# On the single layer boundary integral operator for the Dirac equation

Markus Holzmann  Institut für Angewandte Mathematik
Technische Universität Graz
Steyrergasse 30
8010 Graz
Austria<EMAIL_ADDRESS>
###### Abstract.

This paper is devoted to the analysis of the single layer boundary integral
operator $\mathcal{C}_{z}$ for the Dirac equation in the two- and three-
dimensional situation. The map $\mathcal{C}_{z}$ is the strongly singular
integral operator having the integral kernel of the resolvent of the free
Dirac operator $A_{0}$ and $z$ belongs to the resolvent set of $A_{0}$. In the
case of smooth boundaries fine mapping properties and a decomposition of
$\mathcal{C}_{z}$ in a ’positive’ and ’negative’ part are analyzed. The
obtained results can be applied in the treatment of Dirac operators with
singular electrostatic, Lorentz scalar, and anomalous magnetic interactions
that are combined in a critical way.

##  1\. Introduction

In the analysis of boundary value and transmission problems for partial
differential equations associated potential and boundary integral operators
often play an important role. These objects are well understood for uniformly
elliptic second order differential expressions, cf. the monograph [19] and the
references therein. We remark that the transmission problems are closely
related to differential operators with singular interactions like, e.g.,
$\delta$-potentials. In the recent years it turned out that similar objects
are also of importance in the study of boundary value and transmission
problems for the Dirac equation, which is the relativistic counterpart of the
Laplace equation and for which the associated differential expression is of
first order and lacks ellipticity. While the potential operator is
sufficiently well investigated, the subtle examination of the associated
boundary integral operator is less complete, see the review paper [10] and the
references therein. It is the main goal in this article to make some further
contributions to this study that are, in particular, necessary to examine
Dirac operators with singular $\delta$-potentials, where the involved
parameters are combined in a critical way.

To introduce the problem setting in a more detailed way, consider first a
formally symmetric and strongly elliptic second order partial differential
operator $\mathcal{P}$ in $\mathbb{R}^{q}$, $q\geq 2$, let $P_{0}$ be the
self-adjoint realization of $\mathcal{P}$ defined on $H^{2}(\mathbb{R}^{q})$,
and let $E_{z}(x,y)$ be the integral kernel of $(P_{0}-z)^{-1}$,
$z\in\rho(P_{0})$. Moreover, let $\Omega\subset\mathbb{R}^{q}$ be a
sufficiently smooth domain with unit normal vector field $\nu$ that is
pointing outwards of $\Omega$ and let $\mathcal{B}_{\nu}$ be the conormal
derivative at $\partial\Omega$ associated with $\mathcal{P}$. Then, the single
layer potential $SL(z)$ and the double layer potential $DL(z)$ applied to a
sufficiently smooth function $\varphi:\partial\Omega\rightarrow\mathbb{C}$
evaluated at $x\in\mathbb{R}^{q}\setminus\partial\Omega$ are

$\begin{split}SL(z)\varphi(x)&=\int_{\partial\Omega}E_{z}(x,y)\varphi(y)\textup{d}\sigma(y),\quad
DL(z)\varphi(x)=\int_{\partial\Omega}\partial_{\mathcal{B}_{\nu},y}E_{z}(x,y)\varphi(y)\textup{d}\sigma(y).\end{split}$

It is known that all solutions of the partial differential equation

(1.1)
$(\mathcal{P}-z)f=g\quad\text{in}\quad\mathbb{R}^{q}\setminus\partial\Omega$

can be described with the help of $SL(z)$ and $DL(z)$. To analyze boundary
value or transmission problems associated with the above equation, it is
common to employ the Dirichlet trace operator $\gamma_{D}$ and
$\mathcal{B}_{\nu}$ to $SL(z)$ and $DL(z)$. Of particular interest is the
single layer boundary integral operator $\mathcal{S}(z):=\gamma_{D}SL(z)$. Its
properties are closely related to the solvability of the Dirichlet boundary
value problem for (1.1), and relevant properties, as, e.g., its ellipticity,
are inherited from those of $\mathcal{P}$, cf. [19, Chapter 7]. Moreover, a
good understanding of $\mathcal{S}$ can be useful for a thorough understanding
of different problems related to $\mathcal{P}$ in an operator theoretic
language, for instance to show the self-adjointness and compute spectral
properties of operators associated with boundary value or transmission
problems for $\mathcal{P}$ including perturbations of $P_{0}$ by singular
interactions, cf. [11, 16, 17, 19].

It is the main goal in this paper to provide some properties of the
counterpart of the single layer boundary integral operator for the Dirac
operator in the two- and three-dimensional situation. For $m\geq 0$ the free
Dirac operator in dimension two is given by

(1.2)
$A_{0}f=-i\sigma_{1}\partial_{1}f-i\sigma_{2}\partial_{2}f+m\sigma_{3}f,\quad{\rm
dom\,}A_{0}=H^{1}(\mathbb{R}^{2};\mathbb{C}^{2}),$

where $\sigma_{1},\sigma_{2},\sigma_{3}\in\mathbb{C}^{2\times 2}$ are the
Pauli spin matrices defined in (2.1), and in dimension three it is

(1.3)
$A_{0}f=-i\alpha_{1}\partial_{1}f-i\alpha_{2}\partial_{2}f-i\alpha_{3}\partial_{3}f+m\beta
f,\quad{\rm dom\,}A_{0}=H^{1}(\mathbb{R}^{3};\mathbb{C}^{4}),$

where $\alpha_{1},\alpha_{2},\alpha_{3},\beta\in\mathbb{C}^{4\times 4}$ are
the Dirac matrices in (2.2). Here, we used the notation
$H^{k}(\mathbb{R}^{q};\mathbb{C}^{l})=H^{k}(\mathbb{R}^{q})\otimes\mathbb{C}^{l}$
for the $L^{2}$-based Sobolev space of $k$ times weakly differentiable vector
valued functions. The free Dirac operator is used to describe the propagation
of a spin $\frac{1}{2}$ particle in vacuum taking effects of the special
theory of relativity into account. Moreover, the two-dimensional Dirac
operator appears in the mathematical description of graphene [1]. It is known
that $A_{0}$ is self-adjoint and that its spectrum is

