$\widetilde{\mathcal{W}}^{\pm}_{E,\lambda}(\tau)=\big{(}\xi(\tau),\Omega^{\pm}_{E,\lambda}(\tau)\big{)}$
denote the orbits corresponding to the unstable/stable manifolds emanating
from $S^{-}_{E}$ and $(\frac{\pi}{2},-\cos^{-1}(E)-2\pi N)$ (this is just
$S^{+}_{E}$ shifted down by $2\pi N$) with initial condition determined by
$\xi(0)=0$.

By smooth dependence of solutions on parameters, the function

$\Phi(E,\lambda)\,=\,\Omega^{-}_{E,\lambda}(0)-\Omega^{+}_{E,\lambda}(0)$

is smooth in $E$ and $\lambda$. Since $E_{N}(\lambda)$ corresponds to a
saddles connector with winding number $N$, we have
$\Phi\big{(}E_{N}(\lambda),\lambda\big{)}=0$. By the implicit function
theorem, if

$\frac{\partial\Phi}{\partial
E}\big{(}E_{N}(\lambda),\lambda\big{)}\,\neq\,0,$

then it follows that $E_{N}$ is a $C^{1}$ function of $\lambda$. To show this,
define $u^{\pm}_{E,\lambda}(\tau)\,=\,\frac{\partial}{\partial
E}\Omega^{\pm}_{E,\lambda}(\tau)$. Then $u^{\pm}_{E,\lambda}$ satisfies the
following differential equation

$\frac{d}{d\tau}u^{\pm}_{E,\lambda}(\tau)\,=\,P^{\pm}_{E,\lambda}(\tau)u^{\pm}_{E,\lambda}(\tau)-2a$

where

$\displaystyle P^{\pm}_{E,\lambda}\,$
$\displaystyle=\,-2a\sin\xi\sin\Omega^{\pm}_{E,\lambda}+2\lambda\cos\xi\cos\Omega^{\pm}_{E,\lambda}.$

Setting
$U^{\pm}_{E,\lambda}(\tau,\tau_{0})=\exp(\int_{\tau_{0}}^{\tau}P^{\pm}_{E,\lambda}(s)ds)$,
the solution to this differential equation is

$u^{\pm}_{E,\lambda}(\tau)\,=\,U^{\pm}_{E,\lambda}(\tau,\tau_{0})u^{\pm}_{E,\lambda}(\tau_{0})\,-\,2a\int_{\tau_{0}}^{\tau}U^{\pm}_{E,\lambda}(\tau,s)ds.$

Taking the limit $\tau_{0}\to-\infty$, we find
$P^{-}_{E,\lambda}(-\infty)=-2a\sqrt{1-E^{2}}$ and so
$U^{-}_{E,\lambda}(\tau,-\infty)=0$. Similarly, taking the limit
$\tau_{0}\to\infty$, we find $P^{+}_{E,\lambda}(\infty)=2a\sqrt{1-E^{2}}$ and
so $U^{+}_{E,\lambda}(\tau,\infty)=0$. Therefore
$U^{\pm}_{E,\lambda}(\tau,\tau_{0})u^{\pm}_{E,\lambda}(\tau_{0})\to 0$ as
$\tau_{0}\to\pm\infty$ since
$|u^{\pm}_{E,\lambda}(\pm\infty)|=|\frac{d}{dE}\cos^{-1}(E)|<\infty$.
Therefore

$\frac{\partial\Phi}{\partial
E}(E,\lambda)\,=\,u^{-}_{E,\lambda}(0)-u^{+}_{E\lambda}(0)\,=\,-2a\int_{-\infty}^{0}U^{-}_{E,\lambda}(0,s)ds-2a\int_{0}^{\infty}U^{+}_{E,\lambda}(0,s)ds.$

Since $E_{N}(\lambda)$ corresponds to a saddles connector with winding number
$N$, we have
$\Omega^{-}_{E_{N}(\lambda),\lambda}(\tau)=\Omega^{+}_{E_{N}(\lambda),\lambda}(\tau)$
for all $\tau\in\mathbb{R}$. Therefore
$U^{-}_{E_{N}(\lambda),\lambda}=U^{+}_{E_{N}(\lambda),\lambda}=:U_{E_{N}(\lambda)}$.
Thus

$\frac{\partial\Phi}{\partial
E}\big{(}E_{N}(\lambda),\lambda\big{)}\,=\,-2a\int_{-\infty}^{\infty}U_{E_{N}(\lambda)}(0,s)ds\,<\,0.$

Now we prove the second part of the theorem: $|E_{N}^{\prime}(\lambda)|<1/a$
for all $\lambda$. Define
$v^{\pm}_{E,\lambda}=\frac{\partial}{\partial\lambda}\Omega^{\pm}_{E_{N}(\lambda),\lambda}$.
Then $v^{\pm}_{E,\lambda}$ satisfies the differential equation

$\frac{d}{d\tau}v^{\pm}_{E,\lambda}\,=\,P^{\pm}_{E,\lambda}(\tau)v(\tau)+2\cos\xi(\tau)\sin\Omega^{\pm}_{E,\lambda}(\tau).$

The solution to this differential equation is

$v^{\pm}_{E,\lambda}(\tau)\,=\,U^{\pm}_{E,\lambda}(\tau,\tau_{0})v^{\pm}_{E,\lambda}(\tau_{0})+\int_{\tau_{0}}^{\tau}U^{\pm}_{E,\lambda}(\tau,s)\big{(}2\cos\xi(s)\sin\Omega^{\pm}_{E,\lambda}(s)\big{)}ds.$

Using the same analysis as above, we have

$\frac{\partial\Phi}{\partial\lambda}\big{(}E_{N}(\lambda),\lambda\big{)}\,=\,\int_{-\infty}^{\infty}U_{E_{N}(\lambda)}(0,s)\big{(}2\cos\xi(s)\sin\Omega_{E_{N}(\lambda)}(s)\big{)}ds$

where
$\Omega_{E_{N}(\lambda)}:=\Omega^{+}_{E_{N}(\lambda),\lambda}=\Omega^{-}_{E_{N}(\lambda),\lambda}$.
Then $E_{N}^{\prime}(\lambda)$ is given by implicit differentiation

$E_{N}^{\prime}(\lambda)\,=\,-\frac{\partial\Phi/\partial\lambda}{\partial\Phi/\partial
E}\big{(}E_{N}(\lambda),\lambda)\,=\,\frac{1}{a}\left(\frac{\int_{-\infty}^{\infty}U_{E_{N}(\lambda)}(0,s)\big{(}\cos\xi(s)\sin\Omega_{E_{N}(\lambda)}(s)\big{)}ds}{\int_{-\infty}^{\infty}U_{E_{N}(\lambda)}(0,s)ds}\right).$

Since the numerator in absolute value is strictly less than the denominator in
absolute value, we have $|E_{N}^{\prime}(\lambda)|<\frac{1}{a}$. ∎

For the following theorem, we define the following closed intervals. Let
$N_{\Theta}\geq 0$ be an integer for $I_{1}$ and $I_{2}$, and let
$N_{\Theta}\leq-1$ be an integer for $I_{3}$ and $I_{4}$.

