac{2}{\delta}\gamma-\pi\left(\frac{1}{2}+\frac{k}{2\delta\gamma}\right)$
$\displaystyle>\,-\frac{2}{\nu\delta^{2}}-\frac{\pi}{2}+\frac{\pi\nu}{2}k$
$\displaystyle\text{since }\gamma>-\frac{1}{\nu\delta},$ $\displaystyle>\,0.$

Now we prove (1). From above, we can find a lower bound for $C^{2}$. For
$\eta\in[a_{k},b_{k}]$, we have

$1-\eta\,\geq\,1-b_{k}\,=\,1-\frac{\lambda}{k+\lambda+\delta\gamma}\,>\,1-\frac{\lambda}{1+\lambda-1/\nu}\,=\,\frac{\nu-1}{\nu+\lambda-1}\,\geq\,\frac{\nu-1}{\nu+\Lambda-1}.$

Therefore a lower bound for $C^{2}$ is

$C^{2}\,>\,\left(-\frac{2}{\nu\delta^{2}}-\frac{\pi}{2}+\frac{\pi\nu}{2}k\right)^{2}\left(\frac{\nu-1}{\nu+\Lambda-1}\right)^{2}.$

An upper bound for $A^{2}$ is

$A^{2}\,=\,\frac{4\eta^{4}}{g^{2}(\eta)}\,<\,4\delta^{2}\eta^{2}\,\leq\,4\delta^{2}b_{k}^{2}\,=\,4\delta^{2}\left(\frac{\lambda}{k+\lambda-\delta\gamma}\right)^{2}\,<\,4\delta^{2}\left(\frac{\Lambda}{1+\Lambda}\right)^{2}.$

An upper bound for $B^{2}$ is

$\displaystyle B^{2}\,$
$\displaystyle=\,\frac{4}{g^{2}(\eta)}(1-\eta)^{2}\big{(}k^{2}\eta^{2}-2k\lambda\eta(1-\eta)+\lambda^{2}(1-\eta)^{2}\big{)}$
$\displaystyle<\,\frac{4\delta^{2}}{\eta^{2}}\big{(}k^{2}\eta^{2}-2k\lambda\eta(1-\eta)+\lambda^{2}(1-\eta)^{2}\big{)}$
$\displaystyle=\,4\delta^{2}\left(k^{2}-2k\lambda\frac{(1-\eta)}{\eta}+\lambda^{2}\frac{(1-\eta)^{2}}{\eta^{2}}\right)$
$\displaystyle\leq\,4\delta^{2}\left(k^{2}-2k\lambda\frac{1-b_{k}}{b_{k}}+\lambda^{2}\frac{(1-a_{k})^{2}}{a_{k}^{2}}\right)$
$\displaystyle=\,4\delta^{2}\left(k^{2}-2k\lambda\frac{k+\delta\gamma}{\lambda}+\lambda^{2}\frac{(k-\delta\gamma)^{2}}{\lambda^{2}}\right)$
$\displaystyle=\,4\delta^{2}\big{(}-4k\delta\gamma+\delta^{2}\gamma^{2}\big{)}$
$\displaystyle\leq\,4\delta^{2}\left(\frac{4}{\nu}k+\frac{1}{\nu^{2}}\right).$

Therefore $A^{2}+B^{2}<C^{2}$ holds if the following inequality holds for all
$k\geq 1$.

$4\delta^{2}\left(\frac{\Lambda}{1+\Lambda}\right)^{2}+4\delta^{2}\left(\frac{4}{\nu}k+\frac{1}{\nu^{2}}\right)\,<\,\left(-\frac{2}{\nu\delta^{2}}-\frac{\pi}{2}+\frac{\pi\nu}{2}k\right)^{2}\left(\frac{\nu-1}{\nu+\Lambda-1}\right)^{2}.$

Let $u(k)$ and $v(k)$ denote the LHS and RHS, respectively. We want to show
$u(k)<v(k)$ for all $k\geq 1$. $u(k)$ is linear in $k$, while $v(k)$ is
quadratic in $k$. Plug in the values $\delta=1.01$, $\nu=3$, $\Lambda\approx
0.000383$ into $u(k)$ and $v(k)$. At $k=1$, we have $u(1)\approx 5.89$ and
$v(1)\approx 6.19$. Moreover, $u^{\prime}(1)=16\frac{\delta^{2}}{\nu}\approx
5.44$, while $v^{\prime}(1)\approx 23.4$. Thus $u(k)<v(k)$ for all $k\geq 1$.
∎

######  Remark 2.18.

_It’s clear from the proof of the previous lemma that the chosen values for
$\mathcal{A}$, $\nu$, and $\Lambda$ are not strict. For example, one can
increase $\Lambda$ by increasing $\nu$. $\square$ _

######  Proposition 2.19.

Let $\mathcal{A}$, $\delta$, $\nu$, and $\Lambda$ be as in Lemma _2.16_. The
winding number for $\varepsilon=0$, satisfies $w_{0}\leq 1$ for all
$A_{*}\in[0,\mathcal{A}),\gamma\in(-\frac{1}{\nu\delta},0)$, and
$\lambda\in[\frac{3}{2}Z_{*},\Lambda]$.

###### Proof.

This follows by Lemmas 2.14 and 2.16. The lower barrier is piecewise linear
and constructed by the horizontal lines $\Omega=-\frac{\pi}{2}$ for
$\eta\in[0,a_{k}]$ and $\Omega=-\frac{3\pi}{2}$ for $\eta\in[b_{k},1]$ along
with the line segment connecting the points $(a_{k},-\frac{\pi}{2})$ and
$(b_{k},-\frac{3\pi}{2})$ in the $(\eta,\Omega)$ plane. ∎

The following proposition is the analogue of Proposition 2.13 for $k\geq 1$.

######  Proposition 2.20.

Let $\mathcal{A}$, $\delta$, $\nu$, and $\Lambda$ be as in Lemma _2.16_. Fix
$A_{*}\in[0,\mathcal{A}),\gamma\in(-\frac{1}{\nu\delta},0)$,
$\lambda\in[\frac{3}{2}Z_{*},\Lambda]$, and $k\geq 1$. For each integer $N\geq
1$, there is a unique $\varepsilon_{N}\geq 0$ such that
$\mathcal{W}^{-}_{\varepsilon_{N}}$ is a saddles connector with winding number
$w_{\varepsilon_{N}}=N$. Moreover, they are monotonic:
$0\leq\varepsilon_{1}\leq\varepsilon_{2}\leq\dotsb$.

###### Proof.

Use the same argument as in the proof of Proposition 2.13, but replace
Proposition 2.11 with Proposition 2.19. ∎

For positive integers $k$, the previous proposition establishes winding
numbers for all $N\geq 1$. From Proposition 2.9, we know that there are no
negative-valued winding numbers. This leaves open the possibility for the
dynamical system to have a winding number of $0$. However, for the usual
Coulomb problem on Minkowski spacetime, a winding number of zero is not
obtained, see section 7.4 of [36] (and also section 4 of [23]). The next
proposition finds sufficient conditions guaranteeing that the dynamical system
does not have winding number $0$ when $k$ is positive.

