is necessarily of the form $\zeta_{p}^{\pm}(k,\cdot)$ with an admissible
$(k,\pm)$. It belongs to the domain of $D_{\omega,\lambda}^{f}$ if its
asymptotic form for $x\to 0$, obtained from (D.24), is proportional to $f$.
This yields conditions described in 1.-3.

Function $c_{p,\pm}$ is meromorphic. In the region
$|\mathrm{Re}(\mu)|<\frac{1}{2}$ functions
$\frac{\omega}{\Gamma(1+\mu\mp\mathrm{i}\lambda)}$ and
$\frac{\lambda\mp\mathrm{i}\mu}{\Gamma(1-\mu\mp\mathrm{i}\lambda)}$ do not
simultaneously vanish anywhere, while $\frac{\Gamma(2\mu+1)}{\Gamma(-2\mu+1)}$
is holomorphic and nowhere vanishing. Hence $c_{p,\pm}$ is not of the
indeterminate form $\frac{0}{0}$ anywhere. ∎

Let us note that eigenfunctions corresponding to real eigenvalues (which exist
only for $|\mathrm{Im}(\lambda)|>\frac{1}{2}$) decay at infinity only as fast
as $x^{-|\mathrm{Im}(\lambda)|}$, not exponentially.

Consider a homogeneous operator $D_{p}$ with $p\in\mathcal{E}^{\pm}$ and its
deformations $D_{\omega,\lambda}^{f}$, with $f$ parametrized by $\kappa$ so
that $D_{\omega,\lambda}^{f}=D_{p}$ for $\kappa=0$. Then for $\kappa=0$ the
point spectrum of $D_{\omega,\lambda}^{f}$ is $\mathbb{C}_{\pm}$, but for
every $\kappa\neq 0$ it is disjoint from $\mathbb{C}_{\pm}$.

##  Appendix A 1-dimensional Dirac operators

###  A.1 General formalism

By a 1d Dirac operator on the halfline we will mean a differential operator of
the form

$D=\begin{bmatrix}a(x)&-\partial_{x}\\\ \partial_{x}&b(x)\end{bmatrix},$ (A.1)

where $a,b$ are smooth functions on $\mathbb{R}_{+}=]0,\infty[$. In this
subsection we treat it as a formal operator acting, say, on the space of
distributions on $\mathbb{R}_{+}$ valued in $\mathbb{C}^{2}$. We first
describe a few integral kernels closely related to $D$.

Let $k\in\mathbb{C}$ and

$\xi(k,x)=\begin{bmatrix}\xi_{\uparrow}(k,x)\\\
\xi_{\downarrow}(k,x)\end{bmatrix},\qquad\zeta(k,x)=\begin{bmatrix}\zeta_{\uparrow}(k,x)\\\
\zeta_{\downarrow}(k,x)\end{bmatrix}$ (A.2)

be a pair of linearly independent solutions of the Dirac equation:

$(D-k)\xi(k,\cdot)=(D-k)\zeta(k,\cdot)=0.$ (A.3)

Let

$d(k,x):=\det\left[\xi(k,x),\zeta(k,x)\right]=\det\begin{bmatrix}\xi_{\uparrow}(k,x)&\zeta_{\uparrow}(k,x)\\\
\xi_{\downarrow}(k,x)&\zeta_{\downarrow}(k,x).\end{bmatrix}$ (A.4)

Then $d(k,x)$ does not depend on $x$, so that one can write $d(k)$ instead. We
define

$\displaystyle
G^{\leftrightarrow}(k;x,y):=d(k)^{-1}\xi(k,x)\zeta(k,y)^{\mathrm{T}}-d(k)^{-1}\zeta(k,x)\xi(k,y)^{\mathrm{T}}$
(A.5)
$\displaystyle=d(k)^{-1}\begin{bmatrix}\xi_{\uparrow}(k,x)\zeta_{\uparrow}(k,y)&\xi_{\uparrow}(k,x)\zeta_{\downarrow}(k,y)\\\\[4.30554pt]
\xi_{\downarrow}(k,x)\zeta_{\uparrow}(k,y)&\xi_{\downarrow}(k,x)\zeta_{\downarrow}(k,y)\end{bmatrix}-d(k)^{-1}\begin{bmatrix}\zeta_{\uparrow}(k,x)\xi_{\uparrow}(k,y)&\zeta_{\uparrow}(k,x)\xi_{\downarrow}(k,y)\\\\[4.30554pt]
\zeta_{\downarrow}(k,x)\xi_{\uparrow}(k,y)&\zeta_{\downarrow}(k,x)\xi_{\downarrow}(k,y)\end{bmatrix}.$

Note that $G^{\leftrightarrow}(k,x,y)$ is uniquely defined by

$(D-k)G^{\leftrightarrow}(k;x,y)=0,\qquad
G^{\leftrightarrow}(k;x,x)=\begin{bmatrix}0&1\\\ -1&0\end{bmatrix}.$ (A.6)

We will call it the canonical bisolution.

We also have the forward and backward Green’s operators given by the kernels

$\displaystyle G^{\rightarrow}(k;x,y)=$ $\displaystyle
G^{\leftrightarrow}(k;x,y){\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathbb{R}_{+}}(x-y),$ (A.7a) $\displaystyle
G^{\leftarrow}(k;x,y)=$
$\displaystyle-G^{\leftrightarrow}(k;x,y){\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathbb{R}_{+}}(y-x).$ (A.7b)

They are uniquely defined by

$\displaystyle(D-k)G^{\rightarrow}(k;x,y)=\delta(x-y)\begin{bmatrix}1&0\\\
0&1\end{bmatrix},$ $\displaystyle\quad x<y\Rightarrow G^{\rightarrow}(x,y)=0;$
(A.8a)
$\displaystyle(D-k)G^{\leftarrow}(k;x,y)=\delta(x-y)\begin{bmatrix}1&0\\\
0&1\end{bmatrix},$ $\displaystyle\quad x>y\Rightarrow G^{\leftarrow}(x,y)=0.$
(A.8b)

Note that $G^{\leftrightarrow},G^{\leftarrow},G^{\rightarrow}$ do not depend
on the choice of $\xi,\zeta$.

Using the eigensolutions $\xi,\zeta$, we can introduce yet another important
integral kernel:

$\displaystyle G^{\bowtie}(k;x,y):=$
$\displaystyle-d(k)^{-1}\begin{bmatrix}\xi_{\uparrow}(k,x)\zeta_{\uparrow}(k,y)&\xi_{\uparrow}(k,x)\zeta_{\downarrow}(k,y)\\\
\xi_{\downarrow}(k,x)\zeta_{\uparrow}(k,y)&\xi_{\downarrow}(k,x)\zeta_{\downarrow}(k,y)\end{bmatrix}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathbb{R}_{+}}(y-x)$
$\displaystyle-d(k)^{-1}\begin{bmatrix}\zeta_{\uparrow}(k,x)\xi_{\uparrow}(k,y)&\zeta_{\uparrow}(k,x)\xi_{\downarrow}(k,y)\\\
\zeta_{\downarrow}(k,x)\xi_{\uparrow}(k,y)&\zeta_{\downarrow}(k,x)\xi_{\downarrow}(k,y)\end{bmatrix}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathbb{R}_{+}}(x-y).$ (A.9)

It is also Green’s kernel, because it satisfies

$\displaystyle(D-k)G^{\bowtie}(k;x,y)=\delta(x-y)\begin{bmatrix}1&0\\\
0&1\end{bmatrix}.$ (A.10)

$G^{\bowtie}(k;x,y)$ depends on the choice of the pair of 1-dimensional
subspaces $\mathbb{C}\xi(k,\cdot)$, $\mathbb{C}\zeta(k,\cdot)$ of
$\mathrm{Ker}(D-k)$. The resolvents of various closed realizations of $D$ are
often of this form.

Two classes of 1d Dirac operators have special properties. The case
$a(x)=b(x)$ can be fully diagonalized:

$\displaystyle\begin{bmatrix}a(x)&-\partial_{x}\\\
\partial_{x}&a(x)\end{bmatrix}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{bmatrix}1&\mathrm{i}\\\
\mathrm{i}&1\end{bmatrix}\begin{bmatrix}-\mathrm{i}\partial_{x}+a(x)&0\\\
0&\mathrm{i}\partial_{x}+a(x)\end{bmatrix}\frac{1}{\sqrt{2}}\begin{bmatrix}1&-\mathrm{i}\\\
-\mathrm{i}&1\end{bmatrix}.$ (A.11)

We will analyze 1d Dirac-Coulomb operators of this form in Subsection A.3.

The case $a(x)=-b(x)$ can be brought to an antidiagonal form, used in
supersymmetry:

$\displaystyle\begin{bmatrix}a(x)&-\partial_{x}\\\
\partial_{x}&-a(x)\end{bmatrix}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{bmatrix}1&-1\\\
1&1\end{bmatrix}\begin{bmatrix}0&-\partial_{x}-a(x)\\\
\partial_{x}-a(x)&0\end{bmatrix}\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\\
-1&1\end{bmatrix}.$ (A.12)

We will analyze 1d Dirac-Coulomb operators of this form in Subsection A.5.

###  A.2 Homogeneous first order scalar operators

Let $\alpha\in\mathbb{C}$. In this subsection we discuss the differential
operator

$A_{\alpha}=x^{\alpha}\partial_{x}x^{-\alpha}=\partial_{x}-\frac{\alpha}{x}$
(A.13)

acting on scalar functions. It will be a building block of some special 1d
Dirac-Coulomb operators considered in subsections A.3 and A.5.

Let us briefly recall basic results about realizations of $A_{\alpha}$ as a
closed operator in $L^{2}(\mathbb{R}_{+})$ following [3]. Proofs of all
statements stated in this subsection without justification can be found
therein. (In [3] a different convention was used:
$A_{\alpha}=-\mathrm{i}\partial_{x}+\frac{\mathrm{i}\alpha}{x}$. Thus
$A_{\alpha}^{\rm new}=\mathrm{i}A_{\alpha}^{\rm old}$.)

We let $A_{\alpha}^{\min}$ be the closure (in the sense of operators on
$L^{2}(\mathbb{R}_{+})$) of the restriction of $A_{\alpha}$ to
$C_{c}^{\infty}(\mathbb{R}_{+})$ and $A_{\alpha}^{\max}$ the restriction of
$A_{\alpha}$ to $\mathrm{Dom}(A_{\alpha}^{\max})=\\{f\in
L^{2}(\mathbb{R}_{+})\,|\,A_{\alpha}f\in L^{2}(\mathbb{R})\\}$. Operators
$A_{\alpha}^{\min}$ and $-A_{-\overline{\alpha}}^{\max}$ are adjoint to each
other.

######  Proposition 42.

We have $A_{\alpha}^{\min}=A_{\alpha}^{\max}$ if and only if
$|\mathrm{Re}(\alpha)|\geq\frac{1}{2}$. If
$|\mathrm{Re}(\alpha)|<\frac{1}{2}$, then
$\mathrm{Dom}(A_{\alpha}^{\max})=\mathrm{Dom}(A_{\alpha}^{\min})+\mathbb{C}\chi
x^{\alpha}$, where $\chi\in C_{\mathrm{c}}^{\infty}(\mathbb{R}_{+})$ and
$\chi=1$ near $0$. If $\mathrm{Re}(\alpha)\neq\frac{1}{2}$, then
$\mathrm{Dom}(A_{\alpha}^{\min})=H_{0}^{1}(\mathbb{R}_{+})$.

Closed realizations of $A_{\alpha}$ are of two types, described in the
following pair of propositions.

######  Proposition 43.

Let $\mathrm{Re}(\alpha)>-\frac{1}{2}$,

  1. 1.

$\sigma(-\mathrm{i}A_{\alpha}^{\max})=\mathbb{R}\cup\mathbb{C}_{+}$ and one
has

$(A^{\max}_{\alpha}-\mathrm{i}k)^{-1}f(x)=-\int_{x}^{\infty}\mathrm{e}^{\mathrm{i}k(x-y)}\left(\frac{x}{y}\right)^{\alpha}f(y)\mathrm{d}y,\qquad\mathrm{Im}(k)<0.$
(A.14)

  2. 2.

If $k\in\mathbb{C}_{+}$ and $n\geq 1$, then
$\mathrm{Ker}\left((A_{\alpha}^{\max}-\mathrm{i}k)^{n}\right)$ is the space of
functions of the form $x^{\alpha}\mathrm{e}^{\mathrm{i}kx}q(x)$ with $q$
polynomial of degree at most $n-1$. In particular
$\bigcup\limits_{n=0}^{\infty}\mathrm{Ker}\left((A_{\alpha}^{\max}-\mathrm{i}k)^{n}\right)$
is dense in $L^{2}(\mathbb{R}_{+})$.

  3. 3.

If $k\in\mathbb{C}_{+}$, then $A_{\alpha}^{\max}-\mathrm{i}k$ is a Fredholm
operator of index $1$.

######  Proposition 44.

Let $\mathrm{Re}(\alpha)<\frac{1}{2}$.

  1. 1.

$\sigma(-\mathrm{i}A_{\alpha}^{\min})=\mathbb{R}\cup\mathbb{C}_{-}$ and one
has

$(A^{\min}_{\alpha}-\mathrm{i}k)^{-1}f(x)=\int_{0}^{x}\mathrm{e}^{\mathrm{i}k(x-y)}\left(\frac{x}{y}\right)^{\alpha}f(y)\mathrm{d}y,\qquad\mathrm{Im}(k)>0.$
(A.15)

  2. 2.

$A_{\alpha}^{\min}$ has no eigenvectors.

  3. 3.

If $k\in\mathbb{C}_{-}$, then $A_{\alpha}^{\min}-\mathrm{i}k$ is a Fredholm
operator of index $-1$.

Proof of Propositions 43 and 44. Statements 1. are proven in [3].

2\. requires justification only for the first part in the first proposition.
We factorize

$x^{\alpha}\mathrm{e}^{\mathrm{i}kx}q(x)=x^{\mathrm{i}\,\mathrm{Im}(\alpha)}\mathrm{e}^{\mathrm{i}\,\mathrm{Re}(k)x}\left(x^{\mathrm{Re}(\alpha)}\mathrm{e}^{-\mathrm{Im}(k)x}q(x)\right).$
(A.16)

Functions in the parenthesis form a dense set, because for any real numbers
$c>0$, $\beta>-1$ functions
$e^{-\frac{cx}{2}}x^{\frac{\beta}{2}}L_{n}^{(\beta)}(cx)$, with
$L_{n}^{(\beta)}$ Laguerre polynomials, form an orthogonal basis (see e.g.
[41]). Clearly density is unaffected by the prefactor, which amounts to the
action of a certain unitary operator on $L^{2}$.

Let us show 3. We consider first the case $|\mathrm{Re}(\alpha)|<\frac{1}{2}$.
Then we have explicit inverses modulo rank one operators.

If $\mathrm{Im}(k)>0$, then $A^{\min}_{\alpha}-\mathrm{i}k$ is invertible and
its inverse is a right inverse for $A_{\alpha}^{\max}-\mathrm{i}k$. Thus
$A_{\alpha}^{\max}-\mathrm{i}k$ is surjective. We already know that its kernel
is one-dimensional.

If $\mathrm{Im}(k)<0$, then
$(A_{\alpha}^{\max}-\mathrm{i}k)^{-1}:L^{2}(\mathbb{R}_{+})\to\mathrm{Dom}(A_{\alpha}^{\max})$
is continuous. The range of $A_{\alpha}^{\min}-\mathrm{i}k$ is the preimage of
$\mathrm{Dom}(A_{\alpha}^{\min})$, which is a closed subspace of
$\mathrm{Dom}(A_{\alpha}^{\max})$ of codimension one. Hence
$\mathrm{Ran}(A_{\alpha}^{\min}-\mathrm{i}k)$ is a closed subspace of
$L^{2}(\mathbb{R}_{+})$ of codimension one.

To extended the result beyond the strip $|\mathrm{Re}(\alpha)|<\frac{1}{2}$,
note that
$(A_{\alpha}^{\min}-\mathrm{i}k)^{-1}-(A_{\beta}^{\min}-\mathrm{i}k)^{-1}$
(resp.
$(A_{\alpha}^{\max}-\mathrm{i}k)^{-1}-(A_{\beta}^{\max}-\mathrm{i}k)^{-1}$)
has a square-integrable integral kernel for
$\mathrm{Re}(\alpha),\,\mathrm{Re}(\beta)<\frac{1}{2}$ and
$k\in\mathbb{C}_{+}$ (resp.
$\mathrm{Re}(\alpha),\,\mathrm{Re}(\beta)>-\frac{1}{2}$ and
$k\in\mathbb{C}_{-}$). Therefore it is a Hilbert-Schmidt operator, in
particular compact. By Corollary 25, the essential spectrum of
$A_{\alpha}^{\min}$ and $A_{\alpha}^{\max}$ does not depend on $\alpha$. From
the case $|\mathrm{Re}(\alpha)|<\frac{1}{2}$ we know that it is $\mathbb{R}$.
The statement about the value of the index is clear. ∎

######  Proposition 45.

$A_{\alpha}^{\max}$ is the generator of a $C^{0}$-semigroup if and only if
$\mathrm{Re}(\alpha)\geq 0$. If this condition is satisfied, it generates the
semigroup of contractions

$(\mathrm{e}^{tA_{\alpha}^{\max}}f)(x)=x^{\alpha}(x+t)^{-\alpha}f(x+t),\qquad
t\geq 0,$ (A.17)

$-A_{\alpha}^{\min}$ is the generator of a $C^{0}$-semigroup if and only if
$\mathrm{Re}(\alpha)\leq 0$. If this condition is satisfied, it generated a
semigroup of contractions

$(\mathrm{e}^{-tA_{\alpha}^{\min}}f)(x)=x^{\alpha}(x-t)^{-\alpha}f(x-t),\qquad
t\geq 0.$ (A.18)

Here we put $f(x-t)=0$ if $x-t<0$.

If $|\mathrm{Re}(\alpha)|<\frac{1}{2}$, the operators $-A_{\alpha}^{\max}$ and
$A_{\alpha}^{\min}$ are not generators of $C^{0}$-semigroups.

###### Proof.

We present a proof of the statements concerning $A_{\alpha}^{\max}$. The
others can be proven analogously. It is elementary to check that for
$\mathrm{Re}(\alpha)\geq 0$ the right hand side of (A.17) defines a
$C^{0}$-semigroup of contractions with the generator $A_{\max}^{\alpha}$. If
$\mathrm{Re}(\alpha)<0$, we consider the same expression for $f\in
C_{c}^{\infty}(]1,\infty[)$. Then for $t\leq 1$ it is the unique solution of
the Cauchy problem
$\frac{\mathrm{d}}{\mathrm{d}t}f_{t}=A_{\alpha}^{\max}f_{t}$, $f_{0}=f$.
However, there exists no constant $c$ such that $\|f_{t}\|\leq c\|f\|$ for
every $t\in[0,1]$ and $f$. Thus $A_{\alpha}^{\max}$ is not a generator. If
$|\mathrm{Re}(\alpha)|<\frac{1}{2}$, then $\sigma(-A_{\alpha}^{\max})$ is the
right closed complex half-plane, so $-A_{\alpha}^{\max}$ is not a generator. ∎

###  A.3 Dirac-Coulomb Hamiltonians with $\omega=0$

Dirac-Coulomb Hamiltonians with $\omega=0$ can be reduced to operators
$A_{\alpha}$ studied in Subsection A.2. Therefore, they can be analyzed using
elementary functions only.