$\displaystyle I_{1}\,$
$\displaystyle=\,\left[-(2N_{\Theta}+1)(\frac{1}{2}+\kappa)-2a,\,-\frac{1}{2}-\kappa+a\right]$
$\displaystyle I_{2}\,$
$\displaystyle=\,\left[-(2N_{\Theta}+1)(\frac{1}{2}-\kappa)-2a,\,-\frac{1}{2}+\kappa-a\right]$
$\displaystyle I_{3}\,$
$\displaystyle=\,\left[\,\frac{1}{2}+\kappa-a,\,-(2N_{\Theta}+1)(\frac{1}{2}+\kappa)+2a\right]$
$\displaystyle I_{4}\,$
$\displaystyle=\,\left[\,\frac{1}{2}-\kappa-a,\,-(2N_{\Theta}+1)(\frac{1}{2}-\kappa)+2a\right].$

Intervals $I_{1}$ and $I_{2}$ correspond to $\lambda<0$ with $\kappa>0$ and
$\kappa<0$, respectively; their endpoints are determined by Theorem 3.12.
Intervals $I_{3}$ and $I_{4}$ correspond to $\lambda>0$ with $\kappa>0$ and
$\kappa<0$, respectively; their endpoints are determined by Theorem 3.15.

######  Theorem 3.28.

  * $\bullet$

Let $\lambda<0$. For each pair of integers $N_{\Omega}\geq 0$ and
$N_{\Theta}\geq 0$, there is a unique $E_{*}\in(0,1)$ and a unique
$\lambda_{*}\in I_{1}$ if $\kappa>0$ or unique $\lambda_{*}\in I_{2}$ if
$\kappa<0$ such that $E_{*}=E_{N_{\Omega}}(\lambda_{*})$ and
$\lambda_{*}=\lambda_{N_{\Theta}}(E_{*})$ where $E_{N_{\Omega}}$ and
$\lambda_{N_{\Theta}}$ are the functions from Propositions _3.27_ and _3.26_.
There are no saddles connectors with $N_{\Omega}\leq-1$.

  * $\bullet$

Let $\lambda>0$. For each pair of integers $N_{\Omega}\geq 1$ and
$N_{\Theta}\leq-1$, there is a unique $E_{*}\in(0,1)$ and a unique
$\lambda_{*}\in I_{3}$ if $\kappa>0$ or unique $\lambda_{*}\in I_{4}$ if
$\kappa<0$ such that $E_{*}=E_{N_{\Omega}}(\lambda_{*})$ and
$\lambda_{*}=\lambda_{N_{\Theta}}(E_{*})$ where $E_{N_{\Omega}}$ and
$\lambda_{N_{\Theta}}$ are the functions from Propositions _3.27_ and _3.26_.
There are no saddles connectors with $N_{\Omega}\leq 0$.

###### Proof.

We prove it for $\lambda<0$; the proof for $\lambda>0$ is analogous. That
there are no saddles connectors with $N_{\Omega}\leq-1$ follows from
Proposition 3.20. We set out now to establish existence for $N_{\Omega}\geq
0$. To simplify the notation, set $n=N_{\Theta}$ and $m=N_{\Omega}$.

Define the function $\phi_{n,m}\colon[0,1]\to[0,1]$ by
$\phi_{n,m}(E)=E_{m}\big{(}\lambda_{n}(E)\big{)}$. By Propositions 3.26 and
3.27, we have $\phi_{n,m}$ is a $C^{1}$ function on $[0,1]$ and
$|\phi_{n,m}^{\prime}(E)|=|E_{m}^{\prime}\big{(}\lambda_{n}(E)\big{)}\lambda_{n}^{\prime}(E)|<\frac{1}{a}a=1$.
Therefore, since $[0,1]$ is compact, the mean value theorem implies that
$\phi_{n,m}$ is a contraction. Thus there exists a unique $E_{*}\in[0,1]$ such
that $E_{*}=E_{m}\big{(}\lambda_{n}(E_{*})\big{)}$. Similarly, define
$\psi_{n,m}\colon I\to I$ by
$\psi_{n,m}=\lambda_{n}\big{(}E_{m}(\lambda)\big{)}$ where $I=I_{1}$ if
$\kappa>0$ and $I=I_{2}$ if $\kappa<0$. The same argument shows that
$\psi_{n,m}$ is a contraction, and thus there exists a unique $\lambda_{*}\in
I$ such that $\lambda_{*}=\lambda_{n}\big{(}E_{m}(\lambda_{*})\big{)}$.

We have
$E_{m}(\lambda_{*})=E_{m}\big{(}\lambda_{n}\big{(}E_{m}(\lambda_{*})\big{)}\big{)}$.
By uniqueness it follows that $E_{*}=E_{m}(\lambda_{*})$. Hence
$E_{*}\in(0,1)$ by Theorem 3.25. ∎

Theorem 3.28 completes the proof of Theorem 2.1.

##  4 Connection with the hydrogenic spectra

Based on the above results, for fixed $a$ and $\gamma$, the discrete spectrum
of our Dirac hamiltonian is indexed by three integers: $N_{\Omega}$,
$N_{\Theta}$, and $2\kappa$. By contrast, the energy spectrum of special
relativistic hydrogen, i.e., the Dirac hamiltonian for a point-like electron
in ordinary Minkowski space interacting with a point charge at the origin, is
indexed by two integers only, namely the main (or Bohr’s) quantum number,
often denoted by $n$, and the spin-orbit quantum number,333called $\kappa_{j}$
in Thaller; it should not be confused with our $\kappa$, which is the
eigenvalue of the $z$-component of angular momentum, for which Thaller uses
the notation $m_{j}$. i.e., the set of eigenvalues of the spin-orbit operator
$K=\beta(2\mathbf{S}\cdot\mathbf{L}+1)$.

In this section, we classify the energy eigenvalues of our hamiltonian, relate
the numbers $(N_{\Omega},N_{\Theta},\kappa)$ to the quantum numbers of the
special relativistic hydrogen theory, and associate the corresponding energy
eigenstates with the well-known orbitals of the hydrogen atom. In the next
section we report on the results of several numerical experiments which
provide evidence for our theoretical calculations and arguments.