######  Proposition 2.21.

Let $A_{*}\in[0,\tfrac{1}{10}Z_{*})$,
$Z_{*}\in[\frac{1}{2}\alpha_{\mbox{\tiny{S}}}\sqrt{\mathcal{G}},\sqrt{\mathcal{G}})$,
and $\lambda\in[\tfrac{3}{2}Z_{*},\tfrac{1}{2\pi}\alpha_{\mbox{\tiny{S}}}]$.
Then for all $\varepsilon\in[0,1)$ and integers $k\geq 1$, the winding number
satisfies $w_{\varepsilon}\geq 1$.

######  Remark 2.22.

_In this proposition, note that
$\alpha_{\mbox{\tiny{S}}}\sqrt{\mathcal{G}}\approx 3.58\cdot 10^{-24}$ is the
value of $Z_{*}$ for when $Z=1$. In terms of $\gamma$, the range
$Z_{*}\in[\frac{1}{2}\alpha_{\mbox{\tiny{S}}}\sqrt{\mathcal{G}},\sqrt{\mathcal{G}})$
is equivalent to $\gamma\in(-1,\tfrac{1}{2}\alpha_{\mbox{\tiny{S}}}]$. In
terms of elementary charge units $Z\in\mathbb{N}$, the proposition is valid
for $1\leq Z\leq 137$. $\square$ _

###### Proof.

We want to construct an upper barrier for the system. Since $\dot{\Omega}$
decreases as $\varepsilon$ increases, it suffices to find an upper barrier for
$\varepsilon=0$. Therefore, for the remainder of this proof, we set
$\varepsilon=0$.

By the Hartman–Grobman theorem, the unstable manifold emanating from $S^{-}$
has tangent vector $\partial_{\eta}-2\frac{\lambda}{Z_{*}}\partial_{\Omega}$
at $S^{-}$. Therefore the unstable manifold begins below the horizontal line
$\Omega=0$. Now consider the slope field along $\Omega=0$. We have

$\dot{\Omega}|_{\Omega=0}\,=\,\frac{2\eta^{2}}{g(\eta)}+\frac{2\eta^{2}}{g^{2}(\eta)}\gamma(1-\eta).$

This is negative provided $g(\eta)<-\gamma(1-\eta)$, which holds iff
$g^{2}(\eta)<\gamma^{2}(1-\eta)^{2}$, i.e. iff

$a\eta^{2}+b\eta+c<0,$

where

$\displaystyle a\,$ $\displaystyle:=\,1+2A_{*}+Z_{*}^{2}-\gamma^{2}$
$\displaystyle b\,$ $\displaystyle:=\,-2(A_{*}+Z_{*}^{2}-\gamma^{2})$
$\displaystyle c\,$ $\displaystyle:=\,Z_{*}^{2}-\gamma^{2}.$

The assumption $Z_{*}<\sqrt{\mathcal{G}}$ is equivalent to $\gamma^{2}<1$.
Therefore the parabola opens up. We see that $a\eta^{2}+b\eta+c<0$ holds for
all positive $\eta$ less than

$\displaystyle\frac{-b+\sqrt{b^{2}-4ac}}{2a}\,$
$\displaystyle=\,\frac{A_{*}+Z_{*}^{2}-\gamma^{2}+\sqrt{A_{*}^{2}-Z_{*}^{2}+\gamma^{2}}}{1+2A_{*}+Z_{*}^{2}-\gamma^{2}}$
$\displaystyle=\,\frac{A_{*}-Z_{*}^{2}(\frac{1}{\mathcal{G}}-1)+\sqrt{A_{*}^{2}+Z_{*}^{2}(\frac{1}{\mathcal{G}}-1)}}{1+2A_{*}-Z_{*}^{2}(\frac{1}{\mathcal{G}}-1)}$
$\displaystyle>\,\frac{-Z_{*}^{2}(\frac{1}{\mathcal{G}}-1)+Z_{*}\sqrt{\frac{1}{\mathcal{G}}-1}}{1+0.2Z_{*}-Z_{*}^{2}(\frac{1}{\mathcal{G}}-1)}.$

A quick calculation shows that the above expression increases as $Z_{*}$
increases. The minimum value for $Z_{*}$ we are assuming is one-half the
charge of the electron, i.e.
$Z_{*}\geq\frac{1}{2}\alpha_{\mbox{\tiny{S}}}\sqrt{\mathcal{G}}$. Plugging
this into the above expression gives

$\frac{-b+\sqrt{b^{2}-4ac}}{2a}\,>\,\frac{-\frac{1}{4}\alpha_{\mbox{\tiny{S}}}^{2}(1-\mathcal{G})+\frac{1}{2}\alpha_{\mbox{\tiny{S}}}\sqrt{1-\mathcal{G}}}{1+0.1\alpha_{\mbox{\tiny{S}}}\sqrt{\mathcal{G}}-\frac{1}{4}\alpha_{\mbox{\tiny{S}}}^{2}(1-\mathcal{G})}\,\approx\,0.00364\,>\,0.003.$

We conclude that $\dot{\Omega}$ is negative along the horizontal line
$\Omega=0$ for $\eta\in(0,0.003)$. Set

$\eta_{0}\,:=\,\frac{1}{10}\cdot 0.003=0.0003.$

We form a slanted barrier given by the line segment joining the points
$(\eta_{0},0)$ and $(1,-\pi)$ in the $(\eta,\Omega)$ plane. This line segment
is given by

$\Omega_{0}(\eta)\,:=\,-\frac{\pi}{1-\eta_{0}}(\eta-\eta_{0}).$

To show that this line segment is a barrier, we want to show that the slope
field $\frac{\dot{\Omega}}{\dot{\eta}}$ is less than the slope of
$\Omega_{0}(\eta)$. Therefore we want to show

$\dot{\Omega}\big{(}\eta,\Omega_{0}(\eta)\big{)}+\frac{\pi}{1-\eta_{0}}\eta(1-\eta)^{2}\,<\,0\quad\text{
for all }\quad\eta\in[\eta_{0},1).$

First we estimate $\dot{\Omega}\big{(}\eta,\Omega_{0}(\eta)\big{)}$. By
definition, we have

$\dot{\Omega}\big{(}\eta,\Omega_{0}(\eta)\big{)}\,=\,\frac{2\eta^{2}}{g(\eta)}\cos\Omega_{0}(\eta)+\frac{2}{g(\eta)}\big{(}k\eta(1-\eta)-\lambda(1-\eta)^{2}\big{)}\sin\Omega_{0}(\eta)+\frac{2\eta^{2}}{g^{2}(\eta)}\gamma(1-\eta).$