Let us set $W:=\frac{1}{\sqrt{2}}\begin{bmatrix}1&\mathrm{i}\\\
\mathrm{i}&1\end{bmatrix}$. Using (A.11) we obtain for all $\lambda$

$\displaystyle
D_{0,\lambda}^{\min}=W\begin{bmatrix}-\mathrm{i}A_{\mathrm{i}\lambda}^{\min}&0\\\
0&\mathrm{i}A_{-\mathrm{i}\lambda}^{\min}\end{bmatrix}W^{-1},\qquad
D_{0,\lambda}^{\max}=W\begin{bmatrix}-\mathrm{i}A_{\mathrm{i}\lambda}^{\max}&0\\\
0&\mathrm{i}A_{-\mathrm{i}\lambda}^{\max}\end{bmatrix}W^{-1}.$ (A.19)

Consider now the homogeneous holomorphic family. Note first that $\omega=0$
implies $\mu=\pm\mathrm{i}\lambda$. We set
$D_{\lambda}^{\pm}:=D_{0,\lambda,\pm\mathrm{i}\lambda,[\mp\mathrm{i}:1]}$.
Note that
$(0,\lambda,\pm\mathrm{i}\lambda,[\mp\mathrm{i}:1])\in\mathcal{E}^{\pm}$. We
have:

$\displaystyle D_{\lambda}^{+}$
$\displaystyle=W\begin{bmatrix}-\mathrm{i}A_{\mathrm{i}\lambda}^{\max}&0\\\
0&\mathrm{i}A_{-\mathrm{i}\lambda}^{\min}\end{bmatrix}W^{-1},\qquad\mathrm{Re}(\mathrm{i}\lambda)>-\frac{1}{2},$
(A.20a) $\displaystyle D_{\lambda}^{-}$
$\displaystyle=W\begin{bmatrix}-\mathrm{i}A_{\mathrm{i}\lambda}^{\min}&0\\\
0&\mathrm{i}A_{-\mathrm{i}\lambda}^{\max}\end{bmatrix}W^{-1},\qquad\mathrm{Re}(-\mathrm{i}\lambda)>-\frac{1}{2},$
(A.20b)

Below $\sigma_{2}$ is the Pauli matrix $\begin{bmatrix}0&-\mathrm{i}\\\
\mathrm{i}&0\end{bmatrix}$. Matrices $\frac{1\pm\sigma_{2}}{2}$ are its
spectral projections.

######  Proposition 46.

We have $\sigma(D_{\lambda}^{+})=\mathbb{R}\cup\mathbb{C}_{+}$ and

$(D_{\lambda}^{+}-k)^{-1}=(-\mathrm{i}A_{\mathrm{i}\lambda}^{\max}-k)^{-1}\frac{1+\sigma_{2}}{2}-(-\mathrm{i}A_{-\mathrm{i}\lambda}^{\min}+k)^{-1}\frac{1-\sigma_{2}}{2},\qquad\mathrm{Im}(k)<0,$
(A.21)

whereas $\sigma(D_{\lambda}^{-})=\mathbb{R}\cup\mathbb{C}_{-}$ and

$(D_{\lambda}^{-}-k)^{-1}=(-\mathrm{i}A_{\mathrm{i}\lambda}^{\min}-k)^{-1}\frac{1+\sigma_{2}}{2}-(-\mathrm{i}A_{-\mathrm{i}\lambda}^{\max}+k)^{-1}\frac{1-\sigma_{2}}{2},\qquad\mathrm{Im}(k)>0.$
(A.22)

###### Proof.

Follows from identities (A.20b) and Proposition 43, 44. ∎

######  Proposition 47.

$D_{\lambda}^{\pm}-k$ with $k\in\mathbb{C}_{\pm}$ are Fredholm of index $0$.

###### Proof.

Indeed, by Propositions 43 and 44 they are direct sums of two Fredholm
operators with indices $1$ and $-1$. ∎

######  Proposition 48.

Let $k\in\mathbb{C}_{\pm}$. Then
$\bigcup\limits_{n=0}^{\infty}\mathrm{Ker}((D_{\lambda}^{\pm}-k)^{n})$ is a
dense subspace of $L^{2}(\mathbb{R}_{+})\begin{bmatrix}\mp\mathrm{i}\\\
1\end{bmatrix}$.

######  Proposition 49.

$\mathrm{i}D_{\lambda}^{+}$ is the generator of a $C^{0}$-semigroup if and
only if $\mathrm{Im}(\lambda)\leq 0$. Then it generates the semigroup of
contractions

$\mathrm{e}^{\mathrm{i}tD_{\lambda}^{+}}=\mathrm{e}^{tA_{\mathrm{i}\lambda}^{\max}}\frac{1+\sigma_{2}}{2}+\mathrm{e}^{-tA^{\min}_{-\mathrm{i}\lambda}}\frac{1-\sigma_{2}}{2},\qquad
t\geq 0.$ (A.23)

$-\mathrm{i}D_{\lambda}^{-}$ is the generator of a $C^{0}$-semigroup if and
only if $\mathrm{Im}(\lambda)\geq 0$. Then it generates the semigroup of
contractions

$\mathrm{e}^{-\mathrm{i}tD_{\lambda}^{-}}=\mathrm{e}^{-tA_{\mathrm{i}\lambda}^{\min}}\frac{1+\sigma_{2}}{2}+\mathrm{e}^{tA^{\max}_{-\mathrm{i}\lambda}}\frac{1-\sigma_{2}}{2},\qquad
t\geq 0.$ (A.24)

Operators $-\mathrm{i}D_{\lambda}^{+}$ and $\mathrm{i}D_{\lambda}^{-}$ are not
generators of $C^{0}$-semigroups.

###  A.4 Hankel transformation

The following proposition is proven e.g. in [3].

######  Proposition 50.

Let $\mathrm{Re}(m)\geq-1$. We define

$(\mathcal{F}_{m}^{\mathrm{pre}}f)(x)=\int_{0}^{\infty}J_{m}(xy)\sqrt{xy}f(y)\mathrm{d}y,\qquad
f\in C_{c}^{\infty}(\mathbb{R}_{+}),$ (A.25)

where $J_{m}$ is the Bessel function. Then $\mathcal{F}_{m}^{\mathrm{pre}}$
extends to a bounded operator $\mathcal{F}_{m}$ on $L^{2}(\mathbb{R}_{+})$,
known as the Hankel transformation. $\mathcal{F}_{m}$ is a self-transposed
involution, unitary if $m$ is real.

Recall from Subsection 1.1 that the operator $X$ is defined by

$(Xf)(x)=xf(x),\qquad\mathrm{Dom}(X)=\\{f\in
L^{2}(\mathbb{R}_{+})\,|\,xf(x)\in L^{2}(\mathbb{R}_{+})\\}.$

######  Proposition 51.

If $\mathrm{Re}(\alpha)>-\frac{1}{2}$, one has

$\mathcal{F}_{\alpha+\frac{1}{2}}A_{\alpha}^{\max}\mathcal{F}_{\alpha-\frac{1}{2}}=-X,\qquad\mathcal{F}_{\alpha-\frac{1}{2}}A_{-\alpha}^{\min}\mathcal{F}_{\alpha+\frac{1}{2}}=X.$
(A.26)

###### Proof.

Using the identity

$x^{-m}\frac{\mathrm{d}}{\mathrm{d}x}x^{m}J_{m}(x)=J_{m-1}(x)$ (A.27)

one checks that

$(\mathcal{F}_{\alpha+\frac{1}{2}}^{\mathrm{pre}}A_{\alpha}f)(x)=-x(\mathcal{F}_{\alpha-\frac{1}{2}}^{\mathrm{pre}}f)(x)$
(A.28)

for $f\in C_{c}^{\infty}(\mathbb{R}_{+})$. If
$|\mathrm{Re}(\alpha)|<\frac{1}{2}$, (A.28) may be checked to hold also for
$f(x)=\chi(x)x^{\alpha}$. Taking closures we obtain
$\mathcal{F}_{\alpha+\frac{1}{2}}A_{\alpha}^{\max}\subset-X\mathcal{F}_{\alpha-\frac{1}{2}}$,
so
$\mathcal{F}_{\alpha+\frac{1}{2}}A_{\alpha}^{\max}\mathcal{F}_{\alpha-\frac{1}{2}}\subset-X$.
Since
$\mathcal{F}_{\alpha+\frac{1}{2}}A_{\alpha}^{\max}\mathcal{F}_{\alpha-\frac{1}{2}}$
is a closed operator and $C_{c}^{\infty}(\mathbb{R}_{+})$ is a dense subspace
of $\mathrm{Dom}(X)$ with respect to the graph topology, the opposite
inclusion will be established by demonstrating that
$C_{c}^{\infty}(\mathbb{R}_{+})\subset\mathrm{Dom}(\mathcal{F}_{\alpha+\frac{1}{2}}A_{\alpha}^{\max}\mathcal{F}_{\alpha-\frac{1}{2}})$.
Let $f\in C_{c}^{\infty}(\mathbb{R}_{+})$. It is clear that
$\mathcal{F}_{\alpha-\frac{1}{2}}f$ is a smooth function. Using the identity
$\left(x\frac{\mathrm{d}}{\mathrm{d}x}-y\frac{\mathrm{d}}{\mathrm{d}y}\right)\sqrt{xy}J_{m}(xy)=0$
we find

$\frac{\mathrm{d}}{\mathrm{d}x}(\mathcal{F}_{\alpha-\frac{1}{2}}f)(x)=\frac{1}{x}\int_{0}^{\infty}\sqrt{xy}J_{m}(xy)\frac{\mathrm{d}}{\mathrm{d}y}\frac{f(y)}{y}\mathrm{d}y.$
(A.29)

Since $\frac{\mathrm{d}}{\mathrm{d}y}\frac{f(y)}{y}$ is in
$L^{2}(\mathbb{R}_{+})$, we get that
$\frac{\mathrm{d}}{\mathrm{d}x}(\mathcal{F}_{\alpha-\frac{1}{2}}f)(x)$ is
square-integrable over $[1,\infty[$. Next we use the series expansion of
$J_{m}$ to find that for small $x$

$(\mathcal{F}_{\alpha-\frac{1}{2}}f)(x)=\frac{\int_{0}^{\infty}y^{\alpha}f(y)\mathrm{d}y}{2^{\alpha-\frac{1}{2}}\Gamma(\alpha+\frac{1}{2})}x^{\alpha}+O(x^{\alpha+1}).$
(A.30)

Hence $\mathcal{F}_{\alpha-\frac{1}{2}}f\in\mathrm{Dom}(A_{\alpha}^{\max})$,
so
$f\in\mathrm{Dom}(A_{\alpha}^{\max}\mathcal{F}_{\alpha-\frac{1}{2}})=\mathrm{Dom}(\mathcal{F}_{\alpha+\frac{1}{2}}A_{\alpha}^{\max}\mathcal{F}_{\alpha-\frac{1}{2}})$.
We proved the first equality in (A.26). The other one may be obtained by
taking the transpose. ∎

Following [12] (see also [3]), we consider the formal differential operator

$L_{m^{2}}=-\partial_{x}^{2}+\frac{m^{2}-\frac{1}{4}}{x^{2}}.$ (A.31)

We let $L_{m^{2}}^{\min}$ be the closure of its restriction to
$C_{\mathrm{c}}^{\infty}(\mathbb{R}_{+})$ and $L_{m^{2}}^{\max}$ be the
restriction to $\mathrm{Dom}(L_{m^{2}}^{\max})=\\{f\in
L^{2}(\mathbb{R}_{+})\,|\,L_{m^{2}}f\in L^{2}(\mathbb{R}_{+})\\}$. If
$\mathrm{Re}(m)>-1$, operator $H_{m}$ is defined as the restriction of
$L_{m^{2}}$ to $\mathrm{Dom}(L_{m^{2}}^{\min})+\mathbb{C}\chi
x^{m+\frac{1}{2}}$, where $\chi$ is a smooth function equal to one in a
neighborhood of zero. We remark that $H_{\frac{1}{2}}$ and $H_{-\frac{1}{2}}$
are the Dirichlet Laplacian and the Neuman Laplacian, respectively.
Furthermore, $H_{m}$ can be diagonalized as follows:

$H_{m}=\mathcal{F}_{m}X^{2}\mathcal{F}_{m}.$ (A.32)

###  A.5 Dirac-Coulomb Hamiltonians with $\lambda=0$

Dirac-Coulomb Hamiltonians with $\lambda=0$ can be analyzed without Whittaker
functions, just with Bessel functions.

Let us set $U:=\frac{1}{\sqrt{2}}\begin{bmatrix}1&-1\\\ 1&1\end{bmatrix}$.
Using (A.12) we obtain for all $\omega$

$\displaystyle D_{\omega,0}^{\min}=U\begin{bmatrix}0&-A_{\omega}^{\min}\\\
A_{-\omega}^{\min}&0\end{bmatrix}U^{-1},\qquad
D_{\omega,0}^{\max}=U\begin{bmatrix}0&-A_{\omega}^{\max}\\\
A_{-\omega}^{\max}&0\end{bmatrix}U^{-1}.$ (A.33)

Using Proposition 42 we rewrite the operators
$D^{\prime\pm}_{\omega}:=D_{\omega,0,\pm\omega,[\mp 1:1]}$ as

$\displaystyle D^{\prime+}_{\omega}$
$\displaystyle=U\begin{bmatrix}0&-A_{\omega}^{\max}\\\
A_{-\omega}^{\min}&0\end{bmatrix}U^{-1},\quad\mathrm{Re}(\omega)>-\frac{1}{2};$
(A.34a) $\displaystyle D^{\prime-}_{\omega}$
$\displaystyle=U\begin{bmatrix}0&-A_{\omega}^{\min}\\\
A_{-\omega}^{\max}&0\end{bmatrix}U^{-1},\quad-\mathrm{Re}(\omega)>-\frac{1}{2}.$
(A.34b)

######  Proposition 52.

Introduce

$\mathcal{W}_{\omega}^{\prime\pm}:=\frac{1}{2}\begin{bmatrix}\mathcal{F}_{\pm\omega\pm\frac{1}{2}}{+}\mathcal{F}_{\pm\omega\mp\frac{1}{2}}&\mathcal{F}_{\pm\omega\pm\frac{1}{2}}{-}\mathcal{F}_{\pm\omega\mp\frac{1}{2}}\\\
\mathcal{F}_{\pm\omega\pm\frac{1}{2}}{-}\mathcal{F}_{\pm\omega\mp\frac{1}{2}}&\mathcal{F}_{\pm\omega\pm\frac{1}{2}}{+}\mathcal{F}_{\pm\omega\mp\frac{1}{2}}\end{bmatrix},\quad\pm\mathrm{Re}(\omega)>-\frac{1}{2}.$
(A.35)

Then $\mathcal{W}_{\omega}^{\prime\pm}$ are involutions and we have the
following diagonalizations

$D_{\omega}^{\prime\pm}=\mathcal{W}_{\omega}^{\prime\pm}\begin{bmatrix}\mp
X&0\\\ 0&\pm X\end{bmatrix}\mathcal{W}_{\omega}^{\prime\pm}.$ (A.36)

###### Proof.

We insert (A.26) into (A.34):

$\displaystyle
D_{\omega}^{\prime+}=U\begin{bmatrix}\mathcal{F}_{\omega+\frac{1}{2}}&0\\\
0&\mathcal{F}_{\omega-\frac{1}{2}}\end{bmatrix}\begin{bmatrix}0&X\\\
X&0\end{bmatrix}\begin{bmatrix}\mathcal{F}_{\omega+\frac{1}{2}}&0\\\
0&\mathcal{F}_{\omega-\frac{1}{2}}\end{bmatrix}U^{-1},$ (A.37a) $\displaystyle
D_{\omega}^{\prime-}=U\begin{bmatrix}\mathcal{F}_{-\omega-\frac{1}{2}}&0\\\
0&\mathcal{F}_{-\omega+\frac{1}{2}}\end{bmatrix}\begin{bmatrix}0&-X\\\
-X&0\end{bmatrix}\begin{bmatrix}\mathcal{F}_{-\omega-\frac{1}{2}}&0\\\
0&\mathcal{F}_{-\omega+\frac{1}{2}}\end{bmatrix}U^{-1}.$ (A.37b)

Then we use

$\displaystyle\begin{bmatrix}0&\pm X\\\ \pm
X&0\end{bmatrix}=U^{-1}\begin{bmatrix}\mp X&0\\\ 0&\pm X\end{bmatrix}U.$
(A.38)

∎

######  Corollary 53.

We have

$\displaystyle(D^{\prime+}_{\omega})^{2}=U\begin{bmatrix}H_{\omega+\frac{1}{2}}&0\\\
0&H_{\omega-\frac{1}{2}}\end{bmatrix}U^{-1},$ (A.39a)
$\displaystyle(D^{\prime-}_{\omega})^{2}=U\begin{bmatrix}H_{-\omega-\frac{1}{2}}&0\\\
0&H_{-\omega+\frac{1}{2}}\end{bmatrix}U^{-1}.$ (A.39b)

######  Remark 54.

At least formally, operators $D_{\omega}^{\prime\pm}$, $X\sigma_{2}$ (declared
to be odd) and $(D_{\omega}^{\prime\pm})^{2}$, $X^{2}$, $A$ (declared to be
even) furnish a representation of the Lie superalgebra $\mathfrak{osp}(1|2)$.
We leave a detailed description of this representation for a future study.

##  Appendix B Dirac Hamiltonian in $d$ dimensions

Separation of variables of a spherically symmetric Dirac Hamiltonian in
dimension 3 is described in many texts and belongs to the standard curriculum
of relativistic quantum mechanics [15, p. 267]. Of course, it is even more
straightforward to solve a rotationally symmetric Dirac Hamiltonian in
dimension 2. However, to our knowledge, the first treatment in any dimension
is due to Gu, Ma and Dong [28].

In this appendix we show that a spherically symmetric Dirac Hamiltonian in an
arbitrary dimension can be reduced to 1 dimension. Unlike in [28], we arrive
at the radial Dirac equation by relatively simple algebraic computations which
do not involve a detailed analysis of representations of the Lie algebra
$\mathfrak{so}(d)$.

The main role in this separation is played by a certain operator $\kappa$ that
commutes with the Dirac operator. This operator in dimension 3 goes back to
Dirac himself. It seems that for the first time it has been generalized to
other dimensions in [28]. We analyze this operator in detail.

Recall that operators belonging to the center of the envelopping algebra of
$\mathfrak{so}(d)$ are called Casimir operators of $\mathfrak{so}(d)$. One of
them, built in a standard way as a bilinear form in the generators, will be
called the square of angular momentum or simply the quadratic Casimir (even
though it is not the only Casimir bilinear in generators: these form a vector
space generically of dimension $1$, and of dimension $2$ if $d=4$). $\kappa$
does not coincide with the quadratic Casimir. One can ask whether $\kappa$ is
also a Casimir operator. We will analyse this question in detail. It turns out
that the answer is positive in even, and negative in odd dimensions.

###  B.1 Laplacian in $d$ dimensions

Spherical coordinates can be interpreted as a map

$\displaystyle\mathbb{R}^{d}\backslash\\{0\\}\ni x$
$\displaystyle\mapsto(r,\hat{x})\in\mathbb{R}_{+}\times\mathbb{S}^{d-1},$
(B.1a) $\displaystyle\hat{x}=\frac{x}{|x|},$ $\displaystyle\quad r=|x|.$
(B.1b)

It induces a unitary map

$L^{2}(\mathbb{R}^{d})\to L^{2}(\mathbb{R}_{+},r^{d-1})\otimes
L^{2}(\mathbb{S}^{d-1}).$ (B.2)

We also have the obvious map

$L^{2}(\mathbb{R}_{+},r^{d-1})\ni f\mapsto r^{\frac{d-1}{2}}f\in
L^{2}(\mathbb{R}_{+}).$ (B.3)

The product of (B.3) and (B.2) will be denoted

$U:L^{2}(\mathbb{R}^{d})\to L^{2}(\mathbb{R}_{+})\otimes
L^{2}(\mathbb{S}^{d-1}).$ (B.4)

The momentum is defined as

$p_{i}:=-\mathrm{i}\partial_{i}.$

We also introduce the radial momentum

$\displaystyle R$
$\displaystyle:=\frac{x}{2|x|}p+p\frac{x}{2|x|}=\frac{x}{|x|}p-\mathrm{i}\frac{d-1}{2|x|}.$
(B.5)

Here is the radial momentum and its square in spherical coordinates:

$\displaystyle R$
$\displaystyle=-\mathrm{i}\partial_{r}-\mathrm{i}\frac{d-1}{2r},$ (B.6a)
$\displaystyle R^{2}$
$\displaystyle=-\partial_{r}^{2}-\frac{d-1}{r}\partial_{r}+\left(\frac{1}{4}-\Big{(}\frac{d-2}{2}\Big{)}^{2}\right)\frac{1}{r^{2}}.$
(B.6b)

After applying $U$ we obtain

$URU^{-1}=-\mathrm{i}\partial_{r}.$

In the standard way we introduce the angular momentum and its square:

$\displaystyle L_{ij}$ $\displaystyle:=x_{i}p_{j}-x_{j}p_{i},$ (B.7a)
$\displaystyle L^{2}$ $\displaystyle:=\sum_{i<j}L_{ij}^{2}.$ (B.7b)

They furnish the standard representation of the Lie algebra $\mathfrak{so}(d)$
on $\mathbb{S}^{d-1}$:

$\displaystyle[L_{ij},x_{k}]$
$\displaystyle=-\mathrm{i}\delta_{jk}x_{i}+\mathrm{i}\delta_{ik}x_{j},$ (B.8a)
$\displaystyle[L_{ij},p_{k}]$
$\displaystyle=-\mathrm{i}\delta_{jk}p_{i}+\mathrm{i}\delta_{ik}p_{j},$ (B.8b)
$\displaystyle\Big{[}L_{ij},L_{kl}\Big{]}$
$\displaystyle=-\mathrm{i}\delta_{jk}L_{il}-\mathrm{i}\delta_{il}L_{jk}+\mathrm{i}\delta_{ik}L_{jl}+\mathrm{i}\delta_{jl}L_{ik}.$
(B.8c)

The angular momentum squared $L^{2}$ is the quadratic Casimir operator of
$\mathfrak{so}(d)$.