In the limit $a\to 0$, the angular hamiltonian (2.25), takes the simple form

$\mathfrak{a}_{\kappa}\,:=\,\lim_{a\to
0}T_{ang}\,=\,i\sigma_{2}\partial_{\theta}+\frac{\kappa}{\sin\theta}\sigma_{1}.$
(4.42)

From [4], if $\lambda$ is an eigenvalue of $T_{ang}$, then $k:=\lim_{a\to
0}\lambda$ is an eigenvalue of $\mathfrak{a}_{\kappa}$ since $\lambda$ is
analytic in $a$. Note that $\mathfrak{a}_{\kappa}$ is independent of $E$
unlike $T_{ang}$.

In the limit $a\to 0$, the formal limit of the radial hamiltonian (2.30)
coincides with the radial hamiltonian arising in the special relativistic
Hydrogen problem (e.g. see [59, eq. (7.105)]):

$\mathfrak{h}_{k}\,:=\,\lim_{a\to
0}H_{rad}\,=\,\left(\begin{array}[]{cc}m+\frac{\gamma}{r}&-\partial_{r}+\frac{k}{r}\\\\[20.0pt]
\partial_{r}+\frac{k}{r}&-m+\frac{\gamma}{r}\end{array}\right).$ (4.43)

Therefore $k$ can be identified with the _spin-orbit coupling_.

The spectrum of $\mathfrak{a}_{\kappa}$ is completely understood[4, Appendix
A].444Note that the sign of $\kappa$ in [4, eq. (5)] is opposite of ours
appearing in eq. (4.42). This ambiguity in the sign follows from choosing
either $\pm(Et-\kappa\varphi)$ in eq. (2.16). In particular, for all half-
integers $\kappa\in\mathbb{Z}+\frac{1}{2}$, the operator
$\mathfrak{a}_{\kappa}$ is essentially self-adjoint and has a discrete
spectrum indexed by a nonzero integer which we call $N$ (see [4, eq. (7)]):

$k\,=\,-\mbox{sgn}(N)\left(|N|+|\kappa|-\frac{1}{2}\right),$ (4.44)

as well as a complete set of eigenvectors $\vec{S}_{N,\kappa}$ that are
explicitly known and can be expressed in terms of Jacobi polynomials.

$\vec{S}_{N,\kappa}(\theta)\,:=\,\sin^{\kappa+\frac{1}{2}}\theta\left(\begin{array}[]{c}\qquad-\sqrt{\cot\frac{\theta}{2}}P_{|N|-1}^{\kappa-\frac{1}{2},\kappa+\frac{1}{2}}(\cos\theta)\\\
\mbox{sgn}(N)\sqrt{\tan\frac{\theta}{2}}P_{|N|-1}^{\kappa+\frac{1}{2},\kappa-\frac{1}{2}}(\cos\theta)\end{array}\right).$
(4.45)

Note that the above holds for $\kappa>0$. To find the eigenvectors for
$\kappa<0$, recognize that if $\vec{S}$ is an eigenvector of
$\mathfrak{a}_{\kappa}$, then $i\sigma_{2}\vec{S}$ is an eigenvector of
$\mathfrak{a}_{-\kappa}$.

From the above and the definition of $\Theta$ given by eq. (2.32), it follows
that the saddles connectors of the $\Theta$-system that correspond to the
eigenvectors of the angular hamiltonian are given explicitly by the formula

$\Theta_{N,\kappa}(\theta)\,=\,-2\,\mbox{sgn}(N)\tan^{-1}\left(\frac{P_{|N|-1}^{|\kappa|+\frac{1}{2},|\kappa|-\frac{1}{2}}(\cos\theta)}{P_{|N|-1}^{|\kappa|-\frac{1}{2},|\kappa|+\frac{1}{2}}(\cos\theta)}\tan\frac{\theta}{2}\right).$
(4.46)

The above formula holds for $\kappa>0$. For $\kappa<0$, the formula is the
same except one switches the numerator and denominator. We choose our branch
of $\tan^{-1}$ so that inital values of the saddles connectors agrees with the
saddle equilibrium points within the fundamental domain chosen in the
beginning of section 3.4. That is,

$\Theta_{N,\kappa}(0)\,=\,\left\\{\begin{array}[]{cc}0,&\kappa>0\\\
\pi,&\kappa<0\end{array}\right.$ (4.47)

Then the final values of those connectors are

$\Theta_{N,\kappa}(\pi)\,=\,\left\\{\begin{array}[]{cc}\quad-\mbox{sgn}(N)\pi,&\kappa>0\\\
\pi-\mbox{sgn}(N)\pi,&\kappa<0\end{array}\right.\qquad[\mbox{mod}2\pi]$ (4.48)

The branch of $\tan^{-1}$ needs to be chosen in such a way that $\theta_{N,k}$
is continuous on $[0,\pi]$. Since the Jacobi polynomials
$P_{|N|-1}^{\alpha,\beta}(\cos\theta)$ have $|N|-1$ distinct roots within
$(0,\pi)$, it follows that we need to add $-\mbox{sgn}(N)2\pi$ to eq. (4.48)
for each root to determine the actual endpoint of $\Theta_{N,\kappa}$. That is

$\Theta_{N,\kappa}(\pi)\,=\,-\mbox{sgn}(N)(|N|-1)2\pi+\left\\{\begin{array}[]{cc}\quad-\mbox{sgn}(N)\pi,&\kappa>0\\\
\pi-\mbox{sgn}(N)\pi,&\kappa<0\end{array}\right.$ (4.49)

Thus we establish a correspondence between the integer $N$ and the winding
number $N_{\Theta}$ of the $\Theta$-saddles connectors in the case $a=0$:

$N_{\Theta}\,\sim\,\left\\{\begin{array}[]{cc}N-1,&N\geq 1\\\ \quad
N,&N\leq-1,\end{array}\right.$ (4.50)

where ‘$\sim$’ simply means a correspondence.

We now turn our attention to the radial hamiltonian
$\mathfrak{h}_{k}=\lim_{a\to 0}H_{rad}$. This operator is closely related to
the radial hamiltonian $h_{k}$ of the special-relativistic Hydrogen problem,
as formulated by Dirac[17]:

$h_{k}=-i\sigma_{2}\partial_{r}+m\sigma_{3}+\frac{k}{r}\sigma_{1}+\frac{\gamma}{r}I_{2\times
2}.$ (4.51)

The only difference between $h_{k}$ and $\mathfrak{h}_{k}$ is their domains:
Since $H_{rad}$ is defined on the double-sheeted Sommerfeld space, this is
inherited by its $a\to 0$ limit $\mathfrak{h}_{k}$, which is still defined on
two copies of Minkowski space glued together along a timelike line. In
particular the $r$ variable in $\mathfrak{h}_{k}$ has the range
$(-\infty,\infty)$. By contrast, the $r$ variable in $h_{k}$ goes from $0$ to
$\infty$. The eigenvalue problem for $h_{k}$ was shown to be exactly solvable
by Gordon [27], who proved the discrete spectrum to coincide exactly with the
Bohr-Sommerfeld spectrum, and found explicit formulas for the eigenfunctions
in terms of generalized Laguerre polynomials.