We estimate each term separately. In the following, we use the bounds

$g(\eta)\,>\,\frac{\eta}{1.01}\quad\text{ and }\quad
g(\eta)\,<\,\eta+\sqrt{\mathcal{G}}.$

The first inequality was shown in the proof of Lemma 2.14. The second
inequality follows since
$g(\eta)<(1-Z_{*})\eta+Z_{*}<\eta+\sqrt{\mathcal{G}}$. Using the above bounds,
we have

$\displaystyle\frac{2\eta^{2}}{g(\eta)}\cos\Omega_{0}(\eta)\,$
$\displaystyle=\,\frac{2\eta^{2}}{g(\eta)}\cos\Omega_{0}(\eta)\mathds{1}_{[\eta_{0},\eta_{1}]}(\eta)+\frac{2\eta^{2}}{g(\eta)}\cos\Omega_{0}(\eta)\mathds{1}_{[\eta_{1},1]}(\eta)$
$\displaystyle<\,2.02\eta\cos\Omega_{0}(\eta)\mathds{1}_{[\eta_{0},\eta_{1}]}(\eta)+\frac{2\eta^{2}}{\eta+\sqrt{\mathcal{G}}}\cos\Omega_{0}(\eta)\mathds{1}_{[\eta_{1},1]}(\eta).$

In above, $\mathds{1}$ denotes the indicator function and
$\eta_{1}:=\frac{1+\eta_{0}}{2}=0.5015$. Note that $\cos\Omega_{0}(\eta)$ is
positive on $(\eta_{0},\eta_{1})$ and negative on $(\eta_{1},1)$ which
justifies the above inequality.

Continuing with our estimate of
$\dot{\Omega}\big{(}\eta,\Omega_{0}(\eta)\big{)}$, we have

$\frac{2}{g(\eta)}k\eta(1-\eta)\sin\Omega_{0}(\eta)\,<\,\frac{2}{\eta+\sqrt{\mathcal{G}}}\eta(1-\eta)\sin\Omega_{0}(\eta),$

and

$-\frac{2}{g(\eta)}\lambda(1-\eta)^{2}\sin\Omega_{0}(\eta)\,<\,-\frac{2.02}{\eta}\frac{1}{2\pi}\alpha_{\mbox{\tiny{S}}}(1-\eta)^{2}\sin\Omega_{0}(\eta),$

and

$\frac{2\eta^{2}}{g^{2}(\eta)}\gamma(1-\eta)\,<\,-\frac{\eta^{2}}{(\eta+\sqrt{\mathcal{G}})^{2}}\alpha_{\mbox{\tiny{S}}}(1-\eta).$

We conclude that for $\eta\in[\eta_{0},1]$, we have

$\displaystyle\dot{\Omega}\big{(}\eta,\Omega_{0}(\eta)\big{)}\,$
$\displaystyle<\,2.02\eta\cos\Omega_{0}(\eta)\mathds{1}_{[\eta_{0},\eta_{1}]}(\eta)+\frac{2\eta^{2}}{\eta+\sqrt{\mathcal{G}}}\cos\Omega_{0}(\eta)\mathds{1}_{[\eta_{1},1]}(\eta)$
$\displaystyle\>\>\>\>+\frac{2}{\eta+\sqrt{\mathcal{G}}}\eta(1-\eta)\sin\Omega_{0}(\eta)-\frac{2.02}{\eta}\frac{1}{2\pi}\alpha_{\mbox{\tiny{S}}}(1-\eta)^{2}\sin\Omega_{0}(\eta)$
$\displaystyle\>\>\>\>-\frac{\eta^{2}}{(\eta+\sqrt{\mathcal{G}})^{2}}\alpha_{\mbox{\tiny{S}}}(1-\eta)$
$\displaystyle=:\,h(\eta).$

Notice that $h(\eta)$ is solely a function of $\eta$ since
$\sqrt{\mathcal{G}}$ and $\alpha_{\mbox{\tiny{S}}}$ are treated as numerical
constants in this proof.

Therefore, to show that the line segment $\Omega_{0}(\eta)$ is a barrier, it
suffices to show

$j(\eta)\,:=\,h(\eta)+\frac{\pi}{1-\eta_{0}}\eta(1-\eta)^{2}\,<\,0\quad\text{
for all }\quad\eta\in[\eta_{0},1).$

A plot of $j(\eta)$ shows that it’s negative for $\eta\in[\eta_{0},1)$. To
prove this rigorously, we appeal to algebraic techniques. First consider the
interval $[\eta_{0},\eta_{1}].$ Estimating via Taylor series, we have for
$\eta\in[\eta_{0},\eta_{1}]$

$\displaystyle h(\eta)\,\leq\,\tilde{h}(\eta)\,$
$\displaystyle:=\,2.02\eta\left(1-\frac{\Omega_{0}(\eta)^{2}}{2!}+\frac{\Omega_{0}(\eta)^{4}}{4!}\right)$
$\displaystyle\>\>\>\>+\frac{2}{\eta+\sqrt{\mathcal{G}}}\eta(1-\eta)\left(\Omega_{0}(\eta)-\frac{\Omega_{0}(\eta)^{3}}{3!}\right)-\frac{2.02}{\eta}\frac{1}{2\pi}\alpha_{\mbox{\tiny{S}}}(1-\eta)^{2}\Omega_{0}(\eta)$
$\displaystyle\>\>\>\>-\frac{\eta^{2}}{(\eta+\sqrt{\mathcal{G}})^{2}}\alpha_{\mbox{\tiny{S}}}(1-\eta).$

Define
$\tilde{j}(\eta):=\tilde{h}(\eta)+\frac{\pi}{1-\eta_{0}}\eta(1-\eta)^{2}$. It
suffices to show $\tilde{j}(\eta)$ is negative for
$\eta\in[\eta_{0},\eta_{1}]$. Clearing the denominator, set

$\tilde{J}(\eta)\,:=\,\eta(\eta+\sqrt{\mathcal{G}})^{2}\tilde{j}(\eta).$

Then $\tilde{j}(\eta)<0$ on $[\eta_{0},\eta_{1}]$ iff $\tilde{J}(\eta)<0$ on
$[\eta_{0},\eta_{1}]$. Since $\tilde{J}(\eta_{1})\approx-0.064<0$, it suffices
to show $\tilde{J}(\eta)$ has no zeros between $\eta_{0}$ and $\eta_{1}$.
Indeed, $\tilde{J}(\eta)$ is an eighth degree polynomial, and an application
of Sturm’s theorem [5, §2.2.2] (which can be implemented algorithmically via a
computer program) shows that $\tilde{J}(\eta)$ has no zeros between $\eta_{0}$
and $\eta_{1}$. (When applying Sturm’s theorem, it’s important to work with
rational coefficients. Therefore the numerical values in $\tilde{J}(\eta)$
have to be bounded by rational numbers.) For the interval $[\eta_{1},1]$, one
proceeds analogously but begins the Taylor series at $\Omega=\frac{\pi}{2}$
instead of $\Omega=0$. ∎

We combine the results of this section into our main theorem (cf. the main
result in [23]). The connection of winding numbers with the usual
spectroscopic notation of the Hydrogen spectrum, as outlined in section 4 of
[23], applies to our current situation. This correspondence is essential for
our numerical analysis in the next section.