The representation (B.7a) is decomposed into subspaces of spherical harmonics
of the order $\ell$. The representation of $\mathfrak{so}(d)$ of this type
will be called spherical of degree $\ell$. On this representation we have

$\displaystyle L^{2}=$
$\displaystyle\ell(\ell+d-2)=\Big{(}\ell+\frac{d-2}{2}\Big{)}^{2}-\Bigl{(}\frac{d-2}{2}\Bigr{)}^{2}.$
(B.9)

The Laplacian on $\mathbb{R}^{d}$ in the spherical coordinates is

$\displaystyle-\Delta$
$\displaystyle=-\partial_{r}^{2}-\frac{(d-1)}{r}\partial_{r}+\frac{L_{d}^{2}}{r^{2}},$
(B.10a)
$\displaystyle=R^{2}+\Big{(}-\frac{1}{4}+\Big{(}\ell+\frac{d-2}{2}\Big{)}^{2}\Big{)}\frac{1}{r^{2}}.$
(B.10b)

Sandwiching it with $U$ we obtain

$U(-\Delta)U^{-1}=-\partial_{r}^{2}+\Big{(}-\frac{1}{4}+\Big{(}\ell+\frac{d-2}{2}\Big{)}^{2}\Big{)}\frac{1}{r^{2}}.$
(B.11)

######  Remark 55.

Discussion above is valid even for $d=1$, with $\mathbb{S}^{0}:=\\{\pm 1\\}$.
This case is peculiar in that the only allowed values of $\ell$ are $0$ and
$1$, corresponding to even and odd functions. $d=2$ is also special: $\ell$
takes arbitrary integer values, while for $d\geq 3$ one has $\ell\geq 0$.

###  B.2 Dirac operator in $d$ dimensions

Let $\alpha_{i}$, $i=1,\ldots,d$ and $\beta$ be the Clifford matrices acting
irreducibly in a finite dimensional space $\mathcal{K}$. They satisfy the
Clifford relations

$[\alpha_{i},\alpha_{j}]_{+}=2\delta_{ij},\quad[\alpha_{i},\beta]_{+}=0,\quad\beta^{2}=1.$
(B.12)

We recall that $\dim(\mathcal{K})=2^{\lfloor\frac{d+1}{2}\rfloor}$ and that
for even $d$ one has
$\beta=\pm\mathrm{i}^{\frac{d}{2}}\prod\limits_{i=1}^{d}\alpha_{i}$. The two
sign choices give non-isomorphic representations of the Clifford algebra. By
averaging arguments, $\mathcal{K}$ admits a positive definite hermitian form
such that $\beta$ and $\alpha_{i}$ are unitary and hence hermitian. This form
is unique up to positive scalars; we fix one once and for all.

Using the Einstein summation convention unless there is a summation sign, we
introduce the following operators on
$L^{2}(\mathbb{R}^{d})\otimes\mathcal{K}$:

$\displaystyle D$ $\displaystyle:=\alpha_{i}p_{i},$ (B.13a) $\displaystyle T$
$\displaystyle:=-\mathrm{i}\sum_{i<j}\alpha_{i}\alpha_{j}L_{ij}+\frac{d-1}{2},$
(B.13b) $\displaystyle S$ $\displaystyle:=\frac{\alpha_{i}x_{i}}{|x|},$
(B.13c) $\displaystyle\kappa$ $\displaystyle:=\beta T=T\beta.$ (B.13d)

######  Proposition 56.

We have

$\displaystyle SD$ $\displaystyle=R+\frac{\mathrm{i}}{|x|}T,$ $\displaystyle
DS$ $\displaystyle=R-\frac{\mathrm{i}}{|x|}T,$ (B.14a) $\displaystyle SR$
$\displaystyle=RS,$ $\displaystyle ST$ $\displaystyle=-TS,$ (B.14b)
$\displaystyle S\beta$ $\displaystyle=-\beta S$ $\displaystyle\beta D$
$\displaystyle=-D\beta,$ (B.14c) $\displaystyle DT$ $\displaystyle=-TD,$
$\displaystyle D\kappa$ $\displaystyle=\kappa D.$ (B.14d)

###### Proof.

Let us prove the first identity of (B.14d). We have

$\displaystyle[\alpha_{i}\alpha_{j}L_{ij},\alpha_{k}p_{k}]_{+}$ (B.15)
$\displaystyle=$
$\displaystyle[\alpha_{i}\alpha_{j},\alpha_{k}]L_{ij}p_{k}+\alpha_{k}\alpha_{i}\alpha_{j}[L_{ij},p_{k}]_{+}.$

Using

$\displaystyle\alpha_{j}\alpha_{k}\alpha_{i}$
$\displaystyle=\alpha_{k}\alpha_{i}\alpha_{j}+2\delta_{jk}\alpha_{i}-2\delta_{ij}\alpha_{k},$
(B.16a) $\displaystyle\alpha_{i}\alpha_{j}\alpha_{k}$
$\displaystyle=\alpha_{k}\alpha_{i}\alpha_{j}+2\delta_{jk}\alpha_{i}-2\delta_{ik}\alpha_{j},$
(B.16b)

we obtain

$\displaystyle 3\alpha_{k}\alpha_{i}\alpha_{j}[L_{ij},p_{k}]_{+}$
$\displaystyle=$
$\displaystyle\alpha_{k}\alpha_{i}\alpha_{j}[L_{ij},p_{k}]_{+}+\alpha_{j}\alpha_{k}\alpha_{i}[L_{ki},p_{j}]_{+}+\alpha_{i}\alpha_{j}\alpha_{k}[L_{jk},p_{i}]_{+}$
$\displaystyle=$
$\displaystyle\alpha_{k}\alpha_{i}\alpha_{j}\big{(}[L_{ij},p_{k}]_{+}+[L_{ki},p_{j}]_{+}+[L_{jk},p_{i}]_{+}\big{)}$
$\displaystyle+$ $\displaystyle
2\alpha_{i}[L_{ji},p_{j}]_{+}-2\alpha_{i}[L_{ij},p_{j}]_{+}-2\alpha_{i}[L_{ij},p_{i}]_{+}$
$\displaystyle=$ $\displaystyle
6\alpha_{i}[L_{ji},p_{j}]_{+}=-6\mathrm{i}\alpha_{j}p_{j}(d-1)-12\alpha_{i}L_{ij}p_{j}.$
(B.17)

Moreover,

$\displaystyle[\alpha_{i}\alpha_{j},\alpha_{k}]L_{ij}p_{k}$
$\displaystyle=2(\alpha_{i}\delta_{jk}-\delta_{ik}\alpha_{j})L_{ij}p_{k}$
$\displaystyle=4\alpha_{i}L_{ij}p_{j}.$ (B.18)

Now the sum of $\frac{\mathrm{i}}{6}$(B.17) and $\frac{\mathrm{i}}{2}$(B.18)
is $(d-1)\alpha_{i}p_{i}$. ∎

###  B.3 Decomposition into incoming and outgoing Dirac waves

Let

$\Pi_{\pm}:=\frac{1}{2}(1\pm S)$

be the spectral projections of $S$ onto $\pm 1$. Define

$\mathcal{H}_{\pm}:=\Pi_{\pm}(L^{2}(\mathbb{R}^{d})\otimes\mathcal{K}),\
\text{so that }\
L^{2}(\mathbb{R}^{d})\otimes\mathcal{K}=\mathcal{H}_{+}\oplus\mathcal{H}_{-}.$
(B.19)

For an operator $B$ on $L^{2}(\mathbb{R}^{d})\otimes\mathcal{K}$ let us write

$B_{\pm\pm}=\Pi_{\pm}B\Pi_{\pm},\qquad B_{\pm\mp}=\Pi_{\pm}B\Pi_{\mp}.$ (B.20)

Clearly,

$\displaystyle S=\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix},$
$\displaystyle\qquad R=\begin{bmatrix}R_{++}&0\\\ 0&R_{--}\end{bmatrix},$
(B.21a) $\displaystyle\beta=\begin{bmatrix}0&\beta_{+-}\\\
\beta_{-+}&0\end{bmatrix},$ $\displaystyle\qquad T=\begin{bmatrix}0&T_{+-}\\\
T_{-+}&0\end{bmatrix},$ (B.21b) $\displaystyle
D=\begin{bmatrix}R_{++}&\frac{\mathrm{i}}{|x|}T_{+-}\\\
-\frac{\mathrm{i}}{|x|}T_{-+}&-R_{--}\end{bmatrix},$
$\displaystyle\qquad\kappa=\begin{bmatrix}\beta_{+-}T_{-+}&0\\\
0&\beta_{-+}T_{+-}\end{bmatrix}=\begin{bmatrix}T_{+-}\beta_{-+}&0\\\
0&T_{-+}\beta_{+-}\end{bmatrix}.$ (B.21c)

$D$ commutes with the self-adjoint operator $\kappa$. We can therefore reduce
ourselves to the eigenspace of $\kappa$ with eigenvalue $\omega\in\mathbb{R}$,
denoted $\mathcal{H}_{\omega}$ (see subsection B.7 for a description of these
eigenspaces). We can write

$D=\begin{bmatrix}R_{++}&\frac{\mathrm{i}\omega}{|x|}\beta_{+-}\\\
-\frac{\mathrm{i}\omega}{|x|}\beta_{-+}&-R_{--}\end{bmatrix}.$ (B.22)

Using spherical coordinates, we can identify
$L^{2}(\mathbb{R}^{d})\otimes\mathcal{K}$ with
$L^{2}(\mathbb{R}_{+},r^{d-1})\otimes
L^{2}(\mathbb{S}^{d-1})\otimes\mathcal{K}$. Applying (B.3) and treating
$\beta_{\pm\mp}$ as identifications, we can rewrite the above equation as

$D=\begin{bmatrix}-\mathrm{i}\partial_{r}&\frac{\mathrm{i}\omega}{r}\\\
-\frac{\mathrm{i}\omega}{r}&\mathrm{i}\partial_{r}\end{bmatrix}.$ (B.23)

The $d$-dimensional Dirac Hamiltonian can be reduced to 1 dimension (with
$2\times 2$ matrix structure) if it is perturbed by four kinds of radial
terms: the electric potential $V(r)$, the mass $m(r)$ (called also the Lorentz
scalar), the radial vector potential $A(r)$ and the anomalous (Pauli) coupling
to the electric field $E(r)$. The reduction (B.23) leads to

$\displaystyle D_{V,m,A,E}\,:=\,$ $\displaystyle
D+V(r)+m(r)\beta+\frac{A(r)\alpha_{i}x_{i}}{r}+\frac{\mathrm{i}E(r)\beta\alpha_{i}x_{i}}{r}$
$\displaystyle=$
$\displaystyle\begin{bmatrix}-\mathrm{i}\partial_{r}+V(r)+A(r)&\frac{\mathrm{i}\omega}{r}+m(r)-\mathrm{i}E(r)\\\
-\frac{\mathrm{i}\omega}{r}+m(r)+\mathrm{i}E(r)&\mathrm{i}\partial_{r}+V(r)-A(r)\end{bmatrix}.$
(B.24)

We prefer another form, related by a similarity transformation:

$\displaystyle\frac{1}{\sqrt{2}}\begin{bmatrix}1&-\mathrm{i}\\\
\mathrm{i}&-1\end{bmatrix}D_{V,m,A,E}\frac{1}{\sqrt{2}}\begin{bmatrix}1&-\mathrm{i}\\\
\mathrm{i}&-1\end{bmatrix}=\begin{bmatrix}-\frac{\omega}{r}+E(r)+V(r)&-\partial_{r}-\mathrm{i}A(r)-m(r)\\\
\partial_{r}+\mathrm{i}A(r)-m(r)&\frac{\omega}{r}-E(r)+V(r)\end{bmatrix}.$
(B.25)

For $m=A=E=0$ and $V=-\frac{\lambda}{r}$, this is the 1-dimensional Dirac
operator studied in our paper.

We remark that the radial electromagnetic potential $A(r)$ is necessarily pure
gauge. Indeed, it enters the Dirac operator only in the combination
$\partial_{r}+\mathrm{i}A(r)$, which may be written as
$\mathrm{e}^{-\mathrm{i}\phi(r)}\partial_{r}\mathrm{e}^{i\phi(r)}$ for a
function $\phi(r)$ such that $\phi^{\prime}(r)=A(r)$. Coupling $E(r)$ arises
if the Dirac Lagrangian is extended by the Pauli term, proportional to
$\overline{\psi}\frac{\mathrm{i}}{2}\gamma^{\mu}\gamma^{\nu}F_{\mu\nu}\psi$
with a purely electric and radial field strength tensor $F$.

###  B.4 Composite angular momentum

Introduce the spin operators

$\displaystyle\sigma_{ij}$
$\displaystyle:=-\frac{\mathrm{i}}{2}[\alpha_{i},\alpha_{j}].$ (B.26)

$\frac{1}{2}\sigma_{ij}$ yield a representation of $\mathfrak{so}(d)$ on the
spin space $\mathcal{K}$:

$\displaystyle\Big{[}\frac{1}{2}\sigma_{ij},\alpha_{k}\Big{]}$
$\displaystyle=-\mathrm{i}\delta_{jk}\alpha_{i}+\mathrm{i}\delta_{ik}\alpha_{j},$
(B.27a)
$\displaystyle\Big{[}\frac{1}{2}\sigma_{ij},\frac{1}{2}\sigma_{kl}\Big{]}$
$\displaystyle=-\mathrm{i}\delta_{jk}\frac{1}{2}\sigma_{il}-\mathrm{i}\delta_{il}\frac{1}{2}\sigma_{jk}+\mathrm{i}\delta_{ik}\frac{1}{2}\sigma_{jl}+\mathrm{i}\delta_{jl}\frac{1}{2}\sigma_{ik}.$
(B.27b)

Irreducible representations of $\mathfrak{so}(d)$ contained in $\mathcal{K}$
will be called spinor representations. Their quadratic Casimir is given by

$\displaystyle\frac{\sigma^{2}}{4}=\frac{1}{4}\sum_{i<j}\sigma_{ij}^{2}=$
$\displaystyle\frac{d(d-1)}{8}.$ (B.28)

If $d$ is even, then there are two inequivalent spinor representations. They
correspond to the eigenspaces of $\beta$ with eigenvalues $\pm 1$.

If $d$ is odd, then $\mathcal{K}$ is also a direct sum of two spinor
representations, however they are equivalent to one another. The decomposition
of $\mathcal{K}$ into irreducible components exists but is clearly non-unique.
One possible choice corresponds to the eigenvalues $\pm 1$ of $\beta$.

We also have the composite representation of $\mathfrak{so}(d)$ given by

$\displaystyle J_{ij}$ $\displaystyle:=L_{ij}+\frac{1}{2}\sigma_{ij}.$ (B.29)

Clearly,

$\displaystyle[J_{ij},J_{kl}]$
$\displaystyle=-\mathrm{i}\delta_{jk}J_{il}-\mathrm{i}\delta_{il}J_{jk}+\mathrm{i}\delta_{ik}J_{jl}+\mathrm{i}\delta_{jl}J_{ik},$
(B.30a) $\displaystyle[J,x\cdot\alpha]$
$\displaystyle=[J,p\cdot\alpha]=[J,p^{2}]=[J,x^{2}]=0.$ (B.30b)

The quadratic Casimir of this representation, also called the square of the
total angular momentum, is

$\displaystyle J^{2}=$
$\displaystyle\sum_{i<j}J_{ij}^{2}=L^{2}+L\sigma+\frac{\sigma^{2}}{4},$
(B.31a) $\displaystyle\text{where }\qquad L\sigma$
$\displaystyle:=\sum_{i<j}L_{ij}\sigma_{ij}.$ (B.31b)

######  Proposition 57.

We have the following relation:

$\kappa^{2}=J^{2}+\frac{(d-1)(d-2)}{8}.$ (B.32)

###### Proof.

Directly from the definition we have

$\kappa^{2}=\frac{(d-1)^{2}}{4}+(d-1)L\sigma+\sum_{\begin{subarray}{c}i<j\\\
k<l\end{subarray}}L_{ij}L_{kl}\sigma_{ij}\sigma_{kl}.$ (B.33)

To simplify the last term we write

$\sigma_{ij}\sigma_{kl}=\frac{1}{2}[\sigma_{ij},\sigma_{kl}]+\frac{1}{2}[\sigma_{ij},\sigma_{kl}]_{+}.$
(B.34)

A simple expression for the first term is given by (B.27b). The second one is

$\frac{1}{2}[\sigma_{ij},\sigma_{kl}]_{+}=-\alpha_{[i}\alpha_{j}\alpha_{k}\alpha_{l]}+2\delta^{[i}_{[k}\delta^{j]}_{l]}$
(B.35)

in which $[\cdots]$ denotes skew-symmetrization of the enclosed indices. In
order to prove this formula, first note that both sides are skew-symmetric
with respect to the transposition of $i$ and $j$ or $k$ and $l$, so we may
assume that $i\neq j$ and $k\neq l$. We have three cases. If sets
$A=\\{i,j\\}$ and $B=\\{k,l\\}$ are disjoint, then all $\alpha$ matrices
involved anticommute and hence both sides are equal to
$-\alpha_{i}\alpha_{j}\alpha_{k}\alpha_{l}$. If $A\cap B$ has one element, one
checks that both sides vanish. Finally, if $A=B$ then both sides are equal to
$\pm 1$, the sign depending on the order of indices.

Now plug (B.34) and (B.34) into (B.33). The term with
$\alpha_{[i}\alpha_{j}\alpha_{k}\alpha_{l]}$ drops out after summing over
indices because $L_{[ij}L_{kl]}=0$. In the term with (B.27b) we can replace
$L_{ij}L_{kl}$ by $\frac{1}{2}[L_{ij},L_{kl}]$, by skew-symmetry with respect
to $ij\leftrightarrow kl$. After simplifications with (B.28) and (B.31a) we
obtain the claim. ∎

Recall that on $\mathcal{H}_{\omega}$ the operator $\kappa$ acts as
multiplication by $\omega$. We will now characterize $\mathcal{H}_{\omega}$
more closely.

######  Proposition 58.

Let $\omega$ be such that $\mathcal{H}_{\omega}\neq\\{0\\}$. Then there exist
$\ell$ and subspaces $\mathcal{W}_{\ell},\mathcal{W}_{\ell-1}\subset
L^{2}(\mathbb{R}^{d})$ spherical of degree $\ell$ resp. $\ell-1$ such that

$\displaystyle\mathcal{H}_{\omega}$
$\displaystyle\subset(\mathcal{W}_{\ell}{\oplus}\mathcal{W}_{\ell-1})\otimes\mathcal{K},$
(B.36a) $\displaystyle|\omega|$ $\displaystyle=\ell+\frac{d-3}{2},$ (B.36b)
$\displaystyle\left.J^{2}\right|_{\mathcal{H}_{\omega}}$
$\displaystyle=\ell^{2}+\ell(d-3)+\frac{d^{2}-9d+16}{8}.$ (B.36c)

###### Proof.

Exceptional cases $d=1,2$ are easy to analyze separately: one has $\kappa=0$
in the former case and $\kappa=\pm L_{12}+\frac{1}{2}\beta$ (with the sign
depending on the choice of Clifford matrices) in the latter. From now on we
assume that $d\geq 3$. We note that (B.32) and $J^{2}\geq 0$ imply that
$\omega\neq 0$.