Consider the operator $h_{k}$. As described in detail in section 7.4 of
Thaller’s book[59], for all $k\in\mathbb{Z}\setminus\\{0\\}$, $h_{k}$ is
essentially self-adjoint on $C^{\infty}_{c}((0,\infty))$, has a discrete
spectrum in $(0,1)$, and a complete set of eigenvectors, provided
$-\sqrt{3}/2<\gamma<0$. The discrete spectrum is indexed by two integers,
$n\geq 1$ and $k=-n,\dots,-1,1,\dots,n-1$. Let $M:=n-|k|$. Then we have

$E_{M,k}\,=\,\frac{m}{\sqrt{1+\frac{\gamma^{2}}{\left(M+\sqrt{k^{2}-\gamma^{2}}\right)^{2}}}}$
(4.52)

Note that the case $k>0$ and $M=0$ is excluded.

Gordon [27] computed the corresponding eigenfunctions in terms of generalized
Laguerre polynomials: Let $(\phi_{1},\phi_{2})$ be defined by

$u=\sqrt{1+E_{M,k}}(\phi_{1}+\phi_{2}),\qquad
v=\sqrt{1-E_{M,k}}(\phi_{1}-\phi_{2})$ (4.53)

where $(u,v)$ is an eigenfunction of $\mathfrak{h}_{k}$ with eigenvalue
$E_{M,k}$ (and we have set $m=1$). Then, using the abbreviations

$\rho:=\sqrt{k^{2}-\gamma^{2}},\qquad\eta:=\sqrt{1-E_{M,k}^{2}},$ (4.54)

for all nonnegative integers $M$ and real constants $c_{1},c_{2}$ such that

$\mu:=\frac{c_{1}}{c_{2}}=\frac{M}{k+\frac{\gamma}{\eta}},$ (4.55)

we have

$\displaystyle\phi_{1}(r)\,$ $\displaystyle=\,c_{1}e^{-\eta
r}r^{\rho}F(-M+1,2\rho+1,2\eta r)$ (4.56) $\displaystyle\phi_{2}(r)\,$
$\displaystyle=\,c_{2}e^{-\eta r}r^{\rho}F(-M,2\rho+1,2\eta r),$ (4.57)

where $F$ denotes Gauss’s confluent hypergeometric function

$F(\alpha,\beta,x)\,=\,1+\frac{\alpha}{1!\beta}x+\frac{\alpha(\alpha+1)}{2!\beta(\beta+1)}x^{2}+\dots$
(4.58)

Note that when $\alpha$ is a negative integer, the above series terminates,
and $F$ is a polynomial of degree $-\alpha$, which (up to a numerical factor)
is the generalized Laguerre polynomial $L_{-\alpha}^{(\beta-1)}(x)$.

Accordingly, since $\Omega=2\tan^{-1}\frac{v}{u}$, the corresponding solution
to the $a\to 0$ limit of the $\Omega$-equation (2.35) will be

$\Omega(r)=2\tan^{-1}\left(\sqrt{\frac{1-E}{1+E}}\ \frac{\mu
F(-M+1,2\rho+1,2\eta r)-F(-M,2\rho+1,2\eta r)}{\mu F(-M+1,2\rho+1,2\eta
r)+F(-M,2\rho+1,2\eta r)}\right)$ (4.59)

The values of $\Omega$ at $r=0$ and $r=\infty$ can thus be calculated (modulo
$2\pi$):

$\displaystyle\Omega(0)\,$
$\displaystyle=\,2\tan^{-1}\left(\sqrt{\frac{1-E}{1+E}}\
\frac{\mu-1}{\mu+1}\right)\,=\sin^{-1}\left(\frac{-\gamma}{k}\right)$ (4.60)
$\displaystyle\Omega(\infty)\,$
$\displaystyle=\,-2\tan^{-1}\left(\sqrt{\frac{1-E}{1+E}}\right)\,=\,-\cos^{-1}E$
(4.61)

which is in agreement with the analysis of the equilibrium points of the
corresponding dynamical system. We choose our principal branch of $\sin^{-1}$
depending on the sign of $k$:

$\displaystyle\Omega(0)\,$
$\displaystyle=\,\left\\{\begin{array}[]{ll}\sin^{-1}(\frac{-\gamma}{k})&\mbox{
if }k<0\\\ -\pi-\sin^{-1}(\frac{-\gamma}{k})&\mbox{ if
}k>0\end{array}\right..$ (4.64)

On the other hand, from the properties of Laguerre polynomials, it follows
that the denominator of the rational function in (4.59) has $M$ zeros in
$(0,\infty)$ when $k<0$ and $M-1$ zeros in $(0,\infty)$ when $k>0$. That is

######  Proposition 4.1.

The polynomial $\mu F(-M+1,2\rho+r,2\eta r)+F(-M,2\rho+1,2\eta r)$ has

  * $\bullet$

$M\geq 0$ distinct zeros in $(0,\infty)$ when $k<0$ and

  * $\bullet$

$M-1\geq 0$ distinct zeros in $(0,\infty)$ when $k>0$, provided
$|\gamma|<\frac{1}{2}$.

We prove Proposition 4.1 at the end of this section. Note that it explains why
we must have $k\leq n-1$: if $k=n$, then $M-1=-1$, but a polynomial can’t have
a negative number of zeros. The assumption $|\gamma|<\frac{1}{2}$ is merely
technical and used to simplify the proof of Proposition 4.1. It’s expected
Proposition 4.1 should still hold even under the assumption
$|\gamma|<\frac{\sqrt{3}}{2}$. However, note that $|\gamma|<\frac{1}{2}$ is
part of our assumptions in Theorem 2.1.

Thus in order for $\Omega$ to be a continuous function of $r$, the branch of
$\tan^{-1}$ needs to be chosen such that $-2\pi$ gets added to the value every
time $r$ crosses one these zeros. Therefore

$\Omega(\infty)\,=\,-2\pi M-\cos^{-1}E$ (4.65)

which holds for both $k<0$ and $k>0$ (this is why we chose
$\Omega(0)=-\pi-\sin^{-1}(\frac{-\gamma}{k})$ for $k>0$). Thus we establish
the following correspondence between the integer $M$ and the winding number
$N_{\Omega}$ for the $\Omega$-saddles connectors in the case $a=0$.