######  Theorem 2.23.

Suppose $A_{*}\in[0,\tfrac{1}{10}Z_{*})$,
$\gamma\in(-\tfrac{1}{3.03},-\tfrac{1}{2}\alpha_{\mbox{\tiny{S}}})$, and
$\lambda\in[\tfrac{3}{2}Z_{*},\tfrac{1}{2\pi}\alpha_{\mbox{\tiny{S}}})$.

  * •

Fix an integer $k\leq-1$. For each integer $N\geq 0$, there is a unique
$\varepsilon_{N}\geq 0$ such that $\mathcal{W}^{-}_{\varepsilon_{N}}$ is a
saddles connector with winding number $w_{\varepsilon_{N}}=N$. Moreover, they
are monotonic: $0\leq\varepsilon_{0}\leq\varepsilon_{1}\leq\dotsb$. Lastly,
there are no saddles connectors with winding number $\leq-1$.

  * •

Fix an integer $k\geq 1$. For each integer $N\geq 1$, there is a unique
$\varepsilon_{N}\geq 0$ such that $\mathcal{W}^{-}_{\varepsilon_{N}}$ is a
saddles connector with winding number $w_{\varepsilon_{N}}=N$. Moreover, they
are monotonic: $0\leq\varepsilon_{1}\leq\varepsilon_{2}\leq\dotsb$. Lastly,
for $\varepsilon\in[0,1)$, there are no saddles connectors with winding number
$\leq 0$.

###### Proof.

For $k\leq-1$, the result follows by Propositions 2.13 and 2.9. For $k\geq 1$,
the result follows by Propositions 2.20 and 2.21. ∎

######  Remark 2.24.

_In terms of atomic numbers, Theorem 2.23 is valid for $1\leq Z\leq 45$. The
nuclear mass number range is valid for $0\leq
A<\tfrac{1}{10}(\tfrac{1}{\sqrt{\mathcal{G}}}\tfrac{m_{\mathrm{e}}}{m_{\mathrm{p}}}\alpha_{S})Z$,
which overwhelmingly contains the range $Z\leq A<3Z$ for empirically confirmed
nuclei. The anomalous magnetic moment of the electron, written in units of the
Bohr magneton, $\mu_{a}=-a\frac{e\hbar}{2m_{\mathrm{e}}c}$, is valid for
$3\sqrt{\mathcal{G}}\leq a<\frac{1}{Z}$; for our values of $Z$ this range
contains $a=\tfrac{\alpha_{S}}{2\pi}$, the leading contribution derived from
QED. $\square$ _

######  Remark 2.25.

_In Theorem 2.23, we only prove the existence of _non-negative_ energy
eigenvalues. This is not because negative energy eigenvalues don’t necessarily
exist, but rather that the assumption of non-negative energy eigenvalues
simplified the proofs of the propositions in this section. Negative energy
eigenvalues exist as well, which is suggested by our numerical analysis in the
next section, but a proof of their existence would require a modification to
the proofs in this section. $\square$_

##  3 A Numerical Challenge

In this section we present the results of some numerical experimentation we
have carried out, using the computing platform MATLAB [19], regarding the
discrete spectrum of the Dirac hamiltonian for a single electron on a
Reissner–Weyl–Nordstr̈om background spacetime that models the electrostatic
vacuum outside the point nucleus of a hydrogenic ion, with the electron
assumed to have an anomalous magnetic moment. We will point out what we
believe is the main challenge in carrying out this task in a mathematically
rigorous way, and state a conjecture we believe is worthy of a careful study.

The basic idea for the numerical computation of energy eigenvalues picks up on
the approach we took in analyzing the system (2.27), (2.28), namely, to use
the Prüfer transform to reduce the problem to the study of a single nonlinear
first-order ODE (2.32) on $(0,\infty)$ that depends on the unknown energy
eigenvalue $\varepsilon$. By Theorem 2.6 this ODE satisfies an asymptotic
boundary value problem

$\displaystyle\lim_{r\to 0^{+}}\Omega(r)$ $\displaystyle=$ $\displaystyle 0$
(3.42) $\displaystyle\lim_{r\to\infty}\Omega(r)$ $\displaystyle=$
$\displaystyle-2\pi N-\arccos\varepsilon$ (3.43)

where $N$ denotes the winding number of the corresponding orbit connecting the
two saddle-nodes $S^{-}$ and $S^{+}$. The reason two boundary values can be
specified for this first-order equation is that $\varepsilon$ is unknown. From
the saddle-point nature of these two equilibrium points we in fact know that
the solution in question is unstable, so that for any fixed $N\geq 0$ and a
generic value of $\varepsilon\in[-1,1]$, the orbit satisfying (3.42) will
either over-shoot or under-shoot its target, depending on whether
$\varepsilon$ is smaller or larger than a critical value
$\varepsilon_{N}\in(-1,1)$, which as we have shown, is an energy eigenvalue
for the hamiltonian under study. This therefore makes it possible to follow a
bisection algorithm in which, starting from an initial guess for $\varepsilon$
the ODE (2.32) is solved over a suitably long interval in $r$ and with only
condition (3.42) enforced, and based on whether the final value of this
solution is above or below the target (3.43), the length of the interval
containing the eigenvalue is halved repeatedly until it is below a pre-set
tolerance.

######  Remark 3.1.

_The utility of the Prüfer transform in finding numerical approximations for
energy eigenvalues of spherically-symmetric Dirac hamiltonians was noted long
ago in[42], and used also in [33], but this technique has remained relatively
under-appreciated in the mathematical physics community. The paper [24] seems
to be the first to realize that the Prüfer method can be extended to the
axisymmetric case, which yields a system of two coupled ODEs with two unknown
parameters (separation-of-variables constants corresponding to the energy
eigenvalue and (a generalization of) the spin-orbit coupling,) so that the
above iterative algorithm then needs to be alternated between the two ODEs
until both intervals become smaller than the tolerance. This is the approach
taken to obtain the numerical results in [23]. Other variants of this method
have since been employed in related problems [11]. $\square$ _

We use the MATLAB ODE solver routine ode89, which is an implementation of
Verner’s “most robust” Runge–Kutta 9(8) pair with an 8th-order continuous
extension [38]. The default absolute tolerance for this ODE solver is
$10^{-6}$ and the relative tolerance is $10^{-3}$. This is an adaptive method:
during the integration the step size is adapted so that the estimated error
remains below the specified threshold. For the bisection algorithm to find the
energy eigenvalues, the tolerance we used is around the machine precision,
i.e. $10^{-15}$.