$\kappa$ commutes with $\beta$, hence also with $L\sigma$. Therefore we can
decompose $\mathcal{H}_{\omega}$ with respect to the eigenvalues of $L\sigma$.
From (B.13b) we obtain

$\displaystyle L\sigma=$ $\displaystyle\omega\beta-\frac{d-1}{2},$ (B.37)

which has on $\mathcal{H}_{\omega}$ two distinct eigenvalues

$\pm\omega-\frac{d-1}{2}.$ (B.38)

Both sings are realized because $D$ anticommutes with $\beta$ and preserves
$\mathcal{H}_{\omega}$.

Clearly $L^{2}=J^{2}-L\sigma-\frac{\sigma^{2}}{4}$ has on
$\mathcal{H}_{\omega}$ two distinct eigenvalues corresponding to (B.38). As
seen from (B.9), the representation of orbital angular momentum is uniquely
determined by $L^{2}$. Therefore, for some $\ell_{+},\ell_{-}\in\mathbb{N}$,
$\ell_{+}>\ell_{-}$,

$\mathcal{H}_{\omega}\subset(\mathcal{W}_{\ell_{+}}{\oplus}\mathcal{W}_{\ell_{-}})\otimes\mathcal{K}.$
(B.39)

Comparing the identities

$\displaystyle J^{2}$
$\displaystyle=\ell_{\pm}(\ell_{\pm}+d-2)\mp|\omega|-\frac{d-1}{2}+\frac{d(d-1)}{8},$
(B.40) $\displaystyle J^{2}$ $\displaystyle=\omega^{2}-\frac{(d-1)(d-2)}{8},$
(B.41)

we obtain the equation

$|\omega|(|\omega|\pm
1)=\Big{(}\ell_{\pm}+\frac{d-1}{2}\Big{)}\Big{(}\ell_{\pm}+\frac{d-3}{2}\Big{)}.$
(B.42)

whose solutions take the form
$\ell_{+}+\frac{d-3}{2}\in\\{|\omega|,-|\omega|-1\\}$,
$\ell_{-}+\frac{d-3}{2}\in\\{|\omega|-1,-|\omega|\\}$. In both cases the
second solution has to be discarded because $\ell_{\pm}+\frac{d-3}{2}\geq 0$.
Hence (B.36b) holds and $\ell_{-}=\ell_{+}-1$. Then (B.36c) is obtain by
feeding (B.36b) into (B.32). ∎

We remark that the sign of $\omega$ cannot be obtained from the above
calculation. Indeed, the spectrum of $\kappa$ on
$L^{2}(\mathbb{R}^{d})\otimes\mathcal{K}$ is always invariant with respect to
$\omega\mapsto-\omega$. If $d$ is odd, then $\prod\limits_{j=1}^{d}\alpha_{j}$
commutes with $L$ and $\alpha_{i}$, but anticommutes with $\beta$ and hence
with $\kappa$. If $d$ is even, then $\kappa$ anticommutes with the parity
operator

$f(x_{1},\dots,x_{d})\mapsto\beta\alpha_{1}f(-x_{1},\dots,x_{d}),\qquad f\in
L^{2}(\mathbb{R}^{d},\mathcal{K}).$ (B.43)

However, this operation does not preserve the type of angular momentum
representation. Indeed, it anticommutes with $\beta$ and hence exchanges the
two spinor representations.

###  B.5 Analysis in various dimensions

Let us review the lowest dimensions.

$\bf d=1$. There is no angular momentum and one has $\omega=0$.

$\bf d=2$. Unitary irreducible representations of $\mathfrak{so}(2)$ are
enumerated by spin values $m\in\mathbb{R}$. The corresponding quadratic
Casimir is equal to $m^{2}$. There are two types
$\mathcal{K}_{\pm\frac{1}{2}}$ of spinor representations, corresponding to
$m=\pm\frac{1}{2}$. Spherical representations correspond to
$\ell\in\mathbb{Z}$.

One convenient choice of Clifford representation is given by Pauli matrices:
$\alpha_{1}=\sigma_{1}$, $\alpha_{2}=\sigma_{2}$, $\beta=\pm\sigma_{3}$. Then
$\kappa=\pm J$ and hence

$\displaystyle\mathcal{H}_{\omega}=(\mathcal{W}_{\ell}{\otimes}\mathcal{K}_{-\frac{1}{2}})\oplus(\mathcal{W}_{\ell-1}{\otimes}\mathcal{K}_{\frac{1}{2}}).$
(B.44)

with $\omega=\pm(\ell-\frac{1}{2})\in\mathbb{Z}+\frac{1}{2}$. Sign in the
relation between $\omega$ and total angular momentum depends on the choice of
sign in $\beta$, but after fixing Clifford matrices it is one-to-one.

$\bf d=3$. Unitary irreducible representations of $\mathfrak{so}(3)$ are
parametrized by spin $j\in\frac{1}{2}\mathbb{N}$ or the quadratic Casimir
$j(j+1)$. All spinor representations have the spin $\frac{1}{2}$. The
representation on $\mathcal{H}_{\omega}$ has spin $\ell-\frac{1}{2}$. We have
$\omega=\pm\ell\in\\{\pm 1,\pm 2,\dots\\}$, i.e. two distinct values of
$\omega$ correspond to the same total spin.

$\bf d=4$. We have
$\mathfrak{so}(4)\simeq\mathfrak{so}(3)\oplus\mathfrak{so}(3)$. More
explicitly,

$\displaystyle J_{1}^{\pm}:=\frac{1}{2}(\pm J_{12}+J_{34}),$
$\displaystyle\quad J_{2}^{\pm}:=\frac{1}{2}(\pm J_{13}+J_{42}),\quad
J_{3}^{\pm}:=\frac{1}{2}(\pm J_{14}+J_{23})$ (B.45)

span two algebras isomorphic to $\mathfrak{so}(3)$ and commuting with one
another. Let $(J^{\pm})^{2}$ be the corresponding quadratic Casimirs. We have

$J^{2}=2(J^{+})^{2}+2(J^{-})^{2}.$ (B.46)

Thus irreducible representations of $\mathfrak{so}(4)$ are parametrized by
pairs of spins $(j^{+},j^{-})\in(\frac{1}{2}\mathbb{N})^{2}$ with the
quadratic Casimir $2j^{+}(j^{+}+1)+2j^{-}(j^{-}+1)$. We have also the obvious
analogs of (B.45) and (B.46) for $L_{ij}$ and $\frac{1}{2}\sigma_{ij}$.

Representations of $\mathfrak{so}(4)$ on spherical harmonics satisfy

$L_{12}L_{34}+L_{13}L_{42}+L_{14}L_{23}=0.$ (B.47)

Therefore, $(L^{+})^{2}=(L^{-})^{2}$. Hence a spherical representation of
degree $\ell$ corresponds to the pair of spins
$(\frac{\ell}{2},\frac{\ell}{2})$ with the quadratic Casimir
$\ell(\ell+2)=2\frac{\ell}{2}\Big{(}\frac{\ell}{2}+1\Big{)}+2\frac{\ell}{2}\Big{(}\frac{\ell}{2}+1\Big{)}$.
Spinor representations of $\mathfrak{so}(4)$ are of types $(\frac{1}{2},0)$
and $(0,\frac{1}{2})$, distinguished by the eigenvalue of
$\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}$. They satisfy

$\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}\,\sigma_{i}^{\pm}=\mp\sigma_{i}^{\pm},\qquad(\sigma^{\pm})^{2}=\mp\frac{3}{2}\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}.$
(B.48)

Furthermore, we have $\beta=\pm\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}$, with
the sign in this relation distinguishing Clifford representation. Using these
relations we derive

$\kappa=\mp 2(J^{+})^{2}\pm 2(J^{-})^{2}.$ (B.49)

From spherical representations and spinor representation it is possible to
build total angular momentum representations of two types:
$(\frac{\ell}{2},\frac{\ell-1}{2})$ and $(\frac{\ell-1}{2},\frac{\ell}{2})$.
They have the same quadratic Casimir

$J^{2}=\ell(\ell+1)-\frac{1}{2}$ (B.50)

but can be distinguished by $\omega$:

$\omega=\mp\left(\ell+\frac{1}{2}\right)\qquad\text{and}\qquad\omega=\pm\left(\ell+\frac{1}{2}\right).$
(B.51)

The inclusion (B.36a) may now be stated more precisely:

$\displaystyle\mathcal{H}_{\omega}\cong\left(\frac{\ell}{2},\frac{\ell-1}{2}\right)\subset$
$\displaystyle\Big{(}\frac{\ell-1}{2},\frac{\ell-1}{2}\Big{)}{\otimes}\Big{(}\frac{1}{2},0\Big{)}\oplus\Big{(}\frac{\ell}{2},\frac{\ell}{2}\Big{)}{\otimes}\Big{(}0,\frac{1}{2}\Big{)},\qquad\text{for
}\pm\omega<0$ (B.52a)
$\displaystyle\mathcal{H}_{\omega}\cong\left(\frac{\ell-1}{2},\frac{\ell}{2}\right)\subset$
$\displaystyle\Big{(}\frac{\ell-1}{2},\frac{\ell-1}{2}\Big{)}{\otimes}\Big{(}0,\frac{1}{2}\Big{)}\oplus\Big{(}\frac{\ell}{2},\frac{\ell}{2}\Big{)}{\otimes}\Big{(}\frac{1}{2},0\Big{)},\qquad\text{for
}\pm\omega>0.$ (B.52b)

As in dimension $2$, the relation between the total angular momentum
representation and $\omega$, taking valued in
$\\{\pm\frac{3}{2},\pm\frac{5}{2},\dots\\}$, is one-to-one after fixing
Clifford matrices.

For general dimensions we label irreducible representations as in [23, Section
19].

$\bf d=2n+1,\,n\geq 2$. Irreducible representations are in $1-1$
correspondence with labels $(a_{1},\dots,a_{n})\in\mathbb{N}^{n}$. Spherical
harmonics of degree $\ell$ have type $(\ell,0,\dots)$, while spinor
representations have type $(0,\dots,1)$. Their tensor product decomposes as

$\displaystyle(\ell,\dots,0)\otimes(0,\dots,1)$
$\displaystyle=(\ell,\dots,1)\oplus(\ell-1,\dots,1),\qquad\ell\geq 1.$ (B.53)

Thus the only possible types of $\mathcal{H}_{\omega}$ are $(\ell-1,\dots,1)$.
This representation occurs as a subrepresentation only in two tensor products:

$(\ell,\dots,0){\otimes}(0,\dots,1),\qquad(\ell-1,\dots,0){\otimes}(0,\dots,1).$
(B.54)

We have $|\omega|=\ell+n-1$, thus $\omega$ takes values $\\{\pm
n,\pm(n+1),\dots\\}$, with opposite $\omega$ corresponding to the same total
angular momentum. In particular it is not possible to express $\kappa$ as a
polynomial in $J_{ij}$.

$\bf d=2n,\,n\geq 3$. Types of irreducible representations are parametrized by
$(a_{1},\dots,a_{n})\in\mathbb{N}^{n}$. $\ell$th degree spherical harmonics
are of type $(\ell,0,\dots)$. Spinor representations are of two types:
$(0,\dots,1,0)$. and $(0,\dots,0,1)$. We have tensor products decompositions
($\ell\geq 1$):

$\displaystyle(\ell,\dots,0)\otimes(0,\dots,1,0)$
$\displaystyle=(\ell,\dots,1,0)\oplus(\ell-1,\dots,0,1),$ (B.55a)
$\displaystyle(\ell,\dots,0)\otimes(0,\dots,0,1)$
$\displaystyle=(\ell,\dots,0,1)\oplus(\ell-1,\dots,1,0).$ (B.55b)

It follows that $\mathcal{H}_{\omega}$ must be of the type
$(\ell-1,\dots,0,1)$ or $(\ell-1,\dots,1,0)$. These two representations have
the same quadratic Casimir, however they are exchanged by the parity operator
(B.43). Hence they can be distinguished by the sign of the following Casimir
element, defined as the $n$th wedge power of the 2-form $J$:

$\bigwedge_{j=1}^{n}J:=\frac{1}{2^{n}}\epsilon^{i_{1}\dots
i_{2n}}J_{i_{1}i_{2}}\cdots J_{i_{2n-1}i_{2n}}.$ (B.56)

Here $\epsilon$ is the Levi-Civita symbol.

We will show that (B.56) is actually proportional to $\kappa$. Using the fact
that skew-symmetrization of the product of two or more $L_{ij}$ vanishes and
Clifford relations we derive

$\bigwedge_{j=1}^{n}J=\frac{n(2n-2)!}{2^{2n-2}}\left((-\mathrm{i})^{n}\alpha_{1}\cdots\alpha_{2n}\right)\left(L\sigma+\frac{2n-1}{2}\right).$
(B.57)

A Clifford representation is determined up to isomorphism by specifying the
sign in the relation $\beta=\pm(-\mathrm{i})^{n}\alpha_{1}\cdots\alpha_{2n}$.
Then we have

$\bigwedge_{j=1}^{n}J=\pm\frac{n(2n-2)!}{2^{2n-2}}\kappa.$ (B.58)

As in lower even dimensions, for fixed Clifford matrices angular momentum
types are in one-to-one correspondence with the values
$\omega\in\\{\pm\left(n-\frac{1}{2}\right),\pm\left(n+\frac{1}{2}\right),\dots\\}$.

###  B.6 Dirac operators on manifolds

The operator $\kappa$, which is central to the separation of variables of the
radially symmetric Dirac equation, is closely related to the Dirac equation on
the sphere. We would like to give a short discussion of this topic.

Before we discuss the case of a sphere, in this subsection we give a short
introduction to Dirac operators on Riemannian manifolds. We take Clifford
module bundles as central objects. A popular alternative is based on the
concept of a spin structure. Spinor bundles are then constructed by the
associated bundle construction, see [35, p. 7–44, 77–135] for an exposition. A
comparison between the two approaches is presented in [44].

Given a Euclidean vector space $E$ with the scalar product of $u,v\in E$
denoted $u\cdot v$, we let $\mathrm{Cl}(E)$ be the corresponding Clifford
algebra, that is the quotient of the tensor algebra of $E$ by the ideal
generated by elements of the form $u\otimes u-u\cdot u$. Then $\mathbb{R}$ and
$E$ are naturally embedded in $\mathrm{Cl}(E)$ (in concrete matrix
realizations of $\mathrm{Cl}(E)$ the latter embedding is realized by
contraction of vectors with $\alpha$ matrices such as (B.12)). In this
subsection we identify elements of $E$ with their images in $\mathrm{Cl}(E)$.

The automorphism $\alpha$ of $\mathrm{Cl}(E)$ characterized by the equation
$\alpha(u)=-u$ for $u\in E$ is called the main automorphism or the parity.
Elements of $\mathrm{Cl}(E)$ fixed (negated) by $\alpha$ are said to be even
(odd). The transposition is the anti-automorphism of $\mathrm{Cl}(E)$
characterized by $(u_{1}\dots u_{n})^{\mathrm{T}}=u_{n}\dots u_{1}$ for
$u_{1},\dots,u_{n}\in E$.

The spin group $\mathrm{Spin}(E)$ is the group of even invertible elements
$g\in\mathrm{Cl}(E)$ such that

$gug^{-1}\in E\text{ for every }u\in E,\qquad g^{\mathrm{T}}g=1.$ (B.59)

If $g\in\mathrm{Spin}(E)$, then the endomorphism $u\mapsto gug^{-1}$ of $E$
belongs to the special orthogonal group $\mathrm{SO}(E)$. Thus we have a
homomorphism $\mathrm{Spin}(E)\to\mathrm{SO}(E)$. This homomorphism is
surjective with kernel $\\{\pm 1\\}$. Since this is a central subgroup of
$\mathrm{Cl}(E)$, the adjoint action of $\mathrm{Spin}(E)$ on $\mathrm{Cl}(E)$
descends to an action of $\mathrm{SO}(E)$ on $\mathrm{Cl}(E)$. The Lie algebra
$\mathfrak{spin}(E)$ of $\mathrm{Spin}(E)$ is the subspace of $\mathrm{Cl}(E)$
spanned by elements of the form $[u_{1},u_{2}]$ with $u_{1},u_{2}\in E$. We
have an isomorphism $\mathfrak{spin}(E)\cong\mathfrak{so}(E)$, which takes
$[u_{1},u_{2}]$ to the endomorphism

$u_{3}\mapsto[[u_{1},u_{2}],u_{3}]=4u_{1}(u_{2}\cdot u_{3})-4u_{2}(u_{1}\cdot
u_{3}).$ (B.60)

Therefore, $A\in\mathfrak{so}(E)$ is mapped to

$\frac{1}{8}\sum_{ij}[e_{i},e_{j}](e_{i}\cdot Ae_{j}),$ (B.61)

where $e_{i}$ form an orthonormal basis of $E$.

Every $\mathrm{Cl}(E)$-module $\mathcal{V}$ is a direct sum of irreducible
modules. Let $\mathcal{V}$ be an irreducible complex representation. The even
subalgebra of $\mathrm{Cl}(E)$ (and in particular the spin group
$\mathrm{Spin}(E)$) is represented faithfully on $\mathcal{V}$. There exists a
positive-definite hermitian form $(\cdot|\cdot)$ on $M$, called a spinor
scalar product, such that
$(\psi_{1}|c\psi_{2})=(c^{\mathrm{T}}\psi_{1}|\psi_{2})$ for
$c\in\mathrm{Cl}(E)$ and $\psi_{1},\psi_{2}\in\mathcal{V}$. It is unique up to
positive scalars. Furthermore, there exists an antilinear operator $\Theta$ on
$\mathcal{V}$, called a spinor conjugation, such that

$\Theta c\Theta^{-1}=\begin{cases}c&\text{if }n\not\equiv 3\text{ mod }4,\\\
\alpha(c)&\text{if }n\equiv 3\text{ mod
}4,\end{cases}\qquad\Theta^{2}=\begin{cases}1&\text{if
}n\in\\{0,1,2,7\\}\text{ mod }8,\\\ -1&\text{if }n\in\\{3,4,5,6\\}\text{ mod
}8.\end{cases}$ (B.62)

$\Theta$ is unique up to a phase factor.

Now let $M$ be a Riemannian manifold with tangent bundle $TM$ and the Levi-
Civita connection $\nabla$. For every $x\in M$ consider the Clifford algebra
$\mathrm{Cl}(T_{x}M)$. Together these Clifford algebras form a bundle
$\mathrm{Cl}(TM)$ of Clifford algebras over $M$. If $M$ is oriented, we can
locally choose positively oriented orthonormal framings
$\\{e_{i}\\}_{i=1}^{d}$ and put

$\mathrm{vol}_{M}=e_{1}\cdots e_{d}.$ (B.63)

The right hand side does not depend on the choice of framing, hence it defines
a global section of $\mathrm{Cl}(TM)$.

The Levi-Civita connection extends uniquely to a connection on
$\mathrm{Cl}(TM)$ satisfying the Leibniz rule:

$\nabla(c_{1}c_{2})=(\nabla c_{1})c_{2}+c_{1}\nabla c_{2}$ (B.64)

for sections $c_{1},c_{2}$ of $\mathrm{Cl}(TM)$. This connection commutes with
the main automorphism and the transposition. If defined, $\mathrm{vol}_{M}$ is
covariantly constant.

A vector bundle $\Sigma$ whose fiber $\Sigma_{x}$ is a representation of
$\mathrm{Cl}(T_{x}M)$ (with the module structure smoothly varying with $x$) is
called a Clifford module bundle. A connection $\nabla$ on $\Sigma$ will be
called Clifford covariant if it satisfies

$\nabla(c\psi)=(\nabla c)\psi+c\nabla\psi.$ (B.65)

If in addition for every $x\in M$ and every null-homotopic loop $\gamma$ based
at $x$ the holonomy endomorphism
$\mathrm{hol}_{\Sigma,\gamma}\in\mathrm{GL}(\Sigma_{x})$ is an element of
$\mathrm{Spin}(T_{x}M)$, we call $\nabla$ a locally spin connection. If this
is true for all loops, we say that $\nabla$ is a spin connection. A Clifford
module bundle equipped with a spin connection will be called a spinor bundle.