$\sigma(A_{0})=(-\infty,-m]\cup[m,\infty),$

cf. [23]. This shows, in particular, that $A_{0}$ is not semi-bounded from
above or below. For
$z\in\rho(A_{0})=\mathbb{C}\setminus((-\infty,-m]\cup[m,\infty))$ the
resolvent of $A_{0}$ can be expressed via the convolution with a function
$G_{z,q}$ given in (2.3) below, i.e. as an integral operator. With the help of
this function we can formally introduce for a smooth and closed curve
$\Sigma\subset\mathbb{R}^{2}$ or a smooth and closed surface
$\Sigma\subset\mathbb{R}^{3}$ the boundary integral operator acting on
sufficiently smooth functions $\varphi:\Sigma\rightarrow\mathbb{C}^{N}$ as

(1.4) $\mathcal{C}_{z}\varphi(x):=\lim_{\varepsilon\searrow
0}\int_{\Sigma\setminus
B(x,\varepsilon)}G_{z,q}(x-y)\varphi(y)\textup{d}\sigma(y),\quad x\in\Sigma,$

where $B(x,\varepsilon)$ is the ball centered at $x$ with radius $\varepsilon$
and $N=2$ for $q=2$ and $N=4$ for $q=3$. It is known that $\mathcal{C}_{z}$
gives rise to a bounded operator in $L^{2}(\Sigma;\mathbb{C}^{N})$, but in a
similar way as $A_{0}$ also $\mathcal{C}_{z}$ lacks ellipticity [2, 4, 9].

The operator $\mathcal{C}_{z}$ plays an important role in the analysis of
boundary value and transmission problems for the Dirac equation and motivated
by this $\mathcal{C}_{z}$ was studied intensively in the recent years, cf. [2,
5, 8, 9, 10, 12, 13, 14, 21]. In this paper this study is continued and, in
particular, detailed mapping properties and a more detailed analysis of the
contribution of $\mathcal{C}_{z}$ associated with the ’positive’ and the
’negative’ part of its spectrum are provided. As it is shown in the very
recent paper [10] this is necessary to understand Dirac operators perturbed by
singular potentials involving electrostatic, Lorentz scalar, and anomalous
magnetic interactions combined in a critical way. However, we strongly believe
that the results obtained in this paper are also of independent interest and
may be applied in different situations than in [10].

It should be remarked that the above mentioned results are proved in a
different way in dimension two and three. In dimension two $\mathcal{C}_{z}$
is closely related to the Cauchy transform on $\Sigma$, for its analysis we
follow ideas from [9] and apply the theory of periodic pseudodifferential
operators. In dimension three $\mathcal{C}_{z}$ is closely related to the
Riesz transform on $\Sigma$ and results on pseudo-homogeneous kernels from
[20] are employed to show similar results as in dimension two. In particular,
the latter approach would allow to weaken the geometric assumptions on
$\Sigma$ and to consider also higher space dimensions. We remark that in the
recent paper [15], which was written independently of this article, the
operator $\mathcal{C}_{z}$ is studied in the three-dimensional situation with
the help of pseudodifferential techniques and results that are related to the
ones in this paper are recovered there as well.

The paper is organized as follows. In Section 2 we rigorously introduce the
operator $\mathcal{C}_{z}$ formally given by (1.4) and discuss its basic
properties. Then, in Section 3 we recall some results about periodic
pseudodifferential operators and do a more detailed analysis of
$\mathcal{C}_{z}$ in the two-dimensional case. Finally, in Section 4 we revise
basic notions and results on pseudo-homogeneous kernels and use them to
investigate $\mathcal{C}_{z}$ in dimension three.

### Acknowledgement.

The author thanks Jussi Behrndt, Dale Frymark, Christian Stelzer, and Georg
Stenzel for fruitful discussions. Moreover, he gratefully acknowledges
financial support by the Austrian Science Fund (FWF): P33568-N. This
publication is based upon work from COST Action CA 18232 MAT-DYN-NET,
supported by COST (European Cooperation in Science and Technology),
www.cost.eu.

##  2\. Notations and basic properties of $\mathcal{C}_{z}$

Throughout this paper, let $q\in\\{2,3\\}$ be the space dimension and define
the number $N=N(q)$ by $N(2)=2$ and $N(3)=4$. Moreover, we assume that
$\Omega\subset\mathbb{R}^{q}$ is a bounded simply connected domain with
$C^{\infty}$-smooth boundary $\Sigma:=\partial\Omega$ and unit normal vector
field $\nu$ which is pointing outwards of $\Omega$. Let

(2.1) $\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right),\quad\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\\ i&0\\\
\end{array}\right),\quad\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\ 0&-1\\\
\end{array}\right)$

be the Pauli spin matrices and define the $4\times 4$ Dirac matrices by

(2.2) $\alpha_{j}=\left(\begin{array}[]{cc}0&\sigma_{j}\\\ \sigma_{j}&0\\\
\end{array}\right),\quad
j\in\\{1,2,3\\},\quad\beta=\left(\begin{array}[]{cc}I_{2}&0\\\ 0&-I_{2}\\\
\end{array}\right),$

where $I_{n}$ is the $n\times n$-identity matrix. We will often use for
$x=(x_{1},x_{2})\in\mathbb{C}^{2}$ the notation

$\sigma\cdot x=\sigma_{1}x_{1}+\sigma_{2}x_{2}$

and for $x=(x_{1},x_{2},x_{3})\in\mathbb{C}^{3}$

$\alpha\cdot
x=\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}\quad\text{and}\quad\sigma\cdot
x=\sigma_{1}x_{1}+\sigma_{2}x_{2}+\sigma_{3}x_{3}.$

Next, we introduce the function $G_{z,q}$ evaluated at
$x\in\mathbb{R}^{q}\setminus\\{0\\}$ by

(2.3)
$\begin{split}G_{z,2}(x)&=\frac{\sqrt{z^{2}-m^{2}}}{2\pi}K_{1}\big{(}-i\sqrt{z^{2}-m^{2}}|x|\big{)}\frac{(\sigma\cdot
x)}{|x|}\\\
&\qquad\qquad\qquad+\frac{1}{2\pi}K_{0}\big{(}-i\sqrt{z^{2}-m^{2}}|x|\big{)}\big{(}zI_{2}+m\sigma_{3}\big{)},\\\
G_{z,3}(x)&=\left(zI_{4}+m\beta+\left(1-i\sqrt{z^{2}-m^{2}}|x|\right)\frac{i(\alpha\cdot
x)}{|x|^{2}}\right)\frac{1}{4\pi|x|}e^{i\sqrt{z^{2}-m^{2}}|x|},\end{split}$

where we write $K_{j}$ for the modified Bessel functions of the second kind
and choose $\sqrt{w}$ for $w\in\mathbb{C}\setminus[0,\infty)$ such that
$\textup{Im}\sqrt{w}>0$. It is well-known that $G_{z,q}$ is the integral
kernel of the resolvent of the free Dirac operator $A_{0}$ in (1.2) & (1.3);
cf. [9, 10, 23].