$N_{\Omega}\,\sim\,M.$ (4.66)

Now consider the case $a>0$ (i.e. the Dirac hamiltonian on z$G$KN). The bound
states are indexed by three integers $N_{\Theta},N_{\Omega}$, and $2\kappa$
appearing in Theorem 2.1. Using (4.66) and (4.50), we can define a
correspondence between the integers $N_{\Theta},N_{\Omega},2\kappa$ and the
usual spectroscopic notation $n\ell_{j}$ of hydrogenic states. Given
$N_{\Theta}$ and $N_{\Omega}$, define $N$ and $M$ via

$N\,:=\,\left\\{\begin{array}[]{cc}N_{\Theta}+1,&N_{\Theta}\geq 0\\\
\;N_{\Theta},&N_{\Theta}\leq-1,\end{array}\right.\quad\text{ and }\quad
M\,:=\,N_{\Omega}.$ (4.67)

Then $n$, $\ell$, and $j$ appearing in $n\ell_{j}$ are given by

$\displaystyle k\,$
$\displaystyle:=\,-N-\mbox{sgn}(N)\left(|\kappa|-\frac{1}{2}\right)$ (4.68)
$\displaystyle n\,$ $\displaystyle:=\,M+|k|$ (4.69) $\displaystyle j\,$
$\displaystyle:=\,|k|-\frac{1}{2}$ (4.70) $\displaystyle\ell\,$
$\displaystyle:=\,j+\mbox{sgn}(k)\frac{1}{2}$ (4.71) $\displaystyle m_{j}\,$
$\displaystyle:=\,\kappa.$ (4.72)

This establishes the desired correspondence between the winding numbers
appearing in Theorem 2.1 and the usual spectroscopic notation $n\ell_{j}$ of
hydrogenic states. For example, suppose we ask: Which state in z$G$KN
corresponds to $2p_{1/2}$ with $m_{1/2}=-\frac{1}{2}$? Note $\ell=1$ implies
$1=\frac{1}{2}+\mbox{sgn}(k)\frac{1}{2}$. Therefore $k$ is positive and so
$j=\frac{1}{2}$ implies $k=1$. Therefore $N=-1$ implies $N_{\Theta}=-1$.
Lastly, $n=2$ implies $M=1$ and so $N_{\Omega}=1$. Thus the state $2p_{1/2}$
with $m_{1/2}=-\frac{1}{2}$ corresponds to $N_{\Theta}=-1$, $N_{\Omega}=1$,
and $\kappa=-\frac{1}{2}$.

Hydrogenic state | $N_{\Theta}$ | $N_{\Omega}$ | $\kappa$
---|---|---|---
$1s_{1/2}$, $m_{1/2}=\pm\frac{1}{2}$ | $0$ | $0$ | $\pm\frac{1}{2}$
$2s_{1/2}$, $m_{1/2}=\pm\frac{1}{2}$ | $0$ | $1$ | $\pm\frac{1}{2}$
$2p_{1/2}$, $m_{1/2}=\pm\frac{1}{2}$ | $-1$ | $1$ | $\pm\frac{1}{2}$
$2s_{3/2}$, $m_{3/2}=\pm\frac{1}{2}$ | $-2$ | $0$ | $\pm\frac{1}{2}$
$2s_{3/2}$, $m_{3/2}=\pm\frac{3}{2}$ | $-1$ | $0$ | $\pm\frac{3}{2}$
Table 1: Correspondence between hydrogenic states and winding numbers.

_Proof of Proposition _4.1__.

Set $\beta=2\rho+1$ and $x=2\eta r$. And set $P_{M}(x)=F(-M,\beta,x)$ and
$Q_{M}=\mu P_{M-1}+P_{M}$. We first show that $Q_{M}$ has at least $M-1$ real
distinct roots in $(0,\infty)$. Since complex roots come in pairs, this will
imply that the roots of $Q_{M}$ are all real. Seeking a contradiction, suppose
$Q_{M}$ does not have $M-1$ roots in $(0,\infty)$. Since $P_{M}$ are
proportional to the generalized Laguerre polynomials, they form a system of
orthogonal polynomials with respect to the weight function
$w(x)=x^{\beta-1}e^{-x}$ on $(0,\infty)$. Since
$\int_{0}^{\infty}Q_{M}(x)w(x)dx=0$, it follows that $Q_{M}$ changes sign at
least once in $(0,\infty)$; hence $Q_{M}$ has at least one zero of odd
multiplicity in $(0,\infty)$. Let $x_{1},\dotsc,x_{k}$ denote the distinct
zeros of odd multiplicity located in $(0,\infty)$. By assumption $k<M-1$. Set
$Z(x)=(x-x_{1})\dotsb(x-x_{k})$. Then $Z(x)Q_{M}(x)$ is a polynomial with no
zeros of odd multiplicity in $(0,\infty)$ and therefore
$\int_{0}^{\infty}Z(x)Q_{M}(x)w(x)dx\neq 0$. Since $Z(x)$ is a polynomial with
degree $k$, we can write it as $Z(x)=\sum_{i=0}^{k}a_{i}P_{i}(x)$. Therefore
we get a contradiction:

$0\,\neq\,\int_{0}^{\infty}Z(x)Q_{M}(x)w(x)dx\,=\,\int_{0}^{\infty}\left(\sum_{i=0}^{k}a_{i}P_{i}(x)\right)\big{(}\mu
P_{M-1}(x)+P_{M}(x)\big{)}w(x)dx\,=\,0.$

We get zero for the last equality since $k<M-1$. Thus $Q_{M}$ has at least
$M-1$ simple roots in $(0,\infty)$.

Since $Q_{M}$ has at least $M-1$ distinct real roots in $(0,\infty)$, by
Descartes’ rule of signs, it suffices to show that the signs of the
coefficient for $Q_{M}$ (ordered by ascending variable exponent) alternate $M$
times when $k<0$ and alternate $M-1$ times when $k>0$. An induction argument
shows that the coefficients of $P_{M}$ form Pascal’s triangle with alternating
signs. For example
$P_{3}(x)=1-\frac{3}{\beta}x+\frac{3}{\beta(\beta+1)}x^{2}-\frac{1}{\beta(\beta+1)(\beta+2)}x^{3}$.
So for $M=4$, we have

$\displaystyle Q_{4}(x)\,$ $\displaystyle=\,\mu P_{3}(x)+P_{4}(x)$
$\displaystyle=\,(\mu+1)-(3\mu+4)\frac{x}{\beta}+(3\mu+6)\frac{x^{2}}{\beta(\beta+1)}-(\mu+4)\frac{x^{3}}{\beta(\beta+1)(\beta+2)}+\frac{x^{4}}{\beta\dotsb(\beta+3)}$

In general, the zeroth order term for $Q_{M}$ will be $(\mu+1)$, the first
order term will be proportional to $(\mu+\frac{M}{M-1})$, and each term after
that will be proportional to something greater than $(\mu+\frac{M}{M-1})$ in
magnitude. Moreover each term alternates in sign. Thus

  * (1)

for $k<0$, it suffices to show that $\mu+1>0$ since this implies that the
signs for $Q_{M}(x)$ go like $+-+-+\dotsb$, and

  * (2)

for $k>0$, it suffices to show that $\mu+1<0$ and $\mu+\frac{M}{M-1}>0$ since
this implies that the signs for $Q_{M}(x)$ go like $--+-+\dotsb$. Note this
only applies for $M\geq 2$; the case $M=1$ can be verified independently.