######  Lemma 59.

If $\nabla$ is a spin connection, then the holonomy endomorphism
$\mathrm{hol}_{\Sigma,\gamma}\in\mathrm{Spin}(T_{x}M)$ lifts the holonomy
$\mathrm{hol}_{TM,\gamma}\in\mathrm{SO}(T_{x}M)$ of the Levi-Civita
connection.

###### Proof.

By the Clifford covariance (B.65), for any $c\in\mathrm{Cl}(T_{x}M)$ we have

$\mathrm{hol}_{\mathrm{Cl}(TM),\gamma}(c)\mathrm{hol}_{\Sigma,\gamma}\psi=\mathrm{hol}_{\Sigma,\gamma}(c\psi)=\mathrm{hol}_{\Sigma,\gamma}c\,\mathrm{hol}_{\Sigma,\gamma}^{-1}\mathrm{hol}_{\Sigma,\gamma}\psi.$
(B.66)

Thus

$\mathrm{hol}_{\mathrm{Cl}(TM),\gamma}(c)=\mathrm{hol}_{\Sigma,\gamma}c\,\mathrm{hol}_{\Sigma,\gamma}^{-1}.$
(B.67)

As $c$ we can choose $u\in T_{x}M\subset\mathrm{Cl}(T_{x}M)$ and rewrite
(B.67) as

$\mathrm{hol}_{TM,\gamma}u=\mathrm{hol}_{\Sigma,\gamma}u\,\mathrm{hol}_{\Sigma,\gamma}^{-1}.\qed$

From now on we assume that $M$ is orientable. As a consequence, the holonomies
of the Levi-Civita connection are always contained in $SO(T_{x}M)$. (On non-
orientable manifolds they may be contained in $O(T_{x}M)$)

The following lemma allows us to conveniently check whether a given connection
is spin [44].

######  Lemma 60.

Let $\Sigma$ be a bundle of irreducible Clifford modules with a Clifford
covariant connection $\nabla$. Then $\nabla$ is a spin connection if and only
if there exist a spinor scalar product $(\cdot|\cdot)$ and a spinor
conjugation $\Theta$ on $\Sigma$ such that

$\nabla\Theta\psi=\Theta\nabla\psi,\qquad\mathrm{d}(\psi_{1}|\psi_{2})=(\nabla\psi_{1}|\psi_{2})+(\psi_{1}|\nabla\psi_{2}).$
(B.68)

###### Proof.

$\Rightarrow$. We focus on one connected component $M_{0}$ of $M$ and choose a
point $x$ therein. Then we choose a spinor scalar product $(\cdot|\cdot)$ and
a spinor conjugation $\Theta$ in $\Sigma_{x}$. By assumption, they are
invariant under $\mathrm{hol}_{\Sigma,\gamma}$ for every loop based at $x$.
Now parallel transport $(\cdot|\cdot)$ and $\Theta$ to all other fibers over
$M_{0}$. Invariance under holonomies implies that the result is independent of
the choice of paths, smooth and covariantly constant, hence satisfies (B.68).

$\Leftarrow$. Let $\gamma$ be a loop based at $x$ and let
$c\in\mathrm{Cl}(T_{x}M)$. Let $g\in\mathrm{Spin}(T_{x}M)$ be a lift of
$\mathrm{hol}_{TM,\gamma}\in\mathrm{SO}(T_{x}M)$. Arguing as in the proof of
Lemma 59 we see that
$\mathrm{hol}_{\Sigma,\gamma}c\,\mathrm{hol}_{\Sigma,\gamma}^{-1}=gcg^{-1}$.
By irreducibility of $\Sigma_{x}$, this implies that
$\mathrm{hol}_{\Sigma,\gamma}=zg$ for some $z\in\mathbb{C}$. Since both
$\mathrm{hol}_{\Sigma,\gamma}$ and $g$ preserve the scalar product, $|z|=1$.
Since both commute with $\Theta$, $z\in\mathbb{R}$. Thus
$\mathrm{hol}_{\Sigma,\gamma}$ coincides with $g$ or $-g$ and hence belongs to
$\mathrm{Spin}(T_{x}M)$. ∎

######  Lemma 61.

Every spinor bundle is a direct sum of spinor bundles whose fibers are
irreducible Clifford modules.

###### Proof.

Analogous to the proof of $\Rightarrow$ in Lemma 60. ∎

Recall that for a vector bundle $\Sigma$ with a connection $\nabla$, the
expression

$\Omega(\mathbf{U},\mathbf{V}):=\nabla_{\mathbf{U}}\nabla_{\mathbf{V}}-\nabla_{\mathbf{V}}\nabla_{\mathbf{U}}-\nabla_{[\mathbf{U},\mathbf{V}]_{\mathrm{Lie}}}.$
(B.69)

defines an $\mathrm{End}(\Sigma)$-valued $2$-form, called the curvature of
$\nabla$. Here $[\cdot,\cdot]_{\mathrm{Lie}}$ is the Lie bracket of vector
fields $\mathbf{U},\mathbf{V}$. If $\Sigma=TM$ and $\nabla$ is the Levi-Civita
connection, then $\Omega$ is denoted by $R$ and called the Riemann tensor. One
checks that $\left.R(\mathbf{U},\mathbf{V})\right|_{x}$ is an element of
$\mathfrak{so}(T_{x}M)$.

######  Lemma 62.

If $\nabla$ is a spin connection, then its curvature takes the form

$\Omega(\mathbf{U},\mathbf{V})=\frac{1}{8}\sum_{i,j}(e_{i}\cdot
R(\mathbf{U},\mathbf{V})e_{j})[e_{i},e_{j}].$ (B.70)

A partial converse holds: every Clifford covariant connection with curvature
given by the formula above is a locally spin connection.

###### Proof.

The curvature may be extracted from holonomies along infitesimal
parallelograms. Therefore by Lemma 59, the curvature of $\nabla$ at $x$ is an
element of $\mathfrak{spin}(T_{x}M)$, coinciding with
$R(\mathbf{U},\mathbf{V})$ taken in the representation (B.61).

Now we prove the converse. If $\gamma$ is any path from $y$ to $x$ and
$\mathrm{hol}_{\Sigma,\gamma}\in\mathrm{Hom}(\Sigma_{x},\Sigma_{y})$ is the
corresponding parallel transport, then by the Clifford covariance

$\mathrm{hol}_{\Sigma,\gamma}\Omega(\mathbf{U},\mathbf{V})\mathrm{hol}_{\Sigma,\gamma}^{-1}=\frac{1}{8}\sum_{i,j}(e_{i}\cdot
R(\mathbf{U},\mathbf{V})e_{j})[\mathrm{hol}_{\mathrm{T}M,\gamma}(e_{i}),\mathrm{hol}_{\mathrm{T}M,\gamma}(e_{j})]\in\mathfrak{spin}(T_{x}M).$
(B.71)

Let $\gamma_{s}$ be a family of loops $[0,1]\to M$ based at $x$. For
$t\in[0,1]$ let $\gamma_{s}^{t}:=\left.\gamma_{s}\right|_{[0,t]}$. Then

$\mathrm{hol}_{\Sigma,\gamma_{s}}^{-1}\frac{\mathrm{d}}{\mathrm{d}s}\mathrm{hol}_{\Sigma,\gamma_{s}}=\int_{0}^{1}\mathrm{hol}_{\Sigma,\gamma_{s}^{t}}^{-1}\Omega\left(\frac{\partial\gamma_{s}(t)}{\partial
s},\frac{\partial\gamma_{s}(t)}{\partial
t}\right)\mathrm{hol}_{\Sigma,\gamma_{s}^{t}}\mathrm{d}t.$ (B.72)

It follows that for a null-homotopic loop $\gamma$ based at $x$ we have that
$\mathrm{hol}_{\Sigma,\gamma}\in\mathrm{Spin}(T_{x}M)$. ∎

Next we define the Dirac operator on sections of a spinor bundle $\Sigma$. Let
us choose a locally defined orthonormal framing $\\{e_{i}\\}$ of $TM$. Now put

$D\psi=-\mathrm{i}\sum_{i}e_{i}\cdot\nabla_{e_{i}}\psi.$ (B.73)

Here the multiplication by $e_{i}$ is the Clifford multiplication ($e_{i}$
being regarded as a section of $\mathrm{Cl}(TM)$). It is not dificult to check
that $D\psi$ does not depend on the choice of framing, so local expressions on
the right hand side of (B.73) can be glued to obtain a globally defined
differential operator.

If $\Sigma$ is a spinor bundle over an oriented Riemannian manifold $M$, there
exist two distinguished second order differential operators acting on sections
of $\Sigma$: the square of the Dirac operator $D^{2}$ and the Bochner
Laplacian. To describe the latter, let $(\cdot|\cdot)$ be a spinor scalar
product. It yields a scalar product on $T^{*}M\otimes\Sigma$. Now the Bochner
Laplacian, at least formally, is (minus) the operator associated to the
quadratic form

$-(\psi,\Delta\psi):=\int_{M}\big{(}\nabla\psi(x)|\nabla\psi(x)\big{)}\mathrm{d}x.$
(B.74)

Equivalently, the Bochner Laplacian can be defined without invoking the scalar
product by

$\Delta:=\sum_{i}\left(\nabla_{e_{i}}\nabla_{e_{i}}-\nabla_{\nabla_{e_{i}}e_{i}}\right),$
(B.75)

Note that the Bochner Laplacian uses the covariant Hessian

$\mathrm{Hess}(\mathbf{U},\mathbf{V})=\nabla_{\mathbf{U}}\nabla_{\mathbf{V}}-\nabla_{\nabla_{\mathbf{U}}\mathbf{V}},$
(B.76)

which is bilinear over $C^{\infty}(M)$ and satisfies

$\mathrm{Hess}(\mathbf{U},\mathbf{V})-\mathrm{Hess}(\mathbf{V},\mathbf{U})=\Omega(\mathbf{U},\mathbf{V}).$
(B.77)

(B.77) follows from the torsion-freeness of the Levi-Civita connection, that
is

$\nabla_{\mathbf{U}}\mathbf{V}-\nabla_{\mathbf{V}}\mathbf{U}-[\mathbf{U},\mathbf{V}]_{\mathrm{Lie}}=0.$
(B.78)

In the following proposition we recall the celebrated Lichnerowicz formula:

######  Proposition 63.

The square of the Dirac operator and the Bochner Laplacian are related by

$D^{2}=-\Delta+\frac{1}{4}\mathrm{Sc},$ (B.79)

where $\mathrm{Sc}$ is the scalar curvature.

###### Proof.

Let $\mathrm{Hess}^{\mathrm{s}}$ be the symmetric part of the Hessian. We
choose an orthonormal framing $\\{e_{i}\\}$. Then

$\displaystyle(\mathrm{i}D)^{2}$
$\displaystyle=\sum_{i,j}e_{i}\nabla_{e_{i}}e_{j}\nabla_{e_{j}}=\sum_{i,j}\left(e_{i}e_{j}\nabla_{e_{i}}\nabla_{e_{j}}+e_{i}(\nabla_{e_{i}}e_{j})\nabla_{e_{j}}\right)$
(B.80)
$\displaystyle=\sum_{i,j}e_{i}e_{j}\left(\mathrm{Hess}(e_{i},e_{j})+\nabla_{\nabla_{e_{i}}e_{j}}\right)+\sum_{i,j}e_{i}(\nabla_{e_{i}}e_{j})\nabla_{e_{j}}$
$\displaystyle=\sum_{i,j}e_{i}e_{j}\left(\mathrm{Hess}^{\mathrm{s}}(e_{i},e_{j})+\frac{1}{2}\Omega(e_{i},e_{j})\right)+\sum_{i,j}\left(e_{i}e_{j}\nabla_{\nabla_{e_{i}}e_{j}}+e_{i}(\nabla_{e_{i}}e_{j})\nabla_{e_{j}}\right)$
$\displaystyle=\Delta+\frac{1}{32}\sum_{i,j,n,m}[e_{i},e_{j}][e_{n},e_{m}]\left(e_{n}\cdot
R(e_{i},e_{j})e_{m}\right).$

Below we will show that the last two terms in the third line cancel. The last
term in the fourth line may be shown to be equal to
$-\frac{1}{4}\mathrm{Sc}=-\frac{1}{4}\sum_{ij}e_{i}\cdot R(e_{i},e_{j})e_{j}$
using Clifford relations and symmetries of the Riemann tensor.

Connection coefficients are defined by the formula

$\nabla_{e_{i}}e_{j}=\sum_{k}c_{ijk}e_{k}.$ (B.81)

$\nabla_{e_{i}}(e_{j}\cdot e_{k})=0$ and metric compatibility of the
connection give $c_{ijk}+c_{ikj}=0$. We have

$\sum_{j}\left(e_{j}\nabla_{\nabla_{e_{i}}e_{j}}+(\nabla_{e_{i}}e_{j})\nabla_{e_{j}}\right)=\sum_{j,k}c_{ijk}\left(e_{j}\nabla_{e_{k}}+e_{k}\nabla_{e_{j}}\right).$
(B.82)

Now switch the roles of $j,k$ in the second term to see that (B.82) vanishes.
∎

Now suppose that $N$ is an orientable submanifold of $M$ of codimension $1$.
Then there exists a smooth field of unit normal vectors $\nu$. We have the
following relation between the Levi-Civita connection on $M$ and on $N$:

$\displaystyle\nabla^{N}_{\mathbf{U}}\mathbf{V}$
$\displaystyle=\nabla^{M}_{\mathbf{U}}\mathbf{V}-(\nabla^{M}_{\mathbf{U}}\mathbf{V}\cdot\nu)\nu=\nabla^{M}_{\mathbf{U}}\mathbf{V}+(\mathbf{V}\cdot\nabla^{M}_{\mathbf{U}}\nu)\nu,$
(B.83)

where $\mathbf{U},\mathbf{V}$ are tangent to $N$. That is,
$\nabla^{N}_{\mathbf{U}}\mathbf{V}$ is the projection of
$\nabla^{M}_{\mathbf{U}}\mathbf{V}$ onto $TN$.

(B.83) can be rewritten as follows:

$\displaystyle\nabla^{N}_{\mathbf{U}}\mathbf{V}$
$\displaystyle=\nabla^{M}_{\mathbf{U}}\mathbf{V}+\frac{1}{2}\nu\left((\nabla^{M}_{\mathbf{U}}\nu)\mathbf{V}+\mathbf{V}(\nabla^{M}_{\mathbf{U}}\nu)\right)=\nabla^{M}_{\mathbf{U}}\mathbf{V}+\frac{1}{2}[\nu\nabla^{M}_{\mathbf{U}}\nu,\mathbf{V}].$

where now $\mathbf{V},$ $\nu$ and $\nabla_{\mathbf{U}}^{M}\nu$ are treated as
sections of the Clifford bundle $\mathrm{Cl}(TM)$. This is immediately
generalized to general Clifford fields

$\nabla^{N}_{\mathbf{U}}=\nabla^{M}_{\mathbf{U}}+\frac{1}{2}[\nu\nabla^{M}_{\mathbf{U}}\nu,\cdot]$
(B.84)

Now assume that $\Sigma^{M}$ is a Clifford module bundle over $M$ with a
Clifford covariant connection $\nabla^{M}$. The restriction of $\Sigma^{M}$ to
$N$, denoted $\Sigma^{N}$, is a bundle of Clifford modules. (B.84) motivates
defining the following connection on $\Sigma^{N}$:

$\nabla^{N}_{\mathbf{U}}=\nabla^{M}_{\mathbf{U}}+\frac{1}{2}\nu\nabla^{M}_{\mathbf{U}}\nu.$
(B.85)

By construction, $\nabla^{N}$ is Clifford covariant.

######  Lemma 64.

If $\Sigma^{M}$ is a spinor bundle, so is $\Sigma^{N}$.

###### Proof.

By Lemma 61 we may assume that $\Sigma^{M}$ is a bundle of irreducible
Clifford modules. If $d:=\dim(M)$ is even, then $\Sigma^{N}$ splits into
eigenbundles of $\mathrm{vol}_{N}$, which are bundles of irreducible Clifford
modules. If $d$ is odd, $\Sigma^{N}$ is irreducible.

Now choose a spinor scalar product $(\cdot|\cdot)$ and a spinor conjugation
$\Theta^{M}$ on $\Sigma^{M}$. Let $\Theta^{N}=\Theta^{M}$ if $d\in\\{1,2\\}$
mod $4$ and $\Theta^{N}=\nu\Theta^{M}$ if $d\in\\{0,3\\}$ mod $4$, in both
cases restricted to $\Sigma^{N}$. The restriction of $(\cdot|\cdot)$ to
$\Sigma^{N}$ and $\Theta^{N}$ are a spinor scalar product and a spinor
conjugation satisfying (B.68). If $d$ is even, this is still true if we
further restrict to eigenbundles of $\mathrm{vol}_{N}$. The result follows
from Lemma 60. ∎

Assume now that we have a covariantly constant section $\beta$ of
$\mathrm{End}(\Sigma)$ satisfying $\beta^{2}=1$ and anticommuting with
$TM\subset\mathrm{Cl}(TM)$. Let us consider the operator
$\Gamma=-\mathrm{i}\beta\nu$ acting on sections of $\Sigma^{N}$. It satisfies
$\Gamma^{2}=1$ and commutes with all sections of $\mathrm{Cl}(TN)$. Hence its
eigenbundles $\Sigma_{\pm}^{N}$ for eigenvalues $\pm 1$ are also Clifford
module bundles over $N$. Using (B.85) one checks that $\Gamma$ commutes also
with the covariant differentiation, so $\Sigma_{\pm}^{N}$ inherit the spin
connection.

Operator $\mathrm{vol}_{M}$ commutes with covariant differentiation and
anticommutes with $\Gamma$, hence it takes sections of $\Sigma_{\pm}^{N}$ to
sections of $\Sigma_{\mp}^{N}$. If $d$ is odd, $\mathrm{vol}_{M}$ commutes
with Clifford fields and hence defines an isomorphism of spinor bundles
$\Sigma_{+}^{N}\cong\Sigma_{-}^{N}$. If $d$ is even, $\Sigma_{+}^{N}$ and
$\Sigma_{-}^{N}$ are non-isomorphic as Clifford module bundles. In this case,
we can take $\beta:=\pm\mathrm{i}^{\frac{d}{2}}\mathrm{vol}_{M}$ and $\Gamma$
coincides up to phase with the $\mathrm{Cl}(TN)$ section $\mathrm{vol}_{N}$.

If $\Sigma$ is irreducible for $\mathrm{Cl}(TM)$ and $\beta$, by dimensional
consideration, $\Sigma_{\pm}^{N}$ are bundles of irreducible Clifford modules.

###  B.7 Dirac operators on spheres

Now let us consider the sphere $\mathbb{S}^{d-1}$ of radius $1$ (thus we put
$|x|=1$ below). We will apply the formalism of the previous section with
$M:=\mathbb{R}^{d}$ and $N:=\mathbb{S}^{d-1}$. For brevity, we will write
$\mathbb{S}$ for $\mathbb{S}^{d-1}$. We will use the notation of Subsections
B.1 and B.2, such as $S$, $R$, $T$ and $\kappa$.

The normal vector $\nu$ is identified with $S$. The Levi-Civita connection on
$\mathbb{S}$ is

$\nabla_{\mathbf{U}}^{\mathbb{S}}\mathbf{V}=\partial_{\mathbf{U}}\mathbf{V}+(\mathbf{U}\cdot\mathbf{V})x$
(B.86)

for vector fields $\mathbf{U},\mathbf{V}$ tangent to the sphere. Here
$\partial_{\mathbf{U}}=\sum\limits_{i=1}^{d}\mathbf{U}_{i}\partial_{i}$.
Consider the vector space $\mathcal{K}$ from previous subsections.
$\mathbb{S}\times\mathcal{K}$ is a Clifford module bundle with connection

$\nabla_{\mathbf{U}}^{\mathbb{S}}=\partial_{\mathbf{U}}+\frac{1}{2}S\mathbf{U}.$
(B.87)

######  Proposition 65.

The connection (B.87) is a spin connection with curvature

$\Omega(\mathbf{U},\mathbf{V})=\frac{1}{4}[\mathbf{U},\mathbf{V}].$ (B.88)

If $d=2$, the holonomy of $\nabla$ along $\mathbb{S}^{1}$ is equal to $-1$.