Now, we are prepared to rigorously introduce and discuss the properties of the
operator $\mathcal{C}_{z}$, $z\in\rho(A_{0})=(-\infty,-m]\cup[m,\infty)$,
formally given by (1.4), i.e. we consider now the strongly singular integral
operator

(2.4) $\mathcal{C}_{z}\varphi(x):=\lim_{\varepsilon\searrow
0}\int_{\Sigma\setminus
B(x,\varepsilon)}G_{z,q}(x-y)\varphi(y)\textup{d}\sigma(y),\quad\varphi\in
C^{\infty}(\Sigma;\mathbb{C}^{N}),~{}x\in\Sigma;$

here $B(x,\varepsilon)$ is the ball of radius $\varepsilon$ centered at $x$.
The basic properties of $\mathcal{C}_{z}$ are summarized in the following
proposition; in the proof we mostly refer to [10], but parts of the results
were shown before in [9] in dimension $q=2$ and in [2, 3, 7, 8, 14, 21] in
dimension $q=3$. Below, $H^{s}(\Sigma)$, $s\in\mathbb{R}$, are the Sobolev
spaces on $\Sigma$ defined as in [19] and we denote by
$(\cdot,\cdot)_{H^{s}(\Sigma;\mathbb{C}^{N})\times
H^{-s}(\Sigma;\mathbb{C}^{N})}$ the sesquilinear duality product in
$H^{s}(\Sigma;\mathbb{C}^{N})\times H^{-s}(\Sigma;\mathbb{C}^{N})$.

######  Proposition 2.1.

For the operator $\mathcal{C}_{z}$,
$z\in\rho(A_{0})=\mathbb{C}\setminus((-\infty,-m]\cup[m,\infty))$, introduced
in (2.4) and any $s\in\mathbb{R}$ the following holds:

  * (i)

The map $\mathcal{C}_{z}$ gives rise to a bounded operator in
$H^{s}(\Sigma;\mathbb{C}^{N})$.

  * (ii)

For $\varphi\in H^{s}(\Sigma;\mathbb{C}^{N})$ and $\psi\in
H^{-s}(\Sigma;\mathbb{C}^{N})$ one has

$(\mathcal{C}_{z}\varphi,\psi)_{H^{s}(\Sigma;\mathbb{C}^{N})\times
H^{-s}(\Sigma;\mathbb{C}^{N})}=(\varphi,\mathcal{C}_{\overline{z}}\psi)_{H^{s}(\Sigma;\mathbb{C}^{N})\times
H^{-s}(\Sigma;\mathbb{C}^{N})}.$

In particular, for the realization of $\mathcal{C}_{z}$ in
$L^{2}(\Sigma;\mathbb{C}^{N})$ the relation
$\mathcal{C}_{z}^{*}=\mathcal{C}_{\overline{z}}$ holds.

  * (iii)

For $q=2$, one has
$-4(\mathcal{C}_{z}(\sigma\cdot\nu))^{2}=-4((\sigma\cdot\nu)\mathcal{C}_{z})^{2}=I_{2}$
and for $q=3$ the relation
$-4(\mathcal{C}_{z}(\alpha\cdot\nu))^{2}=-4((\alpha\cdot\nu)\mathcal{C}_{z})^{2}=I_{4}$
holds.

###### Proof.

The mapping properties of $\mathcal{C}_{z}$ in item (i) follow from [9,
Proposition 3.3] for $q=2$ and [14, Theorem 4.1] for $q=3$. The claim in
statement (ii) is shown for $s\in[-\frac{1}{2},\frac{1}{2}]$ in [10,
Proposition 4.4] taking into account that the spaces
$H^{s}_{\alpha}(\Sigma;\mathbb{C}^{N})$ in [10] coincide for
$C^{\infty}$-smooth $\Sigma$ with $H^{s}(\Sigma;\mathbb{C}^{N})$. Together
with this a simple density and continuity argument yields the claim for
$s\notin[-\frac{1}{2},\frac{1}{2}]$. Similarly, assertion (iii) can be found
for $s\in[-\frac{1}{2},\frac{1}{2}]$ in [10, equations (4.6) and (4.24)], the
statement for general $s\in\mathbb{R}$ follows then by restriction for
$s>\frac{1}{2}$ and duality for $s<-\frac{1}{2}$. ∎

##  3\. A more detailed analysis in dimension $q=2$

In this section we investigate $\mathcal{C}_{z}$ in space dimension $q=2$ in a
more detailed way. For this, we make use of the theory of periodic
pseudodifferential operators. Their definition and some basic properties are
recalled in Subsection 3.1. With the help of these results we analyze
$\mathcal{C}_{z}$ in Subsection 3.2.

###  3.1. Periodic pseudodifferential operators

In this subsection we follow closely the short exposition on periodic
pseudodifferential operators in [9, Section 2.1], for a more comprehensive
presentation see [22]. In order to define periodic pseudodifferential
operators, some notations are necessary. We set
$\mathbb{T}:=\mathbb{R}/\mathbb{Z}$ and denote by $\mathcal{D}(\mathbb{T})$
the set of all $1$-periodic test functions and by
$\mathcal{D}^{\prime}(\mathbb{T})$ the set of all $1$-periodic distributions.
We will often make use of the functions $e_{n}\in\mathcal{D}(\mathbb{T})$
defined by

(3.1) $e_{n}(t):=e^{2\pi int},\qquad n\in\mathbb{Z}.$

For $f\in\mathcal{D}^{\prime}(\mathbb{T})$ we introduce the Fourier
coefficients by

(3.2) $\widehat{f}(n):=\langle
f,e_{-n}\rangle_{\mathcal{D}^{\prime}(\mathbb{T})\times\mathcal{D}(\mathbb{T})},$

where
$\langle\cdot,\cdot\rangle_{\mathcal{D}^{\prime}(\mathbb{T})\times\mathcal{D}(\mathbb{T})}$
denotes the bilinear duality product in
$\mathcal{D}^{\prime}(\mathbb{T})\times\mathcal{D}(\mathbb{T})$. Set

$\underline{n}:=\begin{cases}1,&\text{ if }n=0,\\\ |n|,&\text{ if
}n\in\mathbb{Z}\setminus\\{0\\}.\end{cases}$

With the help of the Fourier coefficients one can define for $s\in\mathbb{R}$
the Sobolev space $H^{s}(\mathbb{T})$ of order $s$ on $\mathbb{T}$ by

(3.3)
$H^{s}(\mathbb{T}):=\left\\{f\in\mathcal{D}^{\prime}(\mathbb{T}):\sum_{n\in\mathbb{Z}}\underline{n}^{2s}|\widehat{f}(n)|^{2}<\infty\right\\}$

and endow it with the natural inner product

$(f,g)_{H^{s}(\mathbb{T})}:=\sum_{n\in\mathbb{Z}}\underline{n}^{2s}\widehat{f}(n)\overline{\widehat{g}(n)},\qquad
u,v\in H^{s}(\mathbb{T}).$

The Fourier coefficients in (3.2) allow us also to establish periodic
pseudodifferential operators.