Using $k=\mbox{sgn}(k)\sqrt{\rho^{2}+\gamma^{2}}$ and
$\sqrt{1-E^{2}}=-\gamma/\sqrt{(M+\rho)^{2}+\gamma^{2}}$, we have

$\mu\,=\,\frac{M}{k+\gamma/\sqrt{1-E^{2}}}\,=\,\frac{M}{\mbox{sgn}(k)\sqrt{\rho^{2}+\gamma^{2}}-\sqrt{(M+\rho)^{2}+\gamma^{2}}}.$

From this expression, it follows that $-\mu<1$ if $k<0$; this proves (1). Now
we prove (2). Since $k>0$, rationalizing the denominator gives

$-\mu\,=\,\frac{\sqrt{\rho^{2}+\gamma^{2}}+\sqrt{(M+\rho)^{2}+\gamma^{2}}}{M+2\rho}\,<\,\frac{\rho+|\gamma|+M+\rho+|\gamma|}{M+2\rho}\,=\,1+\frac{2|\gamma|}{M+2\rho}.$

Therefore $\mu+1<0$. Lastly, since $\frac{M}{M-1}=1+\frac{1}{M-1}$, it
suffices to show $\frac{2|\gamma|}{M+2\rho}<\frac{1}{M-1}$. Indeed

$\frac{2|\gamma|}{M+2\rho}\,=\,\frac{2|\gamma|}{M+2\sqrt{k^{2}-\gamma^{2}}}\,<\,\frac{2|\gamma|}{M+2\sqrt{1-\gamma^{2}}}\,<\,\frac{1}{M-1}$

where the last inequality holds whenever $|\gamma|<\frac{1}{2}$. ∎

##  5 Numerical results

The following numerical simulations are performed using MATLAB [42]. The
algorithm for finding the energies of saddles connectors is as follows: The
two ODEs (2.37) and (2.35) are numerically integrated (using Runge-Kutta 4-5)
over a suitably large interval symmetric about zero, with initial data
corresponding to the saddle nodes on the left boundary of the corresponding
cylinders. The two desired winding numbers $N_{\Theta}$ and $N_{\Omega}$ are
fixed, as well as the other parameters $\kappa$, $a$ and
$\gamma=-Z\alpha_{S}$. A shooting method is used iteratively on the parameters
$\lambda$ for the $\Theta$ equation and $E$ for the $\Omega$ equation, to fine
tune the values for these parameters via a binary search, until solutions with
the desired winding numbers are found for both equations using the same (up to
a specified tolerance) pair of $(\lambda,E)$ values. MATLAB’s graphic routines
are then used to generate the plots.

###  5.1 Comparison of computed eigenvalues for the angular hamiltonian with
previously known results

The angular hamiltonian (2.25) is identical to the one obtained for Dirac’s
equation on the Kerr-Newman spacetime. It has been extensively studied, see
[4] and references therein.555We note that a different sign convention for the
Chandrasekhar Ansatz (2.16) was used in [4]. As a result, their expression for
the angular hamiltonian looks different from ours. It is easy to see that in
order to get the same expression, one needs to change the signs of the
parameters $a$, $\omega$ (which corresponds to our $E$), and $\kappa$ in eq.
(5) of [4]. It is known to be essentially self-adjoint, its spectrum is
discrete, and consists of simple eigenvalues
$\\{\lambda_{N}\\}_{N\in\mathbb{Z}^{*}}$ that, as can be seen in (2.25) and
(2.27), depend on the parameters $\mu:=am$ and $\nu:=aE$, as well as on the
half-integer $\kappa$. It was shown in [4] that this dependence is analytic,
and that $\lambda_{N}(\kappa;\mu,\nu)$ has a double series expansion in $\mu$
and $\nu$, with coefficients that can be computed recursively.666The index $j$
used in [4] to enumerate the angular eigenvalues has the opposite sign of our
$N$, i.e. their $\lambda_{j}$ is our $\lambda_{-N}$. In particular the authors
of that paper showed that, setting $\alpha:=\nu-\mu$ and $\beta:=\nu+\mu$, one
has

$\lambda_{N}(\kappa;\alpha,\beta)=\sum_{m,n=0}^{\infty}c_{m,n}\alpha^{m}\beta^{n}$
(5.73)

Moreover, they showed that777Correcting for the sign of $\kappa$, see footnote
5.

$\begin{array}[]{lll}c_{0,0}=-\mbox{sgn}(N)(|\kappa|-\frac{1}{2}+|N|)=:k,&c_{1,0}=\frac{-\kappa}{2k+1},&c_{0,1}=\frac{-\kappa}{2k-1},\\\\[5.0pt]
c_{2,0}=\frac{(2k+1)^{2}-4\kappa^{2}}{4(2k+1)^{3}},&c_{1,1}=0,&c_{0,2}=\frac{(2k-1)^{2}-4\kappa^{2}}{4(2k-1)^{3}}\end{array}$
(5.74)

and that all other coefficients can be computed using these and the recursion
relations they had provided.

Here we have demonstrated a different way to compute the eigenvalues $\lambda$
of the angular hamiltonian (2.25), namely, we compute the two eigenvalues
$(\lambda,E)$ of the angular and radial hamiltonians in an iterative
procedure, by finding saddles connectors for the corresponding flows on finite
cylinders. It is interesting to compare these two methods for computing
$\lambda_{N}$.

Figure 7: Comparison of angular eigenvalues computed in two different ways.

In Figure 7 the blue circles are values of $\lambda_{1}$ computed using our
method, with the one given by the first few terms in the expansion (5.73)
(corresponding to $m+n\leq 2$), in which we have substituted for $E$ its
approximate value computed using our method. We see that there is pretty good
agreement between the two all the way to $a\approx 1$. The minor discrepancies
that show for $a>1$ we expect to go away when more terms of the analytic
expansion are taken into account.