###### Proof.

All but the last statement follow from Lemma 64. (B.88) may also be obtained
from a simple direct computation. Now let $d=2$. We parametrize
$\mathbb{S}^{1}$ as $x=(\cos(\alpha),\sin(\alpha))$. Then (B.87) takes the
form

$\nabla_{\frac{\partial}{\partial\alpha}}\psi=\frac{\partial\psi}{\partial\alpha}+\frac{1}{2}\mathrm{vol}_{\mathbb{R}^{2}}\psi.$
(B.89)

It follows that solutions of the parallel transport equation
$\nabla_{\frac{\partial}{\partial\alpha}}\psi=0$ satisfy
$\psi(2\pi)=-\psi(0)$. ∎

Eigenbundles $\mathcal{K}_{\pm}\subset\mathbb{S}\times\mathcal{K}$ of
$\Gamma=-\mathrm{i}\beta S$ to eigenvalues $\pm 1$ are irreducible spinor
bundles, isomorphic if $d$ is odd and non-isomorphic otherwise.

Choose a local orthonormal framing $\\{e_{i}\\}_{i=1}^{d-1}$ of $T\mathbb{S}$.
Denote the Dirac operator on $\mathbb{S}$ by $D_{\mathbb{S}}$ and on
$\mathbb{R}^{d}$ by $D$. Using (B.87) we manipulate its definition to the form

$\displaystyle D_{\mathbb{S}}$
$\displaystyle=-\mathrm{i}\sum_{i}e_{i}\nabla_{e_{i}}\psi=-\mathrm{i}\sum_{i}e_{i}\partial_{e_{i}}\psi-\frac{\mathrm{i}}{2}\sum_{i}e_{i}Se_{i}\psi$
(B.90)
$\displaystyle=D+\mathrm{i}S\partial_{\nu}\psi+\mathrm{i}\frac{d-1}{2}S\psi=D-SR.$

Next let $\\{e_{i}\\}_{i=1}^{d}$ be the canonical basis of $\mathbb{R}^{d}$.
Multiplying the above by $S$ from the left we find

$\displaystyle SD_{\mathbb{S}}$
$\displaystyle=SD-R=-\mathrm{i}\sum_{i,j}x_{i}e_{i}e_{j}\partial_{j}-SR$
(B.91)
$\displaystyle=-\frac{\mathrm{i}}{2}\sum_{i,j}\left([e_{i},e_{j}]+[e_{i},e_{j}]_{+}x_{i}\partial_{j}\right)-R=\mathrm{i}T.$

Hence we have

$D_{\mathbb{S}}=\mathrm{i}ST.$ (B.92)

Let us note that this is exactly the off-diagonal term of $D$ with respect to
decomposition into eigenspaces of $S$. Next observe that

$\kappa=\Gamma D_{\mathbb{S}}=D_{\mathbb{S}}\Gamma.$ (B.93)

It follows that on sections of $\mathcal{K}_{\pm}$, operator $\kappa$ acts as
$\pm D_{\mathbb{S}}$.

We also remark that if $\psi$ is a $\mathcal{K}$-valued polynomial homogeneous
of degree $\ell$ annihillated by $D$, then $(S\mp\mathrm{i})\psi$ is an
eigenvector of $D_{\mathbb{S}}$ to eigenvalue $\pm(\ell+\frac{d-1}{2})$.
Indeed,

$D_{\mathbb{S}}\psi=-SR\psi=\mathrm{i}\left(\ell+\frac{d-1}{2}\right)S\psi,$
(B.94)

which implies

$\displaystyle D_{\mathbb{S}}(S\mp\mathrm{i})\psi$
$\displaystyle=(-S\mp\mathrm{i})D_{\mathbb{S}}\psi$ (B.95)
$\displaystyle=\mathrm{i}\left(\ell+\frac{d-1}{2}\right)(-S\mp\mathrm{i})S\psi=\pm\left(\ell+\frac{d-1}{2}\right)(S\mp\mathrm{i})\psi.$

By the relation between $D_{\mathbb{S}}$ and $\kappa$, this calculation
reproduces the spectrum of $\kappa$ found in Proposition 58.

We claim that a complete set of eigenfunctions of $D_{\mathbb{S}}$ is obtained
by the construction above, similarly as spherical harmonics are obtained by
restricting scalar-valued homogeneous harmonic polynomials, e.g. [1, p.
73–81]. This may be seen from the Stone-Weierstrass theorem and the following
lemma. Besides this application, the lemma elucidates the decomposition of
spaces of spinor-valued polynomials into irreducible representations of
$\mathrm{Spin}(\mathbb{R}^{d})$ and relates eigenvectors of $D_{\mathbb{S}}$
(and hence also of $\kappa$) to harmonic polynomials.

Consistently with our notation, in the following lemma $x$ denotes the element
of the Clifford algebra $x=\sum_{i}x_{i}e_{i}$, whereas $x_{i}$ are real
numbers. $x^{j}$ is the $j$-th power of $x$.

######  Lemma 66.

Let $\mathcal{K}_{\ell}$ be the space of $\mathcal{K}$-valued polynomials
homogeneous of degree $\ell$ and let $\mathcal{K}_{\ell}^{0}$ be the kernel of
$D$ acting in $\mathcal{K}_{\ell}$. Then

$\mathcal{K}_{\ell}=\mathcal{K}_{\ell}^{0}\oplus x\cdot\mathcal{K}_{\ell-1}.$
(B.96)

In particular we have a vector space decomposition

$\mathcal{K}_{\ell}=\bigoplus_{j=0}^{\ell}x^{j}\mathcal{K}_{\ell-j}^{0}.$
(B.97)

Moreover,
$\dim(\mathcal{K}_{\ell}^{0})=\binom{d+\ell-2}{\ell}\dim(\mathcal{K})$, and
(clearly) $\dim(\mathcal{K}_{\ell})=\binom{d+\ell-1}{\ell}\dim(\mathcal{K})$.

Let $\mathcal{H}_{\ell}$ be the space of scalar-valued harmonic polynomials
homogeneous of degree $\ell$. Then

$D:\mathcal{H}_{\ell}\otimes\mathcal{K}\to\mathcal{K}_{\ell-1}^{0}$ (B.98)

is a surjection with kernel $\mathcal{K}^{0}_{\ell}$. In particular there is
an exact sequence

$\dots\rightarrow\mathcal{H}_{\ell+1}\otimes\mathcal{K}\xrightarrow{D}\mathcal{H}_{\ell}\otimes\mathcal{K}\xrightarrow{D}\mathcal{H}_{\ell-1}\otimes\mathcal{K}\rightarrow\dots$
(B.99)

###### Proof.

Let $\\{e_{i}\\}_{i=1}^{d}$ be an orthonormal basis of $\mathbb{R}^{d}$. A
general element of $\mathcal{K}_{\ell}$ has the form

$\psi=\sum_{i_{1},\dots,i_{\ell}=1}^{d}x_{i_{1}}\cdots
x_{i_{\ell}}\psi_{i_{1}\dots i_{\ell}}$ (B.100)

with coefficients $\psi_{i_{1}\dots i_{\ell}}$ fully symmetric. Acting with
$D$ we find that $D\psi=0$ if and only if

$\psi_{1i_{2}\dots i_{\ell}}=\sum_{j\neq 1}e_{1}e_{j}\psi_{ji_{2}\dots
i_{\ell}}.$ (B.101)

It is easy to see that this system of equation may be uniquely solved once
$\psi_{i_{1}\dots i_{\ell}}$ is fixed for all indices $i_{1},\dots,i_{\ell}$
different than $1$. The formula for $\dim(\mathcal{K}^{0}_{\ell})$ follows.

Using the above result it is easy to check that

$\dim(\mathcal{K}_{\ell})=\dim(\mathcal{K}_{\ell}^{0})+\dim(x\cdot\mathcal{K}_{\ell-1}).$
(B.102)

Hence (B.96) will follow once we establish that $\mathcal{K}_{\ell}^{0}\cap
x\cdot\mathcal{K}_{\ell-1}=\\{0\\}$.

We proceed by induction. There is nothing to prove for $\ell=0$. Suppose that
(B.96) holds for $\ell\leq\ell^{\prime}$. Then also (B.97) holds for
$\ell\leq\ell^{\prime}$. Let us put $\ell=\ell^{\prime}+1$ and let
$\psi\in\mathcal{K}_{\ell}^{0}\cap x\cdot\mathcal{K}_{\ell-1}$. By the
induction hypothesis we have

$\psi=\sum_{j=1}^{\ell}x^{j}\psi_{j}$ (B.103)

with $\psi_{j}\in\mathcal{K}_{\ell-j}$. Now let us observe that

$\displaystyle[\mathrm{i}D,x^{2}]=2x,\qquad$ $\displaystyle\text{hence
}[\mathrm{i}D,x^{2k}]=2kx^{2k-1},$ (B.104a)
$\displaystyle[\mathrm{i}D,x]_{+}=2\sum_{i}x_{i}\partial_{i}+d,\qquad$
$\displaystyle\text{hence}[\mathrm{i}D,x^{2k+1}]_{+}=x^{2k}(2\sum_{i}x_{i}\partial_{i}+d+2k).$
(B.104b)

Thus since $\psi$ is annhilated by $D$ and
$\sum_{i}x_{i}\partial_{i}\psi_{j}=(\ell-j)\psi_{j}$, we obtain

$0=\mathrm{i}D\psi=\sum_{j\text{ even}}jx^{j-1}\psi_{j}+\sum_{j\text{
odd}}(\ell+d-1)x^{j-1}\psi_{j}.$ (B.105)

By induction hypothesis, $x^{j-1}\psi_{j}$ and
$x^{j^{\prime}-1}\psi_{j^{\prime}}$ belong to subspaces of
$\mathcal{K}_{\ell-1}$ with trivial intersection if $j\neq j^{\prime}$.
Therefore each term in the above sum has to vanish separately. Since operators
$x^{j-1}$ are injective, all $\psi_{j}$ vanish. Thus $\psi=0$.

For the last part, note that $\mathcal{H}_{\ell}\otimes\mathcal{K}$ is the
space of harmonic $\mathcal{K}$-valued polynomials homogeneous of degree
$\ell$. It is annihilated by $D^{2}$, so $D$ maps it into
$\mathcal{K}_{\ell-1}^{0}$. Statement about the kernel is obvious. Then
$\dim(\mathcal{H}_{\ell}\otimes\mathcal{K})=\dim(\mathcal{K}_{\ell}^{0})+\dim(\mathcal{K}_{\ell-1}^{0})$
follows from the well-known formula

$\dim(\mathcal{H}_{\ell})=\binom{d+\ell-1}{\ell}-\binom{d+\ell-3}{\ell-2}.$
(B.106)

This implies the surjectivity. ∎

As for any oriented Riemannian manifold, two natural second order operators
act on sections of spinor bundles: the square of the Dirac operator
$D^{2}_{\mathbb{S}}$ and the Bochner Laplacian $\Delta_{\mathbb{S}}$. In the
case of spheres we have an additional natural second order operator: the
square of the total angular momentum $J^{2}$. It turns out that for $d\geq 3$
the operator $J^{2}$ is distinct from both $D_{\mathbb{S}}^{2}$ and
$-\Delta_{\mathbb{S}}$. More precisely, we have

$\displaystyle D^{2}_{\mathbb{S}}$
$\displaystyle=-\Delta_{\mathbb{S}}+\frac{(d-1)(d-2)}{4},$ (B.107)
$\displaystyle D^{2}_{\mathbb{S}}$ $\displaystyle=J^{2}+\frac{(d-1)(d-2)}{8}.$
(B.108)

(B.107) is the Lichnerowicz formula for the spheres (indeed, the scalar
curvature of $\mathbb{S}$ is ${(d-1)(d-2)}$). Moreover,
$D^{2}_{\mathbb{S}}=\kappa^{2}$. Hence (B.108) is essentially the formula
(B.32). Thus, as we were surprised to find out, (B.32) is distinct from the
Lichnerowicz formula.

##  Appendix C Mellin transformation

For a Schwartz function $b$ on $\mathbb{R}$ we define its Fourier transform by

$(\mathfrak{F}b)(k)=\int_{-\infty}^{\infty}b(t)\mathrm{e}^{-\mathrm{i}kt}\mathrm{d}t.$
(C.1)

It is extended to the space of Schwartz distributions in the usual way.
Restriction of $\frac{1}{\sqrt{2\pi}}\mathfrak{F}$ to $L^{2}(\mathbb{R})$ is a
unitary operator.

An isomorphism $\iota:C_{c}^{\infty}(\mathbb{R})\to
C_{c}^{\infty}(\mathbb{R}_{+})$ is defined by $(\iota
f)(x)=x^{-\frac{1}{2}}f(\ln(x))$. Dualizing, we extend it to an isomorphism
between spaces of distributions on $\mathbb{R}$ and on $\mathbb{R}_{+}$.
Restriction of $\iota$ to $L^{2}(\mathbb{R})$ is a unitary operator onto
$L^{2}(\mathbb{R}_{+})$. By Schwartz class functions and tempered
distributions on $\mathbb{R}_{+}$ we shall mean smooth functions (respectively
distributions) on $\mathbb{R}_{+}$ which correspond through $\iota$ to
Schwartz class functions (respectively tempered distributions) on
$\mathbb{R}$.

Mellin transform is defined as the composition
$\mathfrak{M}=\mathfrak{F}\iota^{-1}$. It is an isomorphism between spaces of
tempered distributions on $\mathbb{R}_{+}$ and on $\mathbb{R}$. If $f\in
C_{c}^{\infty}(\mathbb{R}_{+})$, then

$(\mathfrak{M}f)(k)=\int_{0}^{\infty}f(x)x^{-\frac{1}{2}-\mathrm{i}k}\mathrm{d}x.$
(C.2)

Recall that $A$, $J$ and $K$ are defined in Subsection 1.1. We note the
following identities:

$\displaystyle\mathfrak{M}Jf(k)=\mathfrak{M}f(-k),$ (C.3a) $\displaystyle
A=\mathfrak{M}^{-1}K\mathfrak{M}.$ (C.3b)

The following lemma will be used in Proposition 73. The Mellin transformation
plays here a secondary role.

######  Lemma 67.

Suppose that $f_{\epsilon}$ is a family of tempered distributions on
$\mathbb{R}_{+}$ with parameter $\epsilon\in]0,1]$ satisfying the following
conditions:

  * •

$\mathfrak{M}f_{\epsilon}\in L^{1}_{\mathrm{loc}}(\mathbb{R})$,

  * •

there exists $g\in L^{1}_{\mathrm{loc}}(\mathbb{R})$ such that
$\mathfrak{M}f_{\epsilon}\to g$ pointwise for $\epsilon\to 0$,

  * •

there exist $c,N\geq 0$ independent of $\epsilon$ such that
$|\mathfrak{M}f_{\epsilon}(k)|\leq c(1+k^{2})^{N}$ for almost every $k$.

Then there exists a tempered distribution $f_{0}$ on $\mathbb{R}_{+}$ such
that $f_{\epsilon}\to f_{0}$ for $\epsilon\to 0$ in the sense of tempered
distributions. Moreover, $\mathfrak{M}f_{0}=g$.

######  Lemma 68.

Let $b$ be a tempered distribution whose Mellin transform is a Borel function.
Put

$(B^{\mathrm{pre}}f)(x)=\int_{0}^{\infty}b(xk)f(k)\mathrm{d}k,\qquad f\in
C_{c}^{\infty}(\mathbb{R}_{+}).$ (C.4)

Then $B^{\mathrm{pre}}f$ is in $L^{2}(\mathbb{R}_{+})$ and $B^{\mathrm{pre}}$
is a closable operator on $L^{2}(\mathbb{R}_{+})$ with closure

$B=\mathfrak{M}b(A)J.$ (C.5)

##  Appendix D Whittaker functions

In this appendix we review some properties of special functions used in this
paper. In particular, we discuss Whittaker functions, which play the central
role in our paper. We follow the conventions of [13] and [10].

###  D.1 Confluent equation

Before discussing the Whittaker equation and its solutions, let us say a few
words about the closely related confluent equation and the hypergeometric
equation.

The confluent equation has the form

$\big{(}z\partial_{z}^{2}+(c-z)\partial_{z}-a\big{)}v(z)=0.$ (D.1)

Let us list three of its standard solutions:

$\displaystyle{}_{1}F_{1}(a;c;z)$ $\displaystyle\qquad\text{characterized by
}\sim 1\text{ near }0;$ (D.2a) $\displaystyle z^{1-c}{}_{1}F_{1}(a+1-c;2-c;z)$
$\displaystyle\qquad\text{characterized by }\sim z^{1-c}\text{ near }0;$
(D.2b) $\displaystyle z^{-a}{}_{2}F_{0}(a,a+1-c;-;-z^{-1})$
$\displaystyle\qquad\text{characterized by }\sim z^{-a}\text{ near }+\infty.$
(D.2c)

We note that the function
${}_{1}\mathbf{F}_{1}(a;c;z)=\frac{1}{\Gamma(c)}{}_{1}F_{1}(a;c;z)$ is
holomorphic in all variables. The other two solutions are defined for
$z\notin]-\infty,0]$; thus ${}_{2}F_{0}(a,b;-;z)$ is defined on
$\mathbb{C}\backslash[0,\infty[$.

We will also need the hypergeometric equation

$\big{(}z(1-z)\partial_{z}^{2}+(c-(a+b+1)z)\partial_{z}-ab\big{)}v(z)=0.$
(D.3)

Among its 6 standard solutions, members of the famous Kummer’s table, let us
list three:

$\displaystyle{}_{2}F_{1}(a,b;c;z)$ $\displaystyle\qquad\text{characterized by
}\sim 1\text{ near }0;$ (D.4a) $\displaystyle
z^{1-c}{}_{2}F_{1}(a+1-c,b-c+1;2-c;z)$ $\displaystyle\qquad\text{characterized
by }\sim z^{1-c}\text{ near }0;$ (D.4b)
$\displaystyle{}_{2}F_{1}(a,b;a+b+1-c;1-z)$
$\displaystyle\qquad\text{characterized by }\sim 1\text{ near }1.$ (D.4c)

${}_{2}F_{1}(a,b;c;z)$ may be defined by a power series convergent for $z$ in
the unit disc. It admits analytic continuation along any path in
$\mathbb{C}\backslash\\{1\\}$. To make it a single-valued function, it is
customary to restrict its domain to $z\in\mathbb{C}\backslash[1,\infty[$. Then
${}_{2}\mathbf{F}_{1}(a,b;c;z)=\frac{1}{\Gamma(c)}{}_{2}F_{1}(a,b;c;z)$ is
holomorphic in all variables. It satisfies

${}_{2}\mathbf{F}_{1}(a,b;0;z)=abz{}_{2}\mathbf{F}_{1}(a+1,b+1;2;z),$ (D.5)

We have the following identities valid for $z\not\in]-\infty,0]$:

$\displaystyle\frac{\sin(\pi c)}{\pi}z^{-a}{}_{2}F_{0}(a,a+1-c;-;-z^{-1})$
(D.6) $\displaystyle=$
$\displaystyle\frac{{}_{1}\mathbf{F}_{1}(a;c;z)}{\Gamma(a+1-c)}-\frac{z^{1-c}{}_{1}\mathbf{F}_{1}a+1-c;2-c;z)}{\Gamma(a)},$
$\displaystyle\frac{\sin(\pi c)}{\pi}{}_{2}\mathbf{F}_{1}(a,b;a+b+1-c;1-z)$
(D.7) $\displaystyle=$
$\displaystyle\frac{{}_{2}\mathbf{F}_{1}(a,b;c;z)}{\Gamma(a-c+1)\Gamma(b-c+1)}-\frac{z^{1-c}{}_{2}\mathbf{F}_{1}(a-c+1,b-c+1;2-c;z)}{\Gamma(a)\Gamma(b)}.$

(D.6) can be taken as an alternative definition of ${}_{2}F_{0}$.