######  Definition 3.1.

A linear map $A$ defined on $\mathcal{D}(\mathbb{T})$ is called periodic
pseudodifferential operator of order $s\in\mathbb{R}$, if there exists a
function $h:\mathbb{T}\times\mathbb{Z}\rightarrow\mathbb{C}$ such that the
following holds:

  * (i)

For any fixed $n\in\mathbb{Z}$ one has $h(\cdot,n)\in\mathcal{D}(\mathbb{T})$.

  * (ii)

The action of $A$ is given by

$Af=\sum_{n\in\mathbb{Z}}h(\cdot,n)\widehat{f}(n)e_{n}$

with the sum converging in $\mathcal{D}^{\prime}(\mathbb{T})$.

  * (iii)

For all $p,q\in\mathbb{N}_{0}$ there exists $c_{p,q}>0$ such that for all
$n\in\mathbb{Z}$ and $t\in\mathbb{R}$

$\left|\left(\frac{\textup{d}^{p}}{\textup{d}t^{p}}\omega^{q}h\right)(t,n)\right|\leq
c_{p,q}(1+|n|)^{s-q}$

holds, where $\omega h(t,n):=h(t,n+1)-h(t,n)$.

The set of all periodic pseudodifferential operators of order $s$ is denoted
by $\Psi^{s}$. Moreover, we set
$\Psi^{-\infty}:=\bigcap_{s\in\mathbb{R}}\Psi^{s}$.

In the following let $\Sigma\subset\mathbb{R}^{2}$ be the boundary of a simply
connected $C^{\infty}$-domain, let $\ell$ be the length of $\Sigma$, and let
$\gamma:[0,\ell]\rightarrow\mathbb{R}^{2}$ be an arc-length parametrization of
$\Sigma$ that is orientated in the counter clockwise way. Similarly as above,
we denote by $\mathcal{D}(\Sigma)$ the set of all $C^{\infty}$-smooth
functions defined on $\Sigma$ and by $\mathcal{D}^{\prime}(\Sigma)$ the set of
all distributions on $\Sigma$. In order to define periodic pseudodifferential
operators on $\Sigma$, we introduce the mapping
$U:\mathcal{D}^{\prime}(\Sigma)\rightarrow\mathcal{D}^{\prime}(\mathbb{T})$ by

(3.4)
$Uf(\varphi):=f\big{(}\ell^{-1}\varphi(\ell^{-1}\gamma^{-1}(\cdot))\big{)},\qquad
f\in\mathcal{D}^{\prime}(\Sigma),~{}\varphi\in\mathcal{D}(\mathbb{T}).$

If $f$ is a regular distribution, then $Uf$ is also a regular distribution
generated by $Uf=f(\gamma(\ell\cdot))$. With the map $U$ one can translate the
Sobolev spaces $H^{s}(\mathbb{T})$ on $\mathbb{T}$ to Sobolev spaces on
$\Sigma$ by

(3.5)
$H^{s}(\Sigma):=\big{\\{}\varphi\in\mathcal{D}^{\prime}(\Sigma):U\varphi\in
H^{s}(\mathbb{T})\big{\\}},\quad s\in\mathbb{R},$

and endow them with the inner product

$(\varphi,\psi)_{H^{s}(\Sigma)}:=(U\varphi,U\psi)_{H^{s}(\mathbb{T})},\qquad\varphi,\psi\in
H^{s}(\Sigma).$

By this construction the map $U$ defined in (3.4) is for any $s\in\mathbb{R}$
a unitary operator from $H^{s}(\Sigma)$ to $H^{s}(\mathbb{T})$ and the
corresponding norm is equivalent to the norm in $H^{s}(\Sigma)$ defined as in
[19].

######  Definition 3.2.

A map $A$ defined on $\mathcal{D}(\Sigma)$ is called periodic
pseudodifferential operator on $\Sigma$ of order $s\in\mathbb{R}$, if there
exists $A_{0}\in\Psi^{s}$ such that $A=U^{-1}A_{0}U$. The set of all periodic
pseudodifferential operators on $\Sigma$ of order $s$ is denoted by
$\Psi^{s}_{\Sigma}$. Moreover, we set
$\Psi^{-\infty}_{\Sigma}:=\bigcap_{s\in\mathbb{R}}\Psi^{s}_{\Sigma}$.

An important feature of periodic pseudodifferential operators is the fact that
they are automatically bounded in Sobolev spaces [22, Theorems 7.3.1 and
7.8.1].

######  Lemma 3.3.

Let $A\in\Psi^{s}_{\Sigma}$ and $B\in\Psi^{t}_{\Sigma}$ for some
$s,t\in\mathbb{R}$. Then, the following is true:

  * (i)

For any $r\in\mathbb{R}$ the map $A$ can be extended to a well-defined and
bounded operator $A:H^{r}(\Sigma)\rightarrow H^{r-s}(\Sigma)$.

  * (ii)

$AB\in\Psi^{s+t}_{\Sigma}$.

  * (iii)

$AB-BA\in\Psi^{s+t-1}_{\Sigma}$.