###  5.2 Breaking of degeneracies in energy eigenvalues

The discrete spectrum of the Dirac equation on Minkowski space dressed with a
static point charge (the coulombic Dirac hamiltonian) is well-known to have
accidental degeneracies, since energy eigenvalues do not depend on the sign of
the spin-orbit coupling (see (4.52)) while the eigenfunctions of course do.
This degeneracy is broken when $a\neq 0$, leading to effects that mimic the
Lamb-shift (of course they do so only qualitatively) as depicted in Figure 8.
In this figure the horizontal axis is the variable $r$, in units of
$\hbar/m_{e}c$, and the vertical axis is $|\Psi|^{2}$.

Figure 8: Breaking of accidental degeneracy for $a=0.0004$.

Another degeneracy present in the spectrum of the coulombic Dirac operator on
Minkowski space, is the lack of dependence of the energy levels on the
eigenvalue of the $z$-component of the angular momentum, usually denoted by
$m_{j}$ (see the previous section for a dictionary), while the corresponding
eigenfunctions do depend on $m_{j}$ through their dependence on spherical
harmonics. This degeneracy is also broken in our model, since the $\Omega$
equation manifestly depends on $\kappa$, which is the corresponding quantity
to $m_{j}$ in this context. The breaking of this degeneracy results in effects
that are akin to hyperfine splitting, as can be seen in Figure 9.

Figure 9: Hyperfine-like structure.

This figure shows the splitting of the two identical $a=0$ ground state
profiles into two distinct profiles with the values for $a>0$ and $\gamma$, as
indicated, but with $\kappa=\frac{1}{2}$ (corresponding to parallel spins for
the electron and the nucleus) and with $\kappa=-\frac{1}{2}$ (the anti-
parallel case). The two curves are almost indistinguishable at the scale of
this figure (the inset shows the difference between the two,) but their
energies are clearly different, as indicated.

###  5.3 Dependence of the spectrum on the atomic number

The dependence of the spectrum of the coulombic Dirac hamiltonian on
$\gamma=-Z\alpha_{S}$, the strength of the Coulomb potential, is well-
documented, e.g. [59], as well as the interesting behavior of the energies of
the excited states when a term corresponding to the anomalous magnetic moment
of the electron is added to the hamiltonian (ibid., p. 219). As discussed
earlier, anomalous magnetic moment effects are already included in our Dirac
on z$G$KN hamiltonian, so one expects a similar pattern of behavior to emerge
in our case as well. As we shall see, there are some similarities, but there
are also some big surprises.

But first, it stands to reason that the profile of the ground state should
show that as $Z$ increases, the electron would tend to hover closer to the
ring singularity. This is easily confirmed by the numerics, as depicted in
Figure 10.

Figure 10: Ground state profile for different $Z$.

Second, in the case of the coulombic Dirac hamiltonian, the energy of the
ground state has the following behavior: it decreases to 0 as
$-\gamma=Z\alpha_{S}\to 1$, and at $-\gamma=1$ the ground state energy curve
(represented by blue stars in Fig. 11) appears to have a vertical tangent,
with no further values found for the ground state energy when $-\gamma$ is
pushed beyond 1.

Figure 11: Onset of oscillatory behavior of energy eigenvalues for
$Z\alpha_{S}>1$.

The ground state energies of our Dirac on z$G$KN hamiltonian, represented by
the circles in Fig. 11, also follow this pattern, but then begin to deviate
from it in two ways: they continue to exist past $-\gamma=1$, and the energy
value starts to oscillate as $-\gamma$ is increased further and further,
apparently with a damped amplitude, thus perhaps converging to a limiting
positive value as $|\gamma|\to\infty$ (assuming a discrete spectrum exists for
all $|\gamma|>1$).

The same pattern of behavior is observed for the excited state energies as
$-\gamma$ is pushed beyond 1. In Fig. 11 the energies of the first excited
states of the coulombic Dirac hamiltonian, i.e. $2s_{1/2}$ (which because of
the accidental degeneracy is the same as the energy of $2p_{1/2}$) are
depicted by the orange stars, which cease to exist past the critical value of
$\gamma$. By contrast, in our z$G$KN case these two excited states continue to
exist in the supercritical range, and exhibit a similar oscillatory behavior
as the ground state did. The energies of $2s_{1/2}$ orbitals are represented
by purple squares, and that of $2p_{1/2}$ by green 5-point stars. Due to this
oscillation, these two energy levels criss-cross one another, resulting in a
Lamb-shift-like value that is sometimes positive and sometimes negative.

###  5.4 The $a$-dependence of the spectrum

At a formal level, the limit $a\to 0$ of the radial hamiltonian (2.30)
coincides with the coulombic Dirac hamiltonian on Minkowski space. This is of
course purely formal, since the domain of these two hamiltonians is vastly
different. Nevertheless, one may expect that for small values of the parameter
$a$, the spectrum of Dirac hamiltonian on z$G$KN is not that different from
that of the standard coulombic Dirac problem, except that it must be doubled
up since we know that the former is symmetric with respect to zero. Because
the parameter $a$ shows up in many different places in (2.30), it is possible
that changing this parameter causes many different effects to happen at the
same time, so that our initial intuition about $ea$ representing the anomalous
magnetic moment is conceivably to be revised. In particular, we recall that
$a$ is hidden in the definition of $\varpi$, and we also know that $\lambda$
must depend on $a$.

We first study the dependence of the ground state energy on $a$. In Fig. 12 we
have plotted the ground state energy for $0<a<2$. The limiting value at $a=0$
appears to coincide with the ground state energy of the coulombic Dirac
problem. The dependence on $a$ seems to follow a power law with exponent $<1$
as $a\searrow 0$. In the other direction, as $a$ increases, a maximum appears,
after which the energy decreases.

Figure 12: Ground state energy plotted against $a$.

We can also perform a similar study for the differences of the energies of the
excited states $2s_{1/2}$ and $2p_{1/2}$ computed for $a>0$, from their
coulombic Dirac values (i.e. for $a=0$). The result is shown in Fig. 13. Note
that the corresponding energy levels for the coulombic Dirac problem are the
same, due to the aforementioned accidental degeneracy, so the same number is
subtracted from both of these. The fact that both curves appear to go through
the origin supports our speculations about the $a\to 0$ limiting. Moreover, we
observe that the two curves cross and therefore the corresponding Lamb-shift-
like difference between these two energy levels changes sign as $a$ is
increased.

Figure 13: Difference between energy of the first two excited states for $a>0$
and for $a=0$, plotted against $a$.