The immediate challenge in numerically solving the initial value problem
(2.32–3.42) is that the coefficients of the ODE are singular at $r=0$. This is
true even in the case $\mathcal{G}=0$. Thus, for the purpose of numerically
solving this ODE, we need to replace (3.42) with $\Omega(r_{0})=0$ for some
very small $r_{0}>0$. Clearly, $r_{0}$ needs to be larger than the machine
precision. Here we have used $r_{0}=10^{-6}$. On the other hand, the function
$f(r)$ appearing in (2.32) has the peculiar behavior near $r=0$, as evident
from (2.24) that, except for an extremely tiny interval of size
$r_{\mbox{\tiny res}}:=\alpha_{\mbox{\tiny{S}}}\sqrt{\mathcal{G}}\approx
10^{-23}$ around $r=0$, it is practically equal to 1, which is its exact value
in the absence of gravity. Clearly, $r_{\mbox{\tiny res}}$ is many order of
magnitudes smaller than the machine precision $10^{-15}$, as a result of
which, no difference with the case $f\equiv 1$ will be discernible in the
computed solutions. We could even use an artificial value for the non-
dimensional parameter $\mathcal{G}$ that is much larger than its physical
value, and still obtain the same result, so long as $r_{0}>r_{\mbox{\tiny
res}}$. On the other hand, a much smaller $r_{0}$ that is comparable to
$r_{\mbox{\tiny res}}$ would result in the numerical scheme encountering such
large gradients that it would not be able to choose a step size to proceed
with the computation. It follows that, keeping $r_{0}$ at its present value,
there is a threshold $\mathcal{G}_{\mbox{\tiny res}}\approx 10^{-10}$ such
that for any $\mathcal{G}<\mathcal{G}_{\mbox{\tiny res}}$, our numerical way
of computing eigenvalues will not be able to distinguish it from
$\mathcal{G}=0$. This approach, therefore, does not tell us much about the
effect that “turning gravity on” would have on the spectral lines of the
hydrogenic ion in presence of the electron anomalous magnetic moment.

Now, a solution of the asymptotic boundary value problem (2.32), (3.42),
(3.43), in addition to the unknown $\varepsilon$ depends on five other
constants: the two integers $k$ and $N$, and real numbers $\gamma$, $\lambda$,
and $\mathcal{G}$ (through the dependence of $A_{*}$ and $Z_{*}$ on
$\mathcal{G}$.) The dependence of the energy eigenvalues on each of these
parameters is of course worthy of study, but from the above discussion it
should be clear that the last of these, the dependence on $\mathcal{G}$,
presents some unique challenges. The first three parameters, $k$, $N$, and
$\gamma$ already feature in the Sommerfeld fine structure formula (1.4), upon
making the substitutions $N:=n-|k|$ and $\gamma:=-Z\alpha_{\mbox{\tiny{S}}}$.
Formally, setting $\lambda=0$ and $\mathcal{G}=0$ in the reduced hamiltonian
(2.25) yields the original Dirac hamiltonian with pure Coulomb potential,
whose energy eigenvalues are given by (1.4). Plotting the $E_{n,k}$ values in
this formula as a function of $Z$ yields a set of curves that are depicted by
dashed lines in Figure 1. The solid lines in this figure are the result of our
numerical computation of energy eigenvalues when
$\mathcal{G}<\mathcal{G}_{\mbox{\tiny res}}$ and $\lambda$ is set to its
physical value.

Figure 1: Electron Orbital Energies vs. the Nuclear Charge, with
$\mathcal{G}<\mathcal{G}_{\mbox{\tiny res}}$

Our result here therefore conforms with (and confirms) Fig. 7.1 in [36].

A feature evident in Fig. 1 is that, even though our main result, Thm. 2.23,
only holds for atomic numbers $Z\leq 45$, and only proves the existence of
positive energy eigenvalues, the discrete spectrum apparently continues to
exist beyond that range of $Z$ values, and far beyond nuclei found in nature
or observed in the lab. It also appears to be the case, for any $n\geq 1$ and
any nonzero $k\in\\{-n,-n+1,\dots,n-1\\}$, that as $Z$ is increased the energy
continues to decrease, eventually crossing the $\varepsilon=0$ line and
becoming negative, and finally touching the continuous spectrum at
$\varepsilon=-1$. The restrictions in our theorem are therefore far from being
optimal. Indeed, a rich list of phenomena seems to occur in this system beyond
the range of $Z$ corresponding to nuclei so far observed in nature, phenomena
that are awaiting to be studied by further careful mathematical analysis of
the hamiltonian (2.25).

##  4 Summary and Outlook

###  4.1 Summary

In this paper we have supplied a complete characterization of the discrete
Dirac spectrum of a point electron with fixed anomalous magnetic moment in the
Reissner–Weyl–Nordström spacetime of a point nucleus, though only within
certain ranges of the atomic number $Z$, the nuclear mass number $A$, and
anomalous magnetic moment $\mu_{a}$. The $(Z,A)$ range contains approximately
50% of the empirically confirmed nuclei, from those of Hydrogen (H) to those
of Rhodium (Rh), and allows us to form a one-to-one correspondence between the
eigenfunctions of our hamiltonian and the orbitals of Hydrogen found in
textbook quantum mechanics. The discrete spectrum is indexed by two integers
$k$ and $N$, where $k$ is the eigenvalue of the spin-orbit operator, and $N$
can be identified with a topological winding number associated with a
dynamical system on a compact cylinder that arises from the energy eigenvalue
problem of our hamiltonian. The correspondence to the orbitals of Hydrogen is
facilitated by defining a principal quantum number $n:=N+|k|$.

We also carried out a numerical study of the energy eigenvalues as a function
of atomic number $Z$; however, we face a challenge: the gravitational coupling
constant is so small that our algorithm cannot distinguish between the general
and special relativistic problems.

Our numerical results conform with those in [36] that, as far as we can tell,
nobody had tried to reproduce since their publication. Incidentally, note that
the eigenvalues of $H$ _appear_ to “dive into the negative continuum” when $Z$
becomes sufficiently large. However, by a result of Weidmann [40], the
interior of the essential spectrum of the Dirac hamiltonian $H$ of an electron
with anomalous magnetic moment is _purely absolutely continuous_ , and this
holds for whether the electron lives in the RWN spacetime of a point nucleus,
or in the Minkowski spacetime equipped with a point nucleus. Hence, the
statement ensuing Fig. 7.1 in [36], that the eigenvalues of the special-
relativistic $H$ would dive into the lower continuum, should not be taken
literally.