There are three types of periodic pseudodifferential operators that we are
going to use frequently. First, if $f\in\mathcal{D}(\Sigma)$, then one has for
the associated multiplication operator

(3.6) $\big{(}\mathcal{D}(\Sigma)\ni\varphi\mapsto
f\cdot\varphi\big{)}\in\Psi_{\Sigma}^{0}.$

Next, consider for a constant $c_{\Lambda}>0$ and $t\in\mathbb{R}$ the map

$L^{t}f:=\left(\frac{4\pi}{\ell}\right)^{t/2}\sum_{n\in\mathbb{Z}}(c_{\Lambda}+|n|)^{t/2}\widehat{f}(n)e_{n},\qquad
f\in\mathcal{D}(\mathbb{T}).$

Then it is not difficult to see that $L^{t_{1}}L^{t_{2}}=L^{t_{1}+t_{2}}$
holds for all $t_{1},t_{2}\in\mathbb{R}$ and $L^{t}\in\Psi^{t/2}$. Hence, we
can define

(3.7) $\Lambda^{t}:=U^{-1}L^{t}U\in\Psi^{t/2}_{\Sigma}.$

Taking the definition of $H^{r}(\Sigma)$ in (3.3) and (3.5) into account it is
not difficult to show that for any $r\in\mathbb{R}$ the map
$\Lambda^{t}:H^{r}(\Sigma)\rightarrow H^{r-t/2}(\Sigma)$ is bijective and that
the realization of $\Lambda^{t}$ as a possibly unbounded operator in
$L^{2}(\Sigma)$ defined on ${\rm dom\,}\Lambda^{t}=H^{\tau}(\Sigma)$ with
$\tau={\rm max\,}\\{\frac{t}{2},0\\}$ is self-adjoint, cf. [9].

The third periodic pseudodifferential operator that will play a crucial role
in our analysis is a multiple of the Cauchy transform on $\Sigma$. We define
formally the integral operator $\mathcal{R}$ acting on
$\varphi\in\mathcal{D}(\Sigma)$ by

(3.8) $\mathcal{R}\varphi(x)=-\frac{2}{\pi}\lim_{\varepsilon\searrow
0}\int_{\Sigma\setminus
B(x,\varepsilon)}\frac{\nu_{1}(y)+i\nu_{2}(y)}{x_{1}+ix_{2}-(y_{1}+iy_{2})}\varphi(y)\textup{d}\sigma(y),\quad
x\in\Sigma.$

We also consider the formal adjoint $\mathcal{R}^{*}$ of $\mathcal{R}$ in
$L^{2}(\Sigma)$, which acts on $\varphi\in\mathcal{D}(\Sigma)$ as

(3.9) $\mathcal{R}^{*}\varphi(x)=\frac{2}{\pi}\lim_{\varepsilon\searrow
0}\int_{\Sigma\setminus
B(x,\varepsilon)}\frac{\nu_{1}(x)-i\nu_{2}(x)}{x_{1}-ix_{2}-(y_{1}-iy_{2})}\varphi(y)\textup{d}\sigma(y),\quad
x\in\Sigma.$

Let $C_{\Sigma}$ be the Cauchy transform on $\Sigma$. As
$\nu(\gamma(s))=(\dot{\gamma}_{2}(s),-\dot{\gamma}_{1}(s))$, $s\in[0,\ell]$,
the map $C_{\Sigma}$ is related to $\mathcal{R}$ by

(3.10) $\mathcal{R}=2C_{\Sigma},$

cf. [22, Section 5.9]. By [9, Proposition 2.8] this implies that

(3.11) $\mathcal{R},\mathcal{R}^{*}\in\Psi_{\Sigma}^{0}.$

In particular, $\mathcal{R},\mathcal{R}^{*}$ give rise to bounded operators in
$H^{s}(\Sigma)$ for all $s\in\mathbb{R}$. Since $\mathcal{R}$ and
$\mathcal{R}^{*}$ are adjoint to each other, a continuity argument shows for
all $\varphi\in H^{s}(\Sigma)$ and $\psi\in H^{-s}(\Sigma)$, $s\in\mathbb{R}$,
that

(3.12) $(\mathcal{R}\varphi,\psi)_{H^{s}(\Sigma)\times
H^{-s}(\Sigma)}=(\varphi,\mathcal{R}^{*}\psi)_{H^{s}(\Sigma)\times
H^{-s}(\Sigma)},$

where $(\cdot,\cdot)_{H^{s}(\Sigma)\times H^{-s}(\Sigma)}$ denotes the
sesquilinear duality product.

###  3.2. Analysis of $\mathcal{C}_{z}$

In the following proposition, which is the starting point for our further
considerations and which is a direct consequence of [9, Proposition 3.3], we
provide a link of $\mathcal{R}$ and $\mathcal{R}^{*}$ defined in (3.8) and
(3.9), respectively, and the operator $\mathcal{C}_{z}$ given in (2.4). Recall
that the map $\Lambda$ is introduced in (3.7).

######  Proposition 3.4.

Let $z\in(-m,m)$. Then there exists an operator
$\mathcal{K}\in\Psi^{-2}_{\Sigma}$ such that

(3.13)
$\mathcal{C}_{z}=\begin{pmatrix}(z+m)\Lambda^{-2}&-\frac{i}{4}\mathcal{R}(\nu_{1}-i\nu_{2})\\\
\frac{i}{4}(\nu_{1}+i\nu_{2})\mathcal{R}^{*}&(z-m)\Lambda^{-2}\end{pmatrix}+\mathcal{K}.$

In particular, for any $r\in\mathbb{R}$ the map $\mathcal{K}$ gives rise to a
compact operator $\mathcal{K}:H^{r}(\Sigma;\mathbb{C}^{2})\rightarrow
H^{r+1}(\Sigma;\mathbb{C}^{2})$ and the realization of $\mathcal{K}$ in
$L^{2}(\Sigma;\mathbb{C}^{2})$ is self-adjoint.

###### Proof.

Denote the operator defined in [9, equation (2.7)] by $\widetilde{\Lambda}$.
Then one easily sees that the map $\Lambda$ in (3.7) satisfies
$\Lambda=\big{(}\frac{4\pi}{\ell}\big{)}^{1/2}\widetilde{\Lambda}$. Moreover,
the tangential vector $(\dot{\gamma}_{1},\dot{\gamma}_{2})$ on $\Sigma$ and
the normal vector are related by
$\nu(\gamma(s))=(\dot{\gamma}_{2}(s),-\dot{\gamma}_{1}(s))$, $s\in[0,\ell]$.
Hence, the representation in (3.13) follows from (3.10) and [9, Proposition
3.3]. Moreover, as $\mathcal{K}:H^{r}(\Sigma;\mathbb{C}^{2})\rightarrow
H^{r+2}(\Sigma;\mathbb{C}^{2})$ is bounded for any $r\in\mathbb{R}$ by Lemma
3.3 and $H^{r+2}(\Sigma;\mathbb{C}^{2})$ is compactly embedded in
$H^{r+1}(\Sigma;\mathbb{C}^{2})$ by Rellich’s embedding theorem, $\mathcal{K}$
is compact from $H^{r}(\Sigma;\mathbb{C}^{2})$ to
$H^{r+1}(\Sigma;\mathbb{C}^{2})$. Eventually, since the realization of
$\mathcal{C}_{z}$ in $L^{2}(\Sigma;\mathbb{C}^{2})$ is self-adjoint for
$z\in(-m,m)$ by Proposition 2.1 (ii), the map $\mathcal{K}$ must be self-
adjoint in $L^{2}(\Sigma;\mathbb{C}^{2})$ as well. Hence, all claims are
shown. ∎