######  Lemma 69.

Let $\epsilon>0$, $\mathrm{Re}(z)<\frac{1}{2}$. Then

${}_{2}F_{1}(a,b+\lambda,c+\lambda;z)=(1-z)^{-a}+O(\lambda^{-1}),\qquad|\arg(\lambda)|\leq\pi-\epsilon,\
|\lambda|\to\infty.$ (D.8)

###### Proof.

Assumption about $\arg(\lambda)$ guarantees that for $|\lambda|$ sufficiently
large $c+\lambda\notin-\mathbb{N}$, so the left hand side of (D.8) is finite.
We apply the Pfaff transformation:

${}_{2}F_{1}(a,b+\lambda,c+\lambda;z)=(1-z)^{-a}{}_{2}F_{1}\left(a,c-b;c+\lambda;\frac{z}{z-1}\right).$
(D.9)

The claim follows from the standard series defining ${}_{2}F_{1}$, because
$\left|\frac{z}{z-1}\right|<1$. ∎

######  Lemma 70.

The following asymptotic expansion holds for $\mathrm{Re}(z)<\frac{1}{2}$,
$z\notin]-\infty,0]$, $s\to\pm\infty$:

${}_{2}\mathbf{F}_{1}(a,b-\mathrm{i}s;c;1-z)\sim\mathrm{sgn}(s)\cdot\left(\frac{(1-z)^{-a}(\mathrm{i}s)^{-a}}{\Gamma(c-a)}+\frac{z^{c-a-b+\mathrm{i}s}(1-z)^{b-c+\mathrm{i}s}(-\mathrm{i}s)^{a-c}}{\Gamma(a)}\right)$
(D.10)

locally uniformly in $a,b,c,z$.

###### Proof.

Using (D.7) we get

$\displaystyle\frac{\sin\left(\pi(a+b-c+1-\mathrm{i}s)\right)}{\pi}{}_{2}\mathbf{F}_{1}(a,b-\mathrm{i}s,c;1-z)$
(D.11) $\displaystyle=$
$\displaystyle\frac{{}_{2}\mathbf{F}_{1}(a,b-\mathrm{i}s;a+b-c+1-\mathrm{i}s;z)}{\Gamma(c-b+\mathrm{i}s)\Gamma(c-a)}-z^{c-a-b+\mathrm{i}s}\frac{{}_{2}\mathbf{F}_{1}(c-b+\mathrm{i}s,c-a;c-a-b+1+\mathrm{i}s;z)}{\Gamma(a)\Gamma(b-\mathrm{i}s)}.$

Then (D.8) gives for large $|s|$:

$\displaystyle\frac{\sin\left(\pi(a+b-c+1-\mathrm{i}s)\right)}{\pi}{}_{2}\mathbf{F}_{1}(a,b-\mathrm{i}s;c;z)$
(D.12) $\displaystyle\sim$
$\displaystyle\frac{(1-z)^{-a}}{\Gamma(c-b+\mathrm{i}s)\Gamma(c-a)\Gamma(a+b-c+1-\mathrm{i}s)}-\frac{z^{c-a-b+\mathrm{i}s}(1-z)^{b-c-\mathrm{i}s}}{\Gamma(a)\Gamma(b-\mathrm{i}s)\Gamma(c-a-b+1+\mathrm{i}s)}.$

Using
$\sin\left(\pi(a+b-c+1-\mathrm{i}s)\right)\sim\frac{1}{2\mathrm{i}}\mathrm{e}^{\pi|s|+\mathrm{i}\pi\,\mathrm{sgn}(s)(a+b-c+1)}$
and Stirling’s formula

$\Gamma(z_{0}+z)\sim\sqrt{\frac{2\pi}{z}}\mathrm{e}^{-z}z^{z+z_{0}},\qquad|\arg(z)|\leq\pi-\epsilon,\
|z|\to\infty$ (D.13)

yields, after algebraic manipulations, formula (D.10). ∎

###  D.2 Hyperbolic-type Whittaker equation

The standard form of the Whittaker equation is

$\displaystyle\Big{(}-\partial_{z}^{2}+\Big{(}m^{2}-\frac{1}{4}\Big{)}\frac{1}{z^{2}}-\frac{\beta}{z}+\frac{1}{4}\Big{)}g=0.$
(D.14)

In this section we briefly describe solutions of the Whittaker equation,
following mostly [13, 10].

We will sometimes call (D.14) the hyperbolic-type Whittaker equation, to
distinguish it from the trigonometric-type Whittaker equation, which differs
by the sign in front of $\frac{1}{4}$.

There are two kinds of standard solutions to the Whittaker equation.

The function $\mathcal{I}_{\beta,m}$ is defined by

$\mathcal{I}_{\beta,m}(z)=z^{\frac{1}{2}+m}{\mathrm{e}^{\mp\frac{z}{2}}}{}_{1}\mathbf{F}_{1}\Big{(}\frac{1}{2}+m\mp\beta;1+2m;\pm
z\Big{)}.$ (D.15)

The standard domain of $\mathcal{I}_{\beta,m}$ is
$\mathbb{C}\backslash]-\infty,0]$. We have

$\displaystyle\mathcal{I}_{-\beta,m}(-z)=\mathrm{e}^{-\mathrm{i}\pi(\frac{1}{2}+m)\mathrm{sgn}(\mathrm{Im}(z))}\mathcal{I}_{\beta,m}(z)\qquad\text{for
}z\in\mathbb{C}\backslash\mathbb{R},$ (D.16a)
$\displaystyle\overline{\mathcal{I}_{\overline{\beta},\overline{\vphantom{\beta}m}}(\overline{z})}=\mathcal{I}_{\beta,m}(z).$
(D.16b)

The case $2m\in\mathbb{Z}$ is called degenerate, and then

$\mathcal{I}_{\beta,-m}(z)=\Big{(}-\beta-m+\frac{1}{2}\Big{)}_{2m}\,\mathcal{I}_{\beta,m}(z).$
(D.17)

The function $\mathcal{K}_{\beta,m}$ is defined by

$\displaystyle\mathcal{K}_{\beta,m}(z)$
$\displaystyle:=z^{\beta}\mathrm{e}^{-\frac{z}{2}}{}_{2}F_{0}\Big{(}\frac{1}{2}+m-\beta,\frac{1}{2}-m-\beta;-;-z^{-1}\Big{)}$
(D.18) $\displaystyle=\frac{\pi}{\sin(2\pi
m)}\Big{(}-\frac{\mathcal{I}_{\beta,m}(z)}{\Gamma(\frac{1}{2}-m-\beta)}+\frac{\mathcal{I}_{\beta,-m}(z)}{\Gamma(\frac{1}{2}+m-\beta)}\Big{)}.$

It satisfies

$\displaystyle\mathcal{K}_{\beta,-m}(z)=\mathcal{K}_{\beta,m}(z),$ (D.19a)
$\displaystyle\overline{\mathcal{K}_{\overline{\beta},\overline{\vphantom{\beta}m}}(\overline{z})}=\mathcal{K}_{\beta,m}(z).$
(D.19b)

The Wronskian of $\mathcal{I}_{\beta,m}(\cdot)$ and
$\mathcal{K}_{\beta,m}(\cdot)$ takes the form

$\mathcal{W}(\mathcal{I}_{\beta,m},\mathcal{K}_{\beta,m})=-\frac{1}{\Gamma\Big{(}\frac{1}{2}+m-\beta\Big{)}}.$
(D.20)

If $\frac{1}{2}+m-\beta\in-\mathbb{N}$, then the Wronskian vanishes and we
have

$\displaystyle\mathcal{K}_{\beta,m}(z)=$
$\displaystyle\mathrm{e}^{\mathrm{i}\pi(\frac{1}{2}+m-\beta)}\Gamma\left(\frac{1}{2}+m+\beta\right)\mathcal{I}_{\beta,m}(z).$
(D.21)

In this case functions in (D.21) essentially coincide with Laguerre
polynomials

$\mathcal{I}_{\pm(\frac{1}{2}+m+n),m}(z)=\frac{n!\;\\!z^{\frac{1}{2}+m}\mathrm{e}^{\mp\frac{z}{2}}}{\Gamma(1+2m+n)}L_{n}^{(2m)}(\pm
z).$ (D.22)

Asymptotics of $\mathcal{I}_{\beta,m}$ for small arguments are of the form

$\mathcal{I}_{\beta,m}(z)=\frac{z^{\frac{1}{2}+m}}{\Gamma(1+2m)}\Big{(}1-\frac{\beta}{1+2m}z+O(z^{2})\Big{)},\text{
if }m\neq-\frac{1}{2},-1,-\frac{3}{2},\dots,\quad z\to 0,$ (D.23)

The function $\mathcal{K}_{\beta,m}$ satisfies, for $z\to 0$,

$\begin{array}[]{rll}\mathcal{K}_{\beta,m}(z)=&z^{\frac{1}{2}}\Big{(}\frac{\Gamma(-2m)}{\Gamma(\frac{1}{2}-m-\beta)}z^{m}+\frac{\Gamma(2m)}{\Gamma(\frac{1}{2}+m-\beta)}z^{-m}\Big{(}1-\frac{\beta}{1-2m}z\Big{)}\Big{)}&\vspace{0,1cm}\\\
&\qquad\qquad\qquad+O(|z|^{\frac{3}{2}+\mathrm{Re}(m)})+O(|z|^{\frac{5}{2}-\mathrm{Re}(m)})&\text{
for }\mathrm{Re}(m)\in[0,1[,\,m\neq 0,\frac{1}{2}\vspace{0,1cm}\\\
\mathcal{K}_{\beta,0}(z)=&-\frac{z^{\frac{1}{2}}}{\Gamma(\frac{1}{2}-\beta)}\big{(}\mathrm{ln}(z)+\psi(\frac{1}{2}-\beta)+2\lambda\big{)}+O(|z|^{\frac{3}{2}}\ln(z)),&\text{
for }m=0,\,\beta\notin\frac{1}{2}+\mathbb{N}\vspace{0,1cm}\\\
\mathcal{K}_{\beta,0}(z)=&(\beta-\frac{1}{2})!(-1)^{\beta-\frac{1}{2}}z^{\frac{1}{2}}+O(|z|^{\frac{3}{2}}),&\text{
for }m=0,\,\beta\in\frac{1}{2}+\mathbb{N}\vspace{0,1cm}\\\
\mathcal{K}_{\beta,\frac{1}{2}}(z)=&\frac{1}{\Gamma(-\beta)}\big{(}-\frac{1}{\beta}+z\ln(z)+z(\psi(1-\beta)+2\lambda-1+\frac{1}{2\beta})\big{)}&\vspace{0,1cm}\\\
&\qquad\qquad\qquad+O(z^{2}\ln(z)),&\text{ for
}m=\frac{1}{2},\beta\notin\mathbb{N}\vspace{0,1cm}\\\
\mathcal{K}_{0,\frac{1}{2}}(z)=&1-\frac{z}{2}+O(z^{2}),&\text{ for
}m=\frac{1}{2},\beta=0\vspace{0,1cm}\\\
\mathcal{K}_{\beta,\frac{1}{2}}(z)=&\beta!(-1)^{\beta-1}z+O(z^{2}),&\text{ for
}m=\frac{1}{2},\beta\in\mathbb{N}^{\times}\vspace{0,1cm}\\\
\mathcal{K}_{\beta,m}(z)=&\frac{\Gamma(2m)}{\Gamma(\frac{1}{2}+m-\beta)}z^{\frac{1}{2}-m}+O(|z|^{\frac{3}{2}-\mathrm{Re}(m)}),&\text{
for }\mathrm{Re}(m)\geq 1.\end{array}$ (D.24)

Here $\gamma$ denotes Euler’s constant and $\psi$ is the digamma function.

Asymptotics for $|z|\to\infty$ are given by ($\epsilon>0$):

$\displaystyle\mathcal{K}_{\beta,m}(z)$
$\displaystyle=z^{\beta}\mathrm{e}^{-\frac{z}{2}}\big{(}1+O(z^{-1})\big{)},\quad|\mathrm{arg}(z)|\leq\frac{3}{2}\pi-\epsilon,$
(D.25a) $\displaystyle\mathcal{I}_{\beta,m}(z)$
$\displaystyle=\frac{z^{-\beta}\mathrm{e}^{\frac{z}{2}}}{\Gamma(\frac{1}{2}+m-\beta)}\left(1+O(z^{-1})\right),\quad|\mathrm{arg}(z)|\leq\frac{\pi}{2}-\epsilon.$
(D.25b)

The analytic continuations of $\mathcal{K}$, and more precisely the functions
$z\mapsto\mathcal{K}_{-\beta,m}(\mathrm{e}^{\pm\mathrm{i}\pi}z)$ are also
solutions of the Whittaker Equation (D.14). One can define
$\mathcal{K}_{-\beta,m}(\mathrm{e}^{\pm i\pi}z)$ as the unique holomorphic
function of $z\in\mathbb{C}\backslash]-\infty,0]$ which coincides with
$\mathcal{K}_{-\beta,m}(-z)$ on $\mathbb{C}_{\mp}$. Then

$\displaystyle\mathcal{K}_{-\beta,m}(\mathrm{e}^{\pm\mathrm{i}\pi}z)$
$\displaystyle=\frac{\pi}{\sin(2\pi
m)}\Big{(}-\frac{\mathrm{e}^{\pm\mathrm{i}\pi(\frac{1}{2}+m)}\mathcal{I}_{\beta,m}(z)}{\Gamma(\frac{1}{2}-m+\beta)}+\frac{\mathrm{e}^{\pm\mathrm{i}\pi(\frac{1}{2}-m)}\mathcal{I}_{\beta,-m}(z)}{\Gamma(\frac{1}{2}+m+\beta)}\Big{)}.$
(D.26)

$\big{(}\mathcal{K}_{\beta,m}(z),\mathcal{K}_{-\beta,m}(\mathrm{e}^{\pm\mathrm{i}\pi}z)\big{)}$
are linearly independent pairs of functions. In particular,
$\mathcal{I}_{\beta,m}(z)$ can be expressed in terms of these functions:

$\displaystyle\mathcal{I}_{\beta,m}(z)=\mathrm{e}^{\pm\mathrm{i}\pi\beta}\Big{(}\frac{\mathrm{e}^{\mp\mathrm{i}\pi(m-\frac{1}{2})}\mathcal{K}_{\beta,m}(z)}{\Gamma(\frac{1}{2}+m+\beta)}+\frac{\mathcal{K}_{-\beta,m}(\mathrm{e}^{\pm\mathrm{i}\pi}z)}{\Gamma(\frac{1}{2}+m-\beta)}\Big{)}.$

###  D.3 Trigonometric-type Whittaker functions

It is convenient, in parallel to (D.14), to consider the trigonometric-type
Whittaker equation

$\displaystyle\Big{(}-\partial_{z}^{2}+\Big{(}m^{2}-\frac{1}{4}\Big{)}\frac{1}{z^{2}}-\frac{\beta}{z}-\frac{1}{4}\Big{)}g(z)=0.$
(D.27)

It can be easily reduced to the hyperbolic-type Whittaker equation.

The function $\mathcal{J}_{\beta,m}$ is defined by the formula

$\mathcal{J}_{\beta,m}(z)=\mathrm{e}^{\mp\frac{\mathrm{i}\pi}{2}(m+\frac{1}{2})}\mathcal{I}_{\mp\mathrm{i}\beta,m}(\mathrm{e}^{\pm\frac{\mathrm{i}\pi}{2}}z).$
(D.28)

It may also be described without invoking analytic continuations beyond the
principal branch:

$\mathcal{J}_{\beta,m}(z)=\begin{cases}\mathrm{e}^{\frac{\mathrm{i}\pi}{2}\left(\frac{1}{2}+m\right)}\mathcal{I}_{\mathrm{i}\beta,m}(-\mathrm{i}z),&-\frac{\pi}{2}<\mathrm{arg}(z)<\pi,\\\
\mathrm{e}^{-\frac{\mathrm{i}\pi}{2}\left(\frac{1}{2}+m\right)}\mathcal{I}_{-\mathrm{i}\beta,m}(\mathrm{i}z),&-\pi<\mathrm{arg}(z)<\frac{\pi}{2}.\end{cases}$
(D.29)

The two expression agree for $|\mathrm{arg}(z)|<\frac{\pi}{2}$, by (D.16a).
Combined with (D.16b) this implies

$\overline{\mathcal{J}_{\overline{\beta},\overline{\vphantom{\beta}m}}}(\overline{z})=\mathcal{J}_{\beta,m}(z).$
(D.30)

We also have a pair of functions $\mathcal{H}_{\beta,m}^{\pm}$ defined by

$\mathcal{H}_{\beta,m}^{\pm}(z)=\mathrm{e}^{\mp\mathrm{i}\frac{\pi}{2}\left({\frac{1}{2}+m}\right)}\mathcal{K}_{\pm\mathrm{i}\beta,m}(\mp\mathrm{i}z)$
(D.31)

initially for $\mathrm{Re}(z)>0$ and extended analytically to
$z\in\mathbb{C}\backslash]-\infty,0]$. By (D.19b), they satisfy

$\overline{\mathcal{H}^{\pm}_{\overline{\beta},\overline{\vphantom{\beta}m}}(\overline{z})}=\mathcal{H}^{\mp}_{\beta,m}(z).$
(D.32)

The following connection formula holds:

$\mathcal{J}_{\beta,m}(z)=\mathrm{e}^{-\pi\beta}\Big{(}\frac{\mathcal{H}_{\beta,m}^{+}(z)}{\Gamma\left({\frac{1}{2}+m+\mathrm{i}\beta}\right)}+\frac{\mathcal{H}_{\beta,m}^{-}(z)}{\Gamma\left({\frac{1}{2}+m-\mathrm{i}\beta}\right)}\Big{)}.$
(D.33)

For the behaviour around $\infty$, we have for $x>0$, $x\to\infty$

$\mathcal{H}_{\beta,m}^{\pm}(x)\sim\mathrm{e}^{\mp\mathrm{i}\frac{\pi}{2}(\frac{1}{2}+m)}\mathrm{e}^{\frac{\pi\beta}{2}}x^{\pm\mathrm{i}\beta}\;\\!\mathrm{e}^{\pm\mathrm{i}\frac{x}{2}}\big{(}1+O(x^{-1})\big{)}.$
(D.34)

###  D.4 Recurrence relations

Whittaker functions satisfy several recurrence relations. There are 6 basic
ones, which we quote after Appendix A5 of [10].