We emphasize the non-perturbative character of our paper, with its insistence
on conceptual clarity that logically leads one to face the spectral questions
we have addressed in this paper. In this regard it differs markedly from
conventional inquiries into the effects of gravity on the hydrogenic spectra,
e.g. [29], [28], which only consider the influence of an external
gravitational field, and only in a perturbative manner. In a similar vein, our
paper also differs markedly from conventional inquiries into the effect of the
electron’s (and other leptons’) anomalous magnetic moment on the spectra, cf.
[18], [14], [9].

###  4.2 Outlook

Having a well-defined general-relativistic Dirac operator for hydrogenic ions
with nuclei that are treated as point particles, and having characterized its
discrete spectrum (the essential spectrum had been identified earlier already,
in [7]), the litmus test for the operator now is the question whether it has
the _correct_ energy spectrum, in the sense that its eigenvalue differences
reproduce the empirical values of the Hydrogen frequencies within an
empirically dictated margin of error. Since no empirical data exist for
$Z>118$, the question is currently decidable only for when $Z\leq 118$. This
question has not been addressed yet.

We emphasize that this question is more difficult to answer than the analogous
question for the special-relativistic Dirac operator with the anomalous
magnetic moment of the electron taken into account, which implicitly _assumes_
that gravitational effects are negligible — precisely the assumption that one
still has to vindicate rigorously. Yet, suppose for the sake of the argument
right now that gravity is negligible. In that case the convergence in the
regime $Z\leq 137$ of the spectrum of the special-relativistic Dirac operator
with $\mu_{a}$ term to the spectrum of the distinguished self-adjoint
extension of the one without it, when $\mu_{a}\to 0$, as noted in [21] and
[33], is a very helpful piece of information to have. It allows one to work
with the explicitly computable spectrum of the distinguished self-adjoint
extension of the special-relativistic Dirac operator without $\mu_{a}$ term,
plus $\mu_{a}$-dependent error bounds, to answer the question. By contrast,
when not assuming that gravity is negligible, the analogous strategy does not
work for general-relativistic problem with fixed $G>0$, because one can not
take the limit $|\mu_{a}|\searrow 0$, only the limit
$|\mu_{a}|\searrow\mu_{\rm crit}$, with $\mu_{\rm crit}$ given in (1.11), but
this does not simplify the task at hand.

However, it is reasonable to conjecture that the spectrum of the general-
relativistic Dirac hamiltonian in presence of the anomalous magnetic moment
for the electron, converges to the corresponding special-relativistic Dirac
spectrum as $\mathcal{G}$ (or equivalently, Newton’s constant of universal
gravitation $G$) is sent to zero. For reasons described in section 3 and in
the Introduction, this conjecture is easy to conceive but (we suspect) not so
easy to prove. After all, our discussion in section 3 of the RWN metric
function $f(r)$ is equally valid if the electron’s anomalous magnetic moment
is absent from the Dirac hamiltonian, and in that case one knows through the
work of Cohen and Powers [8] that the singular behavior of $f(r)$ as $r\to 0$
destroys the essential self-adjointness of the Dirac hamiltonian with $G=0$
for all $Z\leq 118$; the fact that $f(r)$ is practically 1 except for a tiny
vicinity of $r=0$ is thus not a good guide for one’s intuition.

Be that as it may, we expect to be able to take the limit $G\searrow 0$ for
$\mu_{a}$ fixed, which — if the point spectra converge to the special-
relativistic $G=0$ spectra — would reduce the problem to understanding the
special-relativistic Dirac spectrum with anomalous magnetic moment of the
electron taken into account, which in turn can be estimated with the help of
the strategy mentioned a few lines earlier. Yet, the $G\searrow 0$ limit has
not been rigorously established yet. It is an interesting problem that we hope
to return to in a future work.

Two interesting open questions concerning the Dirac hamiltonian of a point
electron with fixed anomalous magnetic moment in either the RWN spacetime of a
point nucleus, or the Minkowski spacetime equipped with a point nucleus, are
suggested by Fig. 1. The first question is based on an observation already
made in [36], namely the apparent level crossings at zero energy. Numerically
we found that the crossings do not happen exactly at $E=0$, but the difference
is so tiny that the deviation from $E=0$ may well be a numerical artifact. It
would be good if it could be proved, or disproved, that level crossings happen
precisely at $E=0$. The second question concerns the manner in which the
eigenvalue curves meet the negative continuum. Do they meet it with a finite
non-zero slope (treating $Z$ as real variable), or with a vanishing slope, or
do they never meet the continuum but get asymptotically arbitrarily close to
it? These are interesting mathematical questions.

Incidentally, supposing an eigenvalue curve actually meets the negative
continuum at $Z=Z_{\circ}$, say, a natural question to ask in that case is
whether $E=-m_{\mathrm{e}}c^{2}$ is then part of the point spectrum at
$Z_{\circ}$. (It certainly could not be part of the discrete spectrum, as
$E=-m_{\mathrm{e}}c^{2}$ belongs to the essential spectrum.) That question was
already answered in [25], though, where a necessary condition was stated for
$E=-m_{\mathrm{e}}c^{2}$ to be an eigenvalue of the Dirac operator of an
electron with anomalous magnetic moment in electrostatic spacetimes of a point
nucleus for a vast class of electromagnetic vacuum laws, see Thm. 3.25 in
[25]. For the RWN spacetime this condition cannot be met in the naked
singularity sector.

Another task for the future, motivated by physics, is to take the finite size
of nuclei into account. At large $Z$ values, even in the range of the
empirical nuclei, one risks stretching the validity of the point nucleus
approximation beyond the estimable. Although the charge radius of a $Z=96$
nucleus is only a factor $\approx 7$ bigger than that of a proton (cf. the
Nuclear Charge Radii website of the International Atomic Energy Commission),
the electron in its ground state is then typically much closer to the nucleus
than when $Z=1$, so that the charge distribution of the nucleus should begin
to have a significant effect on the spectra.

A final thought on the problem of essential self-adjointness for hydrogenic
ions, related to the finite size of the physical nuclei: It is often argued,
cf. [36], that the problem of a lack of essential self-adjointness of the
Dirac operator for hydrogenic ions (without anomalous magnetic moment of the
electron) disappears if the nucleus is equipped with a continuously smeared
out charge distribution rather than being treated as a point charge. However,
this argument misses the point (pardon the pun) that nuclei are spatially
extended bound states made of three point quarks, according to the standard
model of elementary particle physics. Assuming that this standard model
captures the nature of nuclei correctly, nuclei should fundamentally be
associated with point charges when discussing hydrogenic (and other atomic /
ionic) spectra, though the approximation to concentrate all the point charges
at a single location is a simplification that turns into an over-
simplification when $Z>118$ (as far as essential self-adjointness is
concerned; for the spectra it presumably is an over-simplification already for
smaller $Z$ values). Yet, since the charges of point quarks have magnitude not
more than $\frac{2}{3}e$, it is conceivable that essential self-adjointness
will hold for Dirac operators of an electron without anomalous magnetic moment
in the Coulomb field of nuclei modeled as a sum of Dirac distributions of many
point charges with magnitude not more than $\frac{2}{3}e$. We are not aware of
any work that has addressed the hydrogenic Dirac problem for such models of
nuclei. It should not be too difficult to address this problem in Minkowski
spacetime. However, the general-relativistic problem will be formidable.