In the following lemma we state a variant of (3.13) that is particularly
useful in the application in the boundary triple framework, as it is done,
e.g., in [10]. Define the matrix $V\in\mathbb{C}^{2\times 2}$ by

(3.14) $V=\begin{pmatrix}1&0\\\ 0&-i(\nu_{1}-i\nu_{2})\end{pmatrix}.$

Then we have the following result:

######  Lemma 3.5.

Let $\mathcal{R}$ and $\mathcal{R}^{*}$ be defined by (3.8) and (3.9),
respectively, let $\mathcal{C}_{z}$, $z\in(-m,m)$, be given by (2.4), let
$\Lambda$ be as in (3.7), and let $s\in\mathbb{R}$. Then there exists an
operator $\mathcal{K}$ that is compact in $H^{s}(\Sigma;\mathbb{C}^{2})$ such
that the bounded and everywhere defined operator

$4\Lambda
V(\sigma\cdot\nu)\mathcal{C}_{z}(\sigma\cdot\nu)V^{*}\Lambda:H^{s}(\Sigma;\mathbb{C}^{2})\rightarrow
H^{s-1}(\Sigma;\mathbb{C}^{2})$

can be written as

(3.15) $4\Lambda
V(\sigma\cdot\nu)\mathcal{C}_{z}(\sigma\cdot\nu)V^{*}\Lambda=-\Lambda\begin{pmatrix}0&\mathcal{R}^{*}\\\
\mathcal{R}&0\end{pmatrix}\Lambda+4\begin{pmatrix}z-m&0\\\
0&z+m\end{pmatrix}+\mathcal{K}.$

In particular, the realization of $\mathcal{K}$ in
$L^{2}(\Sigma;\mathbb{C}^{2})$ is self-adjoint.

###### Proof.

The claim follows immediately from Proposition 3.4 and a direct calculation
taking the identity

$V(\sigma\cdot\nu)=\begin{pmatrix}0&\nu_{1}-i\nu_{2}\\\ -i&0\end{pmatrix},$

Lemma 3.3 (iii), and (3.6) into account. ∎

In the following proposition some further properties of $\mathcal{R}$ and
$\mathcal{R}^{*}$ that follow from known results about the Cauchy transform
(cf. [22, Section 4.1.3]), (3.10), Lemma 3.3, and [9, Proposition 2.8] are
stated.

######  Proposition 3.6.

Let $\mathcal{R}$ and $\mathcal{R}^{*}$ be defined by (3.8) and (3.9),
respectively. Then, the following holds:

  * (i)

$\mathcal{R}-\mathcal{R}^{*}\in\Psi^{-\infty}_{\Sigma}$. In particular, for
any $s,t\in\mathbb{R}$ the map
$\mathcal{R}-\mathcal{R}^{*}:H^{s}(\Sigma)\rightarrow H^{t}(\Sigma)$ is
compact.

  * (ii)

One has $\mathcal{R}^{2}=4I_{2}$ and $(\mathcal{R}^{*})^{2}=4I_{2}$. In
particular, $\mathcal{R}\mathcal{R}^{*}-4I_{2}\in\Psi^{-\infty}_{\Sigma}$ and
for all $s,t\in\mathbb{R}$ the operator
$\mathcal{R}\mathcal{R}^{*}-4I_{2}:H^{s}(\Sigma)\rightarrow H^{t}(\Sigma)$ is
compact.

  * (iii)

$\Lambda^{-2}\mathcal{R}-\mathcal{R}\Lambda^{-2}\in\Psi^{-2}_{\Sigma}$.

Eventually, we state a result on the sum $\mathcal{R}+\mathcal{R}^{*}$, that
will play a crucial role in [10] in the proof of the self-adjointness and
spectral analysis of Dirac operators with singular potentials. In order to
formulate the result, we define the operators $P_{\pm}\in\Psi^{0}_{\Sigma}$ by

(3.16)
$UP_{+}U^{-1}f=\sum_{n=0}^{\infty}\widehat{f}(n)e_{n}\quad\text{and}\quad
UP_{-}U^{-1}f=\sum_{n=1}^{\infty}\widehat{f}(-n)e_{-n},\qquad
f\in\mathcal{D}(\mathbb{T}),$

where $U$ is the map in (3.4). Remark that $P_{\pm}$ are orthogonal
projections in $H^{s}(\Sigma)$ defined by (3.5) for all $s\in\mathbb{R}$ and
with $\mathcal{H}_{\pm}:={\rm ran\,}P_{\pm}$ one has
$H^{s}(\Sigma)=\mathcal{H}_{+}\oplus\mathcal{H}_{-}$, ${\rm
dim\,}\mathcal{H}_{\pm}=\infty$, and $\mathcal{H}_{\pm}\not\subset
H^{r}(\Sigma)$ for any $r>s$. Moreover, $P_{\pm}$ commute with the operator
$\Lambda^{t}$ defined in (3.7).

######  Theorem 3.7.

Let $\mathcal{R}$ and $\mathcal{R}^{*}$ be defined by (3.8) and (3.9),
respectively, and let $P_{\pm}$ be given as above. Then, there exists
$\mathcal{K}\in\Psi^{-\infty}_{\Sigma}$ such that

$\mathcal{R}+\mathcal{R}^{*}=4P_{+}-4P_{-}+\mathcal{K}.$

###### Proof.

Let $\mathfrak{C}\subset\mathbb{R}^{2}$ be the circle of radius $1$ centered
at the origin and let $U_{\mathfrak{C}}$ be the map in (3.4) defined for
$\mathfrak{C}$ instead of $\Sigma$. Then, by [22, Section 5.8] one has for the
Cauchy transform $C_{\mathfrak{C}}$ on $\mathfrak{C}$

$U_{\mathfrak{C}}C_{\mathfrak{C}}U_{\mathfrak{C}}^{-1}=UP_{+}U^{-1}-UP_{-}U^{-1}.$

Moreover, by [22, Section 5.9 and Theorem 7.6.1] there exists
$\mathcal{K}_{1}\in\Psi^{-\infty}$ such that

$UC_{\Sigma}U^{-1}=U_{\mathfrak{C}}C_{\mathfrak{C}}U_{\mathfrak{C}}^{-1}+\mathcal{K}_{1}.$

With (3.10) and the last two displayed formulas one gets

$U\mathcal{R}U^{-1}=2UC_{\Sigma}U^{-1}=2U_{\mathfrak{C}}C_{\mathfrak{C}}U_{\mathfrak{C}}^{-1}+2\mathcal{K}_{1}=2UP_{+}U^{-1}-2UP_{-}U^{-1}+2\mathcal{K}_{1}.$

Similarly, there exists $\mathcal{K}_{2}\in\Psi^{-\infty}$ such that

$U\mathcal{R}^{*}U^{-1}=2UP_{+}U^{-1}-2UP_{-}U^{-1}+\mathcal{K}_{2}.$

By adding the last two displayed formulas one gets the claim. ∎

Finally, combining the results from Lemma 3.5 and Theorem 3.7, we get the
following decomposition of $\mathcal{C}_{z}$ into a positive and a negative
part.