$\displaystyle\Big{(}\sqrt{z}\partial_{z}+\frac{-\frac{1}{2}-m}{\sqrt{z}}-\frac{\sqrt{z}}{2}\Big{)}\mathcal{I}_{\beta,m}(z)$
$\displaystyle=\Big{(}-\frac{1}{2}-m-\beta\Big{)}\mathcal{I}_{\beta+\frac{1}{2},m+\frac{1}{2}}(z),$
(D.35a)
$\displaystyle\Big{(}\sqrt{z}\partial_{z}+\frac{-\frac{1}{2}+m}{\sqrt{z}}+\frac{\sqrt{z}}{2}\Big{)}\mathcal{I}_{\beta,m}(z)$
$\displaystyle=\mathcal{I}_{\beta-\frac{1}{2},m-\frac{1}{2}}(z),$ (D.35b)
$\displaystyle\Big{(}\sqrt{z}\partial_{z}+\frac{-\frac{1}{2}+m}{\sqrt{z}}-\frac{\sqrt{z}}{2}\Big{)}\mathcal{I}_{\beta,m}(z)$
$\displaystyle=\mathcal{I}_{\beta+\frac{1}{2},m-\frac{1}{2}}(z),$ (D.35c)
$\displaystyle\Big{(}\sqrt{z}\partial_{z}+\frac{-\frac{1}{2}-m}{\sqrt{z}}+\frac{\sqrt{z}}{2}\Big{)}\mathcal{I}_{\beta,m}(z)$
$\displaystyle=\Big{(}\frac{1}{2}+m-\beta\Big{)}\mathcal{I}_{\beta-\frac{1}{2},m+\frac{1}{2}}(z),$
(D.35d)
$\displaystyle\Big{(}z\partial_{z}+\beta-\frac{z}{2}\Big{)}\mathcal{I}_{\beta,m}(z)$
$\displaystyle=\Big{(}\frac{1}{2}+m+\beta\Big{)}\mathcal{I}_{\beta+1,m}(z),$
(D.35e)
$\displaystyle\Big{(}z\partial_{z}-\beta+\frac{z}{2}\Big{)}\mathcal{I}_{\beta,m}(z)$
$\displaystyle=\Big{(}\frac{1}{2}+m-\beta\Big{)}\mathcal{I}_{\beta-1,m}(z);$
(D.35f)
$\displaystyle\Big{(}\sqrt{z}\partial_{z}+\frac{-\frac{1}{2}-m}{\sqrt{z}}-\frac{\sqrt{z}}{2}\Big{)}\mathcal{K}_{\beta,m}(z)$
$\displaystyle=-\mathcal{K}_{\beta+\frac{1}{2},m+\frac{1}{2}}(z),$ (D.36a)
$\displaystyle\Big{(}\sqrt{z}\partial_{z}+\frac{-\frac{1}{2}+m}{\sqrt{z}}+\frac{\sqrt{z}}{2}\Big{)}\mathcal{K}_{\beta,m}(z)$
$\displaystyle=\Big{(}-\frac{1}{2}+m+\beta\Big{)}\mathcal{K}_{\beta-\frac{1}{2},m-\frac{1}{2}}(z),$
(D.36b)
$\displaystyle\Big{(}\sqrt{z}\partial_{z}+\frac{-\frac{1}{2}+m}{\sqrt{z}}-\frac{\sqrt{z}}{2}\Big{)}\mathcal{K}_{\beta,m}(z)$
$\displaystyle=-\mathcal{K}_{\beta+\frac{1}{2},m-\frac{1}{2}}(z),$ (D.36c)
$\displaystyle\Big{(}\sqrt{z}\partial_{z}+\frac{-\frac{1}{2}-m}{\sqrt{z}}+\frac{\sqrt{z}}{2}\Big{)}\mathcal{K}_{\beta,m}(z)$
$\displaystyle=\Big{(}-\frac{1}{2}-m+\beta\Big{)}\mathcal{K}_{\beta-\frac{1}{2},m+\frac{1}{2}}(z),$
(D.36d)
$\displaystyle\Big{(}z\partial_{z}+\beta-\frac{z}{2}\Big{)}\mathcal{K}_{\beta,m}(z)$
$\displaystyle=-\mathcal{K}_{\beta+1,m}(z),$ (D.36e)
$\displaystyle\Big{(}z\partial_{z}-\beta+\frac{z}{2}\Big{)}\mathcal{K}_{\beta,m}(z)$
$\displaystyle=\Big{(}\frac{1}{2}+m-\beta\Big{)}\Big{(}\frac{1}{2}-m-\beta\Big{)}\mathcal{K}_{\beta-1,m}(z).$
(D.36f)

They have an interesting algebraic interpretation—they correspond to the roots
of the Lie algebra of generalized symmetries of the heat equation in 2
dimensions, see [9] (where they are presented using confluent functions, which
as we know are equivalent to Whittaker functions).

These recurrence relations involve 1st order differentiation, so it is
tempting to expect that they are closely related to the Dirac-Coulomb
Hamiltonian. It turns out, however, that the relationship is not direct. By
easy algebraic manipulations involving (D.35a)-(D.35d) and (D.36a)-(D.36d), we
derive an additional pair of recurrence relations described in the following
proposition. In some sense, (D.35) and (D.36) are “lower order” than (D.37)
and (D.38). In fact, in (D.35) and (D.36) the parameters $\mu,\beta$ appear
only in 0th order terms, whereas in (D.37) and (D.38) the differential
operator is multiplied by $\mu$.

######  Proposition 71.

$\displaystyle\Big{(}2\mu\partial_{x}+\frac{2\mu^{2}}{x}-\beta\Big{)}\mathcal{I}_{\beta,\mu+\frac{1}{2}}(x)$
$\displaystyle=\mathcal{I}_{\beta,\mu-\frac{1}{2}}(x),$ (D.37a)
$\displaystyle\Big{(}2\mu\partial_{x}-\frac{2\mu^{2}}{x}+\beta\Big{)}\mathcal{I}_{\beta,\mu-\frac{1}{2}}(x)$
$\displaystyle=(\mu^{2}-\beta^{2})\mathcal{I}_{\beta,\mu+\frac{1}{2}}(x);$
(D.37b)
$\displaystyle\Big{(}2\mu\partial_{x}+\frac{2\mu^{2}}{x}-\beta\Big{)}\mathcal{K}_{\beta,\mu+\frac{1}{2}}(x)$
$\displaystyle=-(\mu+\beta)\mathcal{K}_{\beta,\mu-\frac{1}{2}}(x),$ (D.38a)
$\displaystyle\Big{(}2\mu\partial_{x}-\frac{2\mu^{2}}{x}+\beta\Big{)}\mathcal{K}_{\beta,\mu-\frac{1}{2}}(x)$
$\displaystyle=(-\mu+\beta)\mathcal{K}_{\beta,\mu+\frac{1}{2}}(x).$ (D.38b)

###### Proof.

We first rewrite (D.35a)-(D.35d) and (D.36a)-(D.36d) as follows:

$\displaystyle\Big{(}\sqrt{z}\partial_{z}-\frac{\mu}{\sqrt{z}}-\frac{\sqrt{z}}{2}\Big{)}\mathcal{I}_{\beta,\mu-\frac{1}{2}}(z)$
$\displaystyle=\Big{(}-\mu-\beta\Big{)}\mathcal{I}_{\beta+\frac{1}{2},\mu}(z),$
(D.39a)
$\displaystyle\Big{(}\sqrt{z}\partial_{z}+\frac{\mu}{\sqrt{z}}+\frac{\sqrt{z}}{2}\Big{)}\mathcal{I}_{\beta,\mu+\frac{1}{2}}(z)$
$\displaystyle=\mathcal{I}_{\beta-\frac{1}{2},\mu}(z),$ (D.39b)
$\displaystyle\Big{(}\sqrt{z}\partial_{z}+\frac{\mu}{\sqrt{z}}-\frac{\sqrt{z}}{2}\Big{)}\mathcal{I}_{\beta,\mu+\frac{1}{2}}(z)$
$\displaystyle=\mathcal{I}_{\beta+\frac{1}{2},\mu}(z),$ (D.39c)
$\displaystyle\Big{(}\sqrt{z}\partial_{z}-\frac{\mu}{\sqrt{z}}+\frac{\sqrt{z}}{2}\Big{)}\mathcal{I}_{\beta,\mu-\frac{1}{2}}(z)$
$\displaystyle=\Big{(}\mu-\beta\Big{)}\mathcal{I}_{\beta-\frac{1}{2},\mu}(z);$
(D.39d)
$\displaystyle\Big{(}\sqrt{z}\partial_{z}-\frac{\mu}{\sqrt{z}}-\frac{\sqrt{z}}{2}\Big{)}\mathcal{K}_{\beta,\mu-\frac{1}{2}}(z)$
$\displaystyle=-\mathcal{K}_{\beta+\frac{1}{2},\mu}(z),$ (D.40a)
$\displaystyle\Big{(}\sqrt{z}\partial_{z}+\frac{\mu}{\sqrt{z}}+\frac{\sqrt{z}}{2}\Big{)}\mathcal{K}_{\beta,\mu+\frac{1}{2}}(z)$
$\displaystyle=\Big{(}\mu+\beta\Big{)}\mathcal{K}_{\beta-\frac{1}{2},\mu}(z),$
(D.40b)
$\displaystyle\Big{(}\sqrt{z}\partial_{z}+\frac{\mu}{\sqrt{z}}-\frac{\sqrt{z}}{2}\Big{)}\mathcal{K}_{\beta,\mu+\frac{1}{2}}(z)$
$\displaystyle=-\mathcal{K}_{\beta+\frac{1}{2},\mu}(z),$ (D.40c)
$\displaystyle\Big{(}\sqrt{z}\partial_{z}-\frac{\mu}{\sqrt{z}}+\frac{\sqrt{z}}{2}\Big{)}\mathcal{K}_{\beta,\mu-\frac{1}{2}}(z)$
$\displaystyle=\Big{(}-\mu+\beta\Big{)}\mathcal{K}_{\beta-\frac{1}{2},\mu}(z).$
(D.40d)

Then we compute

$\displaystyle\frac{1}{\sqrt{z}}\big{(}-\eqref{rei1.}+(-\mu+\beta)\eqref{rei2.}-(\mu+\beta)\eqref{rei3.}+\eqref{rei4.}\big{)},$
(D.41)
$\displaystyle\frac{1}{\sqrt{z}}\big{(}(\mu-\beta)\eqref{rei1.}+(-\mu^{2}+\beta^{2})\eqref{rei2.}+(\mu^{2}-\beta^{2})\eqref{rei3.}+(\mu+\beta)\eqref{rei4.}\big{)},$
(D.42)
$\displaystyle\frac{1}{\sqrt{z}}\big{(}-(\mu+\beta)\eqref{rek1.}+(\mu-\beta)\eqref{rek2.}+(\mu+\beta)\eqref{rek3.}+(\mu+\beta)\eqref{rek4.}\big{)},$
(D.43)
$\displaystyle\frac{1}{\sqrt{z}}\big{(}(\mu-\beta)\eqref{rek1.}+(\mu-\beta)\eqref{rek2.}-(\mu-\beta)\eqref{rek3.}+(\mu+\beta)\eqref{rek4.}\big{)}.$
(D.44)

∎

(D.37) and (D.38) are closely related to the Dirac-Coulomb Hamiltonian. To see
this relation let us introduce $\omega$ satisfying
$\omega^{2}=\mu^{2}-\beta^{2}$. Then (D.37) and (D.38) can be rewritten in the
following form:

$\displaystyle 0=$ $\displaystyle
2\mu\partial_{x}\big{(}\mathcal{I}_{\beta,\mu-\frac{1}{2}}(x)+\mathrm{i}\omega\mathcal{I}_{\beta,\mu+\frac{1}{2}}(x)\big{)}$
(D.45)
$\displaystyle+\Big{(}-\frac{2\mu^{2}}{x}+\beta-\mathrm{i}\omega\Big{)}\big{(}\mathcal{I}_{\beta,\mu-\frac{1}{2}}(x)-\mathrm{i}\omega\mathcal{I}_{\beta,\mu+\frac{1}{2}}(x)\big{)},$
$\displaystyle 0=$ $\displaystyle
2\mu\partial_{x}\big{(}\mathcal{I}_{\beta,\mu-\frac{1}{2}}(x)-\mathrm{i}\omega\mathcal{I}_{\beta,\mu+\frac{1}{2}}(x)\big{)}$
(D.46)
$\displaystyle+\Big{(}-\frac{2\mu^{2}}{x}+\beta+\mathrm{i}\omega\Big{)}\big{(}\mathcal{I}_{\beta,\mu-\frac{1}{2}}(x)+\mathrm{i}\omega\mathcal{I}_{\beta,\mu+\frac{1}{2}}(x)\big{)};$
$\displaystyle 0=$ $\displaystyle
2\mu\partial_{x}\big{(}(\mu+\beta)\mathcal{K}_{\beta,\mu-\frac{1}{2}}(x)-\mathrm{i}\omega\mathcal{K}_{\beta,\mu+\frac{1}{2}}(x)\big{)}$
(D.47)
$\displaystyle+\Big{(}-\frac{2\mu^{2}}{x}+\beta-\mathrm{i}\omega\Big{)}\big{(}(\mu+\beta)\mathcal{K}_{\beta,\mu-\frac{1}{2}}(x)+\mathrm{i}\omega\mathcal{K}_{\beta,\mu+\frac{1}{2}}(x)\big{)},$
$\displaystyle 0=$ $\displaystyle
2\mu\partial_{x}\big{(}(\mu+\beta)\mathcal{K}_{\beta,\mu-\frac{1}{2}}(x)+\mathrm{i}\omega\mathcal{K}_{\beta,\mu+\frac{1}{2}}(x)\big{)}$
(D.48)
$\displaystyle+\Big{(}-\frac{2\mu^{2}}{x}+\beta+\mathrm{i}\omega\Big{)}\big{(}(\mu+\beta)\mathcal{K}_{\beta,\mu-\frac{1}{2}}(x)-\mathrm{i}\omega\mathcal{K}_{\beta,\mu+\frac{1}{2}}(x)\big{)}.$

The eigenvalue equations for $\xi_{p}^{\pm}$ and $\zeta_{p}^{\pm}$ follow
directly from these identities.

###  D.5 Integral transforms

Let us compute a useful integral transform of the confluent function:

######  Lemma 72.

Assuming $\mathrm{Re}(b)>0$ and $|\mathrm{Re}(w)|<\mathrm{Re}(z)$, one has

$\displaystyle\int_{0}^{\infty}x^{b-1}\mathrm{e}^{-zx}{}_{1}\mathbf{F}_{1}(a;c;wx)\mathrm{d}x=z^{-b}\Gamma(b){}_{2}\mathbf{F}_{1}(a,b;c;z^{-1}w).$
(D.49)

Furthermore, if $\mathrm{Re}(b)>0$, $\mathrm{Re}(b+1-c)>0$,
$\mathrm{Re}(z)>0$, $w,z^{-1}w\notin]-\infty,0]$, then

$\displaystyle\int_{0}^{\infty}x^{b-1}\mathrm{e}^{-zx}{}_{2}F_{0}\big{(}a,a+1-c;-;-(wx)^{-1}\big{)}(wx)^{-a}\mathrm{d}x$
(D.50) $\displaystyle=$ $\displaystyle
z^{-b}\Gamma(b)\Gamma(1+b-c){}_{2}\mathbf{F}_{1}(a,b;a+b+1-c;1-z^{-1}w).$

###### Proof.

We expand the confluent function in a power series and integrate term by term:

$\displaystyle\int_{0}^{\infty}\sum_{n=0}^{\infty}\frac{(a)_{n}}{\Gamma(c+n)}\mathrm{e}^{-zx}x^{b+n-1}w^{n}\mathrm{d}x$
$\displaystyle=\sum_{n=0}^{\infty}\frac{\Gamma(b+n)(a)_{n}}{\Gamma(c+n)}\frac{w^{n}}{z^{n+b}}=z^{-b}\Gamma(b)\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{\Gamma(c+n)}\left(\frac{w}{z}\right)^{n}.$
(D.51)

This proves the first identity under additional assumption
$|w|<\mathrm{Re}(z)$. The integrand can be majorized by an integrable function
for $|\mathrm{Re}(w)|<\mathrm{Re}(z)$. Therefore, we can extend the identity
by analytic continuation to this domain, yielding (D.49).

(D.49), (D.6) and (D.7) and analytic continuation imply (D.50). ∎

The following identity, valid for $v>0$, $\mathrm{Re}(\epsilon)>0$,
$\mathrm{Re}(m+1-\mathrm{i}s)>0$, follows from (D.49):

$\displaystyle\int_{0}^{\infty}\mathrm{e}^{-\epsilon
x}x^{-\frac{1}{2}-\mathrm{i}s}\mathcal{J}_{\beta,m}(vx)\mathrm{d}x$ (D.52)
$\displaystyle=$ $\displaystyle
v^{m+\frac{1}{2}}\Big{(}\epsilon\pm\mathrm{i}\frac{v}{2}\Big{)}^{-m-1+\mathrm{i}s}\Gamma(m+1-\mathrm{i}s){}_{2}\mathbf{F}_{1}\left(m+\frac{1}{2}\pm\mathrm{i}\beta,m+1-\mathrm{i}s;2m+1;\frac{v}{\frac{v}{2}\mp\mathrm{i}\epsilon}\right).$

######  Proposition 73.

Let $v>0$, $\mathrm{Re}(m)>-1$. Then
$x\mapsto\mathrm{e}^{-0x}\mathcal{J}_{\beta,m}(vx):=\lim\limits_{\epsilon\downarrow
0}\mathrm{e}^{-\epsilon x}\mathcal{J}_{\beta,m}(vx)$ is a tempered
distribution on $\mathbb{R}_{+}$ with the Mellin transform

$\displaystyle\int_{0}^{\infty}\mathrm{e}^{-0x}x^{-\frac{1}{2}-\mathrm{i}s}\mathcal{J}_{\beta,m}(vx)\mathrm{d}x$
(D.53) $\displaystyle=$ $\displaystyle
v^{-\frac{1}{2}+\mathrm{i}s}2^{m+1-\mathrm{i}s}(\pm\mathrm{i})^{-m-1+\mathrm{i}s}\Gamma(m+1-\mathrm{i}s){}_{2}\mathbf{F}_{1}\left(m+\frac{1}{2}\pm\mathrm{i}\beta,m+1-\mathrm{i}s;2m+1;2\pm\mathrm{i}0\right),$

which is bounded by
$c_{\beta,m}\left(\mathrm{e}^{-\frac{\pi}{2}(|s|+s)}|s|^{\mathrm{Im}(\beta)}+\mathrm{e}^{-\frac{\pi}{2}(|s|-s)}|s|^{-\mathrm{Im}(\beta)}\right)$,
with $c_{\beta,m}$ a locally bounded function of $\beta,m$.

Similarly, for any $\mu$ (including $\mu=0$),
$x\mapsto\mathrm{e}^{-0x}\frac{1}{\mu}\big{(}\mathcal{J}_{\beta,\mu-\frac{1}{2}}(vx)+\beta\mathcal{J}_{\beta,\mu+\frac{1}{2}}(vx)\big{)}$
is a tempered distribution on $\mathbb{R}_{+}$, whose Mellin transform can be
computed from (D.53) and is bounded by
$c_{\beta,\mu}\left(\mathrm{e}^{-\frac{\pi}{2}(|s|+s)}|s|^{\mathrm{Im}(\beta)}+\mathrm{e}^{-\frac{\pi}{2}(|s|-s)}|s|^{-\mathrm{Im}(\beta)}\right)$,
with $c_{\beta,\mu}$ a locally bounded function of $\beta,\mu$.

###### Proof.

We use the criterion from Lemma 67. Let $f_{\epsilon}(x)=\mathrm{e}^{-\epsilon
x}\mathcal{J}_{\beta,m}(vx)$. Then
$t\mapsto\mathrm{e}^{\frac{t}{2}}f_{\epsilon}(\mathrm{e}^{t})$ is smooth and
vanishes exponentially for $t\to-\infty$ and superexponentially for
$t\to\infty$. In particular $f_{\epsilon}$ is a tempered distribution on
$\mathbb{R}_{+}$. Its Mellin transform is given by the absolutely convergent
integral (D.52). It is a smooth function with smooth pointwise limit
$\epsilon\to 0$. Required bounds follow from (D.10). This completes the proof
of the first part.

Next, we compute

$\displaystyle\int_{0}^{\infty}\mathrm{e}^{-\epsilon
x}x^{-\frac{1}{2}-\mathrm{i}s}\frac{1}{\mu}\left(\mathcal{J}_{\beta,\mu-\frac{1}{2}}(vx)+\beta\mathcal{J}_{\beta,\mu+\frac{1}{2}}(vx)\right)\mathrm{d}x$
(D.54) $\displaystyle=$ $\displaystyle
v^{\mu}\left(\epsilon\pm\mathrm{i}\frac{v}{2}\right)^{-\mu-\frac{1}{2}+\mathrm{i}s}\Gamma\left(\frac{1}{2}+\mu-\mathrm{i}s\right)$
$\displaystyle\times$
$\displaystyle\frac{1}{\mu}\Bigg{(}{}_{2}\mathbf{F}_{1}\left(\mu+\mathrm{i}\beta,\frac{1}{2}+\mu-\mathrm{i}s;2\mu;\frac{v}{\frac{v}{2}-\mathrm{i}\epsilon}\right)-\mathrm{i}\beta\left(\frac{1}{2}+\mu-\mathrm{i}s\right)\frac{v}{\frac{v}{2}-\mathrm{i}\epsilon}{}_{2}\mathbf{F}_{1}\left(1+\mu+\mathrm{i}\beta,\frac{3}{2}+\mu-\mathrm{i}s;2\mu+2;\frac{v}{\frac{v}{2}-\mathrm{i}\epsilon}\right)\Bigg{)}.$

Expression in the last line is nonsingular for $\mu\to 0$ on the account of
(D.5). Bounds on the growth at infinity are derived as in the first case. ∎

Acknowledgement. The work of J.D. was supported by National Science Center
(Poland) under the grant UMO-2019/35/B/ST1/01651.

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