##  6 Summary and Outlook

In this paper we have supplied a complete characterization of the discrete
spectrum of a Dirac electron in the zero-$G$ Kerr–Newman spacetime, though
only under certain smallness conditions on the electric coupling constant and
on the radius of the ring singularity. This improves on our previous works
[34], [35], [36] where it was shown that the discrete spectrum is non-empty
for more restrictive conditions on the parameters of the problem. The discrete
spectrum is associated with a triplet of integers. One set of integers is the
familiar enumeration of the azimuthal modes, associate with the axisymmetry.
The other two integer families are associated with topological winding numbers
of a coupled system of dynamical equations on finite cylinders. We have
provided a dictionary for how our characterization maps into the familiar
hydrogenic Dirac spectra in the small radius regime of the ring singularity.
We have also provided some numerical illustrations of the low-lying
eigenvalues as functions of the coupling constant and of the radius of the
ring singularity, obtained by implementing the algorithm of our constructive
proof. Empirically, the algorithm converges outside of the estimated
convergence regime, demonstrating that our estimates are far from sharp.
Interestingly, the numerically obtained results show some never-before-seen
oscillatory behavior of the Dirac eigenvalues for large coupling constants
beyond the currently assessible regime of experimental data.

Future work should improve on our assumptions on the parameters of the problem
and obtain sharp results.

Although our constructive proof relies on the iteration of two dynamical
systems that determine the pairs $(\lambda,E)$ jointly, and which is
instrumental in identifying the doubly integer labels for $(\lambda,E)$, from
the computational point of view it is of interest to also try to compute the
energy eigenvalues $E$ directly from the dynamical $\Omega$ system, with the
angular eigenvalues $\lambda$ expressed as analytical functions of $E$ in the
manner determined in [4]. Whether this has computational advantages or not is
an interesting question.

In the future we plan to rigorously investigate the $a\to 0$ limit and
hopefully show that the $a\to 0$ limit of the z$G$KN spectrum converges to the
Bohr-Sommerfeld spectrum as suggested in Figures 12 and 13. In fact, due to
the symmetry of the spectrum, we expect to get two copies of the Bohr-
Sommerfeld spectrum. One copy corresponds to a negative point charge $-e$ in
the $r>0$ sheet, while the other copy corresponds to a positive point charge
$+e$ in the $r<0$ sheet. An interpretation of this negative spectrum is given
in [36].

The KN electromagnetic field can also be generalized to exhibit what elsewhere
we have called a _KN-anomalous magnetic moment_ ,

$\mathbf{A}_{\textsc{KN}}^{\mbox{\tiny{gen}}}=-\frac{r}{r^{2}+a^{2}\cos^{2}\theta}\left(\textsc{q}dt-\frac{\textsc{i}\pi
a^{2}}{c^{2}}\sin^{2}\theta\,d\varphi\right),$ (6.75)

with which one can decorate the zero-$G$ Kerr spacetime of the same ring
radius $a$; here, i is the electrical current which produces a magnetic dipole
moment $\textsc{i}\pi a^{2}/c$ when viewed from spacelike infinity in the
$r>0$ sheet. Although no spacetime with such a generalized KN field has yet
been found for when $G>0$, it is an interesting open question whether one
exists. In any event, in the zero-$G$ limit this family of spacetimes does
exist, and it is not yet known whether the Dirac operator is essentially self-
adjoint or not on its minimal domain. Clearly there are plenty of intriguing
questions that await a rigorous mathematical treatment.

## Acknowledgments

Eric Ling gratefully acknowledges being supported by the Harold H. Martin
Postdoctoral Fellowship at Rutgers University. Shadi Tahvildar-Zadeh thanks
Prof. Siavash Shahshahani for pointing out Ref. [18].

##  Appendix A Errata for reference [34]

While working on this follow-up paper to [34] we found several sign slip-ups
and some other blunders that we correct in this list of errata:

  * -

P.16, Eq.(95): $e$ should be replaced with $-e$.

  * -

P.16, Eq.(101): $m$ should be replaced with $-m$.

  * -

P.16, Eq.(102): change the signs of all four matrix entries.

  * -

P.16, Eq.(103): $i$ should be replaced with $-i$.

  * -

P.16, Eq.(104): change $\mp$ into $\pm$.

  * -

P.17, first two unnumbered equations: change $\lambda$ to $-\lambda$.

  * -

P.17, Eq.(108): exchange $u$ and $v$, and as a result:

  * -

P.17, Eq.(109): the r.h.s. should be replaced with
$\frac{1}{\sqrt{2}}\begin{pmatrix}1&-i\\\ 1&i\end{pmatrix}$.

  * -

P.18, Eq.(114): change $R=\frac{1}{2}\sqrt{u^{2}+v^{2}}$ to
$R=\frac{1}{\sqrt{2}}\sqrt{u^{2}+v^{2}}$.

  * -

P.18, text after Eq.(114): change to $R_{1}=Re^{-i\Omega/2}$ and
$R_{2}=Re^{i\Omega/2}$.

  * -

P.18, Eq.(115): the r.h.s. should be replaced with eq. (2.34) in this paper.

  * -

P.19, Proposition 7.2.: change $|\kappa|\geq\frac{1}{2}$ to
$\kappa\geq\frac{1}{2}$. (The case of $\kappa<0$ requires a different boundary
condition than (123)).

  * -

P.20, in the THEOREM: replace $(-\infty,b]\cup[a,\infty)$ by
$(-\infty,b)\cup(a,\infty)$.

  * -

P.21, above Eq.(131): the fundamental domain is fixed to be
$[x_{-},x_{+}]\times[-\pi,\pi)$. In this paper we let the fundamental domain
be arbitrary $[x_{-},x_{+}]\times[y_{0},y_{0}+2\pi)$.

  * -

P.21, below Eq.(133): replace $s_{-}<n_{-}$ with $s_{-}>n_{-}$ so that it
agrees with assumption (c) in section 3.1.

  * -

P.21, Eq.(137): purge “$(\text{mod}\>2\pi)$”.

  * -

P.24, below Eq.(142): it should read $-\frac{\partial g/\partial\mu}{\partial
g/\partial y}\leq 0$.

  * -

P.27, the expression for $q(T)$: the first $-$ should be a $+$ and the last
$+$ should be a $-$. The conclusion “$\lambda<-a"$ on page 28 is no longer
true given this correction.

  * -

P.32, section 10: $n_{-}$ should be replaced with $-\pi-\cos^{-1}(E)$.

  * -

P.36, the second $\leq$ after $h(\xi)$: this is not valid since $\lambda$ is
negative. Proposition 3.20 in this paper replaces Proposition 7.10 in [34].

  * -

P.36, when examining $j^{\prime}(\xi_{0})$: $\frac{\pi}{2}$ needs to be added
to $\xi_{0}$. Proposition 3.21 in this paper replaces Proposition 7.11 in
[34].

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