######  Corollary 3.8.

Let $\mathcal{C}_{z}$, $z\in(-m,m)$, be defined by (2.4), let $P_{\pm}$ be as
in (3.16), and let $V$ be given by (3.14). Then, there exists an operator
$\mathcal{K}\in\Psi_{\Sigma}^{-1}$ such that

$\mathcal{C}_{z}=\frac{1}{2}(\sigma\cdot\nu)V^{*}\begin{pmatrix}0&P_{-}-P_{+}\\\
P_{-}-P_{+}&0\end{pmatrix}V(\sigma\cdot\nu)+\mathcal{K}.$

In particular, the realization of $\mathcal{K}$ in
$L^{2}(\Sigma;\mathbb{C}^{2})$ is self-adjoint.

##  4\. A more detailed analysis in dimension $q=3$

In this section we are going to prove similar results as in Section 3, but
instead of pseudodifferential operators we use the theory of pseudo-
homogeneous kernels from [20]. Throughout this section we assume that
$\Sigma\subset\mathbb{R}^{3}$ is the boundary of a bounded and simply
connected $C^{\infty}$-smooth domain. First, in Subsection 4.1 we follow
closely [20, Section 4.3] and recall the main definitions and results related
to pseudo-homogeneous kernels, which are employed in Subsection 4.2 to analyze
$\mathcal{C}_{z}$.

###  4.1. Pseudo-homogeneous kernels

With the notion of pseudo-homogeneous kernels one can easily describe mapping
properties of the associated integral operators. In the following, we restrict
our attention to the three dimensional case, but a similar construction can be
done in any space dimension, cf. [20, Section 4.3]. To define pseudo-
homogeneous kernels, one first introduces homogeneous kernels [20, Section
4.3.2].

######  Definition 4.1.

Let $m\in\mathbb{N}_{0}$. A function $K\in
C^{\infty}(\mathbb{R}^{3}\times\mathbb{R}^{3}\setminus\\{0\\})$ is called a
homogeneous kernel of class $-m$, if the following conditions hold:

  * (i)

For all $\alpha,\beta\in\mathbb{N}_{0}^{3}$ there exists a constant
$C_{\alpha,\beta}$ such that

$\sup_{y\in\mathbb{R}^{3}}\sup_{|\zeta|=1}\left|\frac{\partial^{|\alpha|}}{\partial
y^{\alpha}}\frac{\partial^{|\beta|}}{\partial\zeta^{\beta}}K(y,\zeta)\right|\leq
C_{\alpha,\beta}.$

  * (ii)

For any $\beta\in\mathbb{N}_{0}^{3}$ with $|\beta|=m$ the function
$\frac{\partial^{|\beta|}}{\partial\zeta^{\beta}}K(y,\zeta)$ is homogeneous of
degree $-2$ with respect to $\zeta$.

  * (iii)

For any plane $H$ which is constituted by the equation $(h,\zeta)=0$ and any
$m$-tuple of vectors $\zeta_{1},\dots,\zeta_{m}$ in $H$ the condition

$\int_{S^{1}}D_{\zeta}^{m}K(y,\zeta^{\prime})(\zeta_{1},\dots,\zeta_{m})d\zeta^{\prime}=0$

holds, where $S^{1}$ is the intersection of the sphere $S^{2}$ with $H$.

In the following lemma we consider a type of homogeneous kernels that will
play an important role in the analysis of $\mathcal{C}_{z}$ in the next
section.

######  Lemma 4.2.

Let $\kappa\in C^{\infty}(\mathbb{R}^{3})$ such that $\kappa$ and all of its
derivatives are bounded, let $k\in\mathbb{Z}$, and let
$\alpha\in\mathbb{N}_{0}^{3}$ such that $|\alpha|+2k+1>0$. Then the function
$K(y,\zeta)=\kappa(y)\zeta^{\alpha}|\zeta|^{2k-1}$ is a homogeneous kernel of
class $-(|\alpha|+2k+1)$.

###### Proof.

First, one shows as in [20, Example 4.2] that
$\widetilde{K}(y,\zeta)=\zeta^{\alpha}|\zeta|^{2k-1}$ is a homogeneous kernel
of class $-(|\alpha|+2k+1)$. Since the multiplication by a smooth function
depending only on $y$ does not affect the conditions in Definition 4.1, the
claim on $K$ follows. ∎

With the help of homogeneous kernels we can introduce now pseudo-homogeneous
kernels [20, Section 4.3.3].

######  Definition 4.3.

Let $m\in\mathbb{N}_{0}$. A function
$K:\mathbb{R}^{3}\times\mathbb{R}^{3}\setminus\\{0\\}\rightarrow\mathbb{C}$ is
called a pseudo-homogeneous kernel of class $-m$, if for any $s\in\mathbb{N}$
there exist $l\in\mathbb{N}$, homogeneous kernels $K_{m+j}$ of class $-(m+j)$,
$j\in\\{0,\dots,l-1\\}$, and a function $K_{m+l}$ that is $s$ times
differentiable such that

$K(y,\zeta)=K_{m}(y,\zeta)+\sum_{j=1}^{l-1}K_{m+j}(y,\zeta)+K_{m+l}(y,\zeta).$

The important property of pseudo-homogeneous kernels is that one can provide
the mapping properties of the associated integral operators [20, Theorem
4.3.2].

######  Proposition 4.4.

Let $m\in\mathbb{N}_{0}$ and let $K$ be a pseudo-homogeneous kernel of class
$-m$. Then, for any $r\in\mathbb{R}$ the operator that is formally defined by

$\mathcal{K}\varphi(x):=\int_{\Sigma}K(y,x-y)\varphi(y)\textup{d}\sigma(y),\quad\varphi\in
C^{\infty}(\Sigma),~{}x\in\Sigma,$

gives rise to a bounded operator $\mathcal{K}:H^{r}(\Sigma)\rightarrow
H^{r+m}(\Sigma)$.