Another issue that needs to be addressed is whether $\mu_{a}$ changes
significantly with the strength of the Coulomb field ($Z$), or the
gravitational field ($M$). Our use of the classical value for the anomalous
magnetic moment independently of $Z$ and of $M$, which is admissible for
electrons in weak external fields, may be an oversimplification. In any event,
by using a fixed $\mu_{a}(=-\mu_{cl})$ for all $Z$ and $M$ values we have
followed the tradition [6], [15], [36], [7].

## Acknowledgments

We thank Moulik Balasubramanian for bringing [42] to our attention. Eric Ling
was supported by Carlsberg Foundation CF21-0680 and Danmarks
Grundforskningsfond CPH-GEOTOP-DNRF151.

Appendix

##  Appendix A The reduced Dirac hamiltonian as a linear matrix operator

As mentioned in section 2, the Dirac operator $H$ of a (test) electron with
anomalous magnetic moment $\mu_{a}=-\mu_{\text{class}}$ in the RWN spacetime
of a naked point nucleus of charge $Q=Ze$ and mass $M=A(Z,N)m_{\mathrm{p}}$ is
essentially self-adjoint on the domain of $C^{\infty}$ bispinor wave functions
which are compactly supported away from the singularity at $r=0$; see [7]. We
also mentioned that this is so only because the physical value of the
anomalous magnetic moment is larger (actually, much larger) than a critical
value. This is in contrast to the corresponding problem on the Minkowski
spacetime with point nucleus, where no critical anomalous magnetic moment has
to be surpassed. In this appendix we briefly address the existence of a
critical magnetic moment for the Dirac problem in the RWN spacetime versus its
absence in the Minkowski spacetime.

Recall that due to the spherical symmetry and static character of the RWN and
Minkowski spacetimes, the Dirac hamiltonian separates in the spherical
coordinates and their default spin frame [8], resp. [36]. More precisely, $H$
is a direct sum of radial partial-wave Dirac operators
$H^{\mbox{\tiny{rad}}}_{k}:=m_{\mathrm{e}}c^{2}K^{\mbox{\tiny{}}}_{k}$,
$k\in\mathbb{Z}\backslash\\{0\\}$, cf. (2.17),

$\displaystyle
K_{k}^{\text{}}:=\left[\begin{array}[]{cc}f(r)-Z\alpha_{\mbox{\tiny{S}}}\frac{1}{r}&\left[\frac{k}{r}-Z\alpha_{\mbox{\tiny{S}}}^{2}\frac{1}{4\pi}\frac{1}{r^{2}}\right]f(r)-f^{2}(r)\tfrac{d}{dr}\\\
\left[\frac{k}{r}-Z\alpha_{\mbox{\tiny{S}}}^{2}\frac{1}{4\pi}\frac{1}{r^{2}}\right]f(r)+f^{2}(r)\tfrac{d}{dr}&-f(r)-Z\alpha_{\mbox{\tiny{S}}}\frac{1}{r}\end{array}\right],$
(A.46)

with $f(r)$ given in (2.23) for the RWN spacetime, while $f(r)\equiv 1$ for
the Minkowski spacetime. Thus $H$ is essentially self-adjoint when all the
operators $K^{\mbox{\tiny{}}}_{k}$ are, which act on
$g(r):=\big{(}u(r),v(r)\big{)}^{T}\in C^{\infty}_{c}(\mathbb{R}_{+})^{2}$
equipped with a weighted $L^{2}$ norm given by

$\|g\|^{2}:=\int_{0}^{\infty}\frac{1}{f^{2}(r)}\Big{(}|u(r)|^{2}+|v(r)|^{2}\Big{)}\mathrm{d}r;$
(A.47)

cf. (2.18).

Setting $E_{k,d(k)}=:m_{\mathrm{e}}c^{2}\varepsilon_{k,d(k)}$, with
$k\in\mathbb{Z}\backslash\\{0\\}$, and where $d(k)$ denotes a degeneracy
label, the dimensionless eigenvalue problem reads

$K^{\mbox{\tiny{}}}_{k}g(r)=\varepsilon_{k,d(k)}g(r).$ (A.48)

With the essential spectrum equal to $(-\infty,-1]\cup[1,\infty)$ for either
choice of $f$, see [7] and [36], we only need to address the regime
$\varepsilon\in(-1,1)$. Even though the eigenvalue problem does not seem to be
solvable in terms of known functions, special or not, whether $f(r)\equiv 1$
or given by (2.23), Weyl’s limit point / limit circle criterion [39] states
that the issue of essential self-adjointness is decided by studying the
asymptotic problems for when $r\nearrow\infty$ and when $r\searrow 0$.

For $r\nearrow\infty$ (A.48) becomes the free electron problem

$\displaystyle\left[\begin{array}[]{cc}1&-\tfrac{d}{dr}\\\
\tfrac{d}{dr}&-1\end{array}\right]\left[\begin{array}[]{c}u(r)\\\
v(r)\end{array}\right]=\varepsilon\left[\begin{array}[]{c}u(r)\\\
v(r)\end{array}\right];\qquad r\gg 1,$ (A.55)

with $\varepsilon=\varepsilon_{k,d(k)}$, which has the well-known unique (up
to an overall factor) $L^{2}([R,\infty))^{2}$ solution

$\displaystyle\left[\begin{array}[]{c}u(r)\\\
v(r)\end{array}\right]=\left[\begin{array}[]{c}C_{1}\\\
C_{2}\end{array}\right]e^{-\sqrt{1-\varepsilon^{2}}\,r},$ (A.60)

with $C_{1}$ and $C_{2}$ related by $C_{2}=-C_{1}/\sqrt{1-\varepsilon^{2}}$,
and $C_{1}\equiv C$ to be determined subsequently. Note that this free
electron asymptotics holds both for the RWN and for the Minkowski spacetime
problems.

For the other asymptotic regime, as $r\searrow 0$, we need to distinguish the
cases $f(r)\equiv 1$ and $f(r)$ given by (2.23). Beginning with the problem
for the RWN spacetime, after factoring out
$\mathcal{G}^{1/2}\alpha_{\mbox{\tiny{S}}}^{2}Z^{2}\frac{1}{r^{3}}$, equation
(A.48) is asymptotic to the problem

$\left[\begin{array}[]{cc}0&-\frac{\alpha_{\mbox{\tiny{S}}}}{4\pi}-{\mathcal{G}}^{1/2}r\tfrac{d}{dr}\\\
-\frac{\alpha_{\mbox{\tiny{S}}}}{4\pi}+{\mathcal{G}}^{1/2}r\tfrac{d}{dr}&0\end{array}\right]\left[\begin{array}[]{c}u(r)\\\
v(r)\end{array}\right]=\left[\begin{array}[]{c}0\\\ 0\end{array}\right];\;r\ll
1.$ (A.61)

Problem (A.61) has the one-dimensional $L^{2}((0,R])^{2}$ solution

$\left[\begin{array}[]{c}u(r)\\\
v(r)\end{array}\right]=\left[\begin{array}[]{c}B\\\
0\end{array}\right]r^{{\alpha_{\mbox{\tiny{S}}}}/{4\pi{\mathcal{G}}^{1/2}}},$
(A.62)

with $B$ to be determined subsequently. Note that for the physical parameter
values the power of the monomial is enormous, about $1.17\cdot 10^{18}$, so
that the pertinent monomial with the negative of that power, viz. the second
solution of (A.61), is not in $L^{2}((0,R],r^{2}dr)^{2}$. Therefore, for
physical parameter values the operator $K_{k}$ is of limit point type at both
ends, hence essentially self-adjoint on its minimal domain (cf. Theorem 2.6
and the remark after it). Note furthermore that the power is independent of
$Z$, which in a nutshell is the reason for why the Dirac $H$ of an electron
with (empirical-valued) anomalous magnetic moment in the RWN spacetime of a
naked nucleus is essentially self-adjoint for all $Z$ values.

Yet, mathematically one can contemplate a world with different values of
Newton’s constant of universal gravitation, everything else remaining
untouched. Thus one can make $G$, hence $\mathcal{G}$ so large that also the
monomial $r^{-{\alpha_{\mbox{\tiny{S}}}}/{4\pi{\mathcal{G}}^{1/2}}}\in
L^{2}((0,R],r^{2}dr)$. In that case the operator is of limit circle type at
$r=0$, while still of limit point type at $r=\infty$, and therefore not
essentially self-adjoint. Alternatively to enlarging $G$ while keeping
everything else fixed, one could also keep $G$ fixed and replace the anomalous
magnetic moment of the electron by a fictiously small $\mu_{a}$ value (in
magnitude), with the same conclusion of a not essentially self-adjoint $K_{k}$
when $|\mu_{a}|$ is small enough. The critical $|\mu_{a}|$ value is given in
(1.11)

Note finally that the power is $\propto 1/\sqrt{\mathcal{G}}$, hence $\propto
1/\sqrt{G}$, which signals the highly non-trivial character of the problem of
taking the mathematical zero-gravity limit $G\searrow 0$.

We now contrast $r\searrow 0$ asymptotics of the Dirac problem on RWN with its
counterpart for the more familiar Dirac problem of a test electron with
anomalous magnetic moment in the Coulomb field of a point nucleus in Minkowski
space. The radial partial-wave Dirac operators
$H^{\mbox{\tiny{rad}}}_{k}:=m_{\mathrm{e}}c^{2}K^{\mbox{\tiny{}}}_{k}$,
$k\in\mathbb{Z}\backslash\\{0\\}$ now are given by

$K_{k}^{\text{}}:=\left[\begin{array}[]{cc}1-Z\alpha_{\mbox{\tiny{S}}}\frac{1}{r}&k\frac{1}{r}-Z\frac{\alpha_{\mbox{\tiny{S}}}^{2}}{4\pi}\frac{1}{r^{2}}-\tfrac{d}{dr}\\\
k\frac{1}{r}-Z\frac{\alpha_{\mbox{\tiny{S}}}^{2}}{4\pi}\frac{1}{r^{2}}+\tfrac{d}{dr}&-1-Z\alpha_{\mbox{\tiny{S}}}\frac{1}{r},\end{array}\right],$
(A.63)

acting on $g(r):=\big{(}u(r),v(r)\big{)}^{T}\in
C^{\infty}_{c}(\mathbb{R}_{+})^{2}$ equipped with an $L^{2}$ norm given by
(A.47) with $f\equiv 1$. For the regime as $r\searrow 0$, after factoring out
$\frac{1}{r^{2}}$, (A.48) now is asymptotic to the problem

$\left[\begin{array}[]{cc}0&-Z\frac{\alpha_{\mbox{\tiny{S}}}^{2}}{4\pi}-{r^{2}}\tfrac{d}{dr}\\\
-Z\frac{\alpha_{\mbox{\tiny{S}}}^{2}}{4\pi}+r^{2}\tfrac{d}{dr}&0\end{array}\right]\left[\begin{array}[]{c}u(r)\\\
v(r)\end{array}\right]=\left[\begin{array}[]{c}0\\\ 0\end{array}\right];\qquad
r\ll 1,$ (A.64)

which has the unique one-dimensional $L^{2}((0,R])^{2}$ solution

$\left[\begin{array}[]{c}u(r)\\\
v(r)\end{array}\right]=\left[\begin{array}[]{c}B\\\
0\end{array}\right]e^{-Z\frac{\alpha_{\mbox{\tiny{S}}}^{2}}{4\pi}\frac{1}{r}},$
(A.65)

with $B$ to be determined subsequently. Notice that the second solution
$\propto e^{Z\frac{\alpha_{\mbox{\tiny{S}}}^{2}}{4\pi}\frac{1}{r}}$ is not in
$L^{2}((0,R],dr)^{2}$ for any $Z>0$, given $\alpha_{\mbox{\tiny{S}}}>0$. Hence
the Dirac operator is of limit point type at both asymptotic ends, and
therefore essentially self-adjoint on its minimal domain for all $Z>0$, and
for all $\alpha_{\mbox{\tiny{S}}}^{2}>0$ — which means for any nontrivial
anomalous magnetic moment of the electron $\mu_{a}\neq 0$.

Our discussion reveals a technical obstacle that needs to be overcome when
trying to prove the conjecture that the essentially self-adjoint Dirac
hamiltonian for an electron with physical anomalous magnetic moment on the RWN
spacetime converges not only in strong resolvent sense but also in terms of
its spectrum to the corresponding operator on the Minkowski spacetime when one
takes the zero-gravity limit $G\searrow 0$. Manifestly the
$L^{2}((0,R],r^{2}dr)$ function
$r^{{\alpha_{\mbox{\tiny{S}}}}/{4\pi{\mathcal{G}}^{1/2}}}$ does not converge
to the $L^{2}((0,R],dr)$ function
$e^{-Z\frac{\alpha_{\mbox{\tiny{S}}}^{2}}{4\pi}\frac{1}{r}}$ when $G\searrow
0$. The $G\searrow 0$ limit presumably requires a study of the asymptotics
beyond all orders, involving the theory of transseries [13]. Such a study has
to be deferred to some future work.

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