###  4.2. Analysis of $\mathcal{C}_{z}$

Let $-\Delta$ be the free Laplacian defined on $H^{2}(\mathbb{R}^{3})$. We use
for $\mu\in\rho(-\Delta)=\mathbb{C}\setminus[0,\infty)$ the notation
$\mathcal{S}(\mu)$ for the single layer boundary integral operator associated
with $-\Delta-\mu$, which acts on a sufficiently smooth function
$\varphi:\Sigma\rightarrow\mathbb{C}$ as

(4.1)
$\mathcal{S}(\mu)\varphi(x)=\int_{\Sigma}\frac{e^{i\sqrt{\mu}|x-y|}}{4\pi|x-y|}\varphi(y)\textup{d}\sigma(y),\qquad
x\in\Sigma,$

where $\sqrt{\mu}$ is the complex square root satisfying
$\textup{Im}\,\sqrt{\mu}>0$ for $\mu\notin[0,\infty)$, cf. [19]. If $\mu<0$,
then it is well-known that $\mathcal{S}(\mu):H^{s}(\Sigma)\rightarrow
H^{s+1}(\Sigma)$ is bounded and bijective for all $s\in\mathbb{R}$ and that
the realization of $\mathcal{S}(\mu)$ in $L^{2}(\Sigma)$ is non-negative and
self-adjoint; cf. [6, Lemma 2.6] for a similar argument for $\mu=-1$, space
dimension $2$, and $s\geq-\frac{1}{2}$; the property for negative $s$ follows
then by duality due to the formal symmetry of $\mathcal{S}(\mu)$ for $\mu<0$.
For a constant $c_{\Lambda}>0$ we define

(4.2) $\Lambda:=\big{(}\mathcal{S}(-1)^{-1}+c_{\Lambda}\big{)}^{1/2}.$

The following mapping properties $\Lambda$ should be well-known, but for the
sake of completeness we present a proof:

######  Proposition 4.5.

For any $s\in\mathbb{R}$ the operator $\Lambda$ defined in (4.2) gives rise to
a bounded and bijective map $\Lambda:H^{s}(\Sigma)\rightarrow
H^{s-1/2}(\Sigma)$. Moreover, the realization of $\Lambda$ in $L^{2}(\Sigma)$
viewed as an unbounded operator defined on $H^{1/2}(\Sigma)$ is self-adjoint.

###### Proof.

Step 1: First, we show the claim for $s=\frac{1}{2}$ and about the realization
of $\Lambda$ in $L^{2}(\Sigma)$. In the same way as in [6, Lemma 2.6 (ii)] one
finds that the map $\mathcal{S}(-1)^{1/2}:L^{2}(\Sigma)\rightarrow
H^{1/2}(\Sigma)$ is a non-negative and bijective operator that is self-adjoint
in $L^{2}(\Sigma)$. Hence, for any $c_{\Lambda}>0$ the quadratic form

$\begin{split}\mathfrak{a}[\varphi,\psi]&:=\big{(}\mathcal{S}(-1)^{-1/2}\varphi,\mathcal{S}(-1)^{-1/2}\psi\big{)}_{L^{2}(\Sigma)}+c_{\Lambda}(\varphi,\psi)_{L^{2}(\Sigma)},\\\
\varphi,\psi\in{\rm dom\,}\mathfrak{a}&={\rm
ran\,}\mathcal{S}(-1)^{1/2}=H^{1/2}(\Sigma),\end{split}$

that is associated with the self-adjoint operator
$\mathcal{S}(-1)^{-1}+c_{\Lambda}$, is closed and strictly positive. By the
second representation theorem [18, Chapter VI, Theorem 2.23] one has that
$(\mathcal{S}(-1)^{-1}+c_{\Lambda})^{1/2}$ is self-adjoint on the set ${\rm
dom\,}(\mathcal{S}(-1)^{-1}+c_{\Lambda})^{1/2}={\rm
dom\,}\mathfrak{a}=H^{1/2}(\Sigma)$ and, since this operator is strictly
positive, it is bijective from its domain $H^{1/2}(\Sigma)$ into
$L^{2}(\Sigma)$. This implies all claims for $s=\frac{1}{2}$.

Step 2: Next, we show the claim for $s=l+\frac{1}{2}$ with $l\in\mathbb{N}$.
Since $\mathcal{S}(-1):H^{r}(\Sigma)\rightarrow H^{r+1}(\Sigma)$ is bijective
for all $r\in\mathbb{R}$ and $c_{\Lambda}>0$, it is not difficult to see that
$\mathcal{S}(-1)^{-1}+c_{\Lambda}$ gives for any $r\in\mathbb{R}$ rise to a
bounded and bijective operator

$\mathcal{S}(-1)^{-1}+c_{\Lambda}:H^{r+1}(\Sigma)\rightarrow H^{r}(\Sigma).$

In particular, the mapping

$A_{l,r}:=\big{(}\mathcal{S}(-1)^{-1}+c_{\Lambda}\big{)}^{l}:H^{l+r}(\Sigma)\rightarrow
H^{r}(\Sigma)$

is bounded and bijective for any $r\in\mathbb{R}$. Hence, we conclude with the
result from Step 1 and the spectral theorem that

$\Lambda:=A_{l,0}^{-1}(\mathcal{S}(-1)^{-1}+c_{\Lambda})^{1/2}A_{l,1/2}:H^{l+1/2}(\Sigma)\rightarrow
H^{l}(\Sigma)$

is a bounded and bijective map that is a restriction of
$(\mathcal{S}(-1)^{-1}+c_{\Lambda})^{1/2}$.

Step 3: To conclude, we mention first that the statement for $s=-l$ with
$l\in\mathbb{N}_{0}$ follows from the results in Steps 1 & 2 by duality and
the formal symmetry of $\Lambda$. An interpolation argument yields the claim
for arbitrary $s\in\mathbb{R}$. ∎

The main object in this section is the operator $\mathcal{R}$ formally acting
on a sufficiently smooth function $\varphi:\Sigma\rightarrow\mathbb{C}^{2}$ as

(4.3) $\mathcal{R}\varphi(x)=\lim_{\varepsilon\searrow 0}\int_{\Sigma\setminus
B(0,\varepsilon)}r(x,y)\varphi(y)\textup{d}\sigma(y),$

where the integral kernel $r$ is the $\mathbb{C}^{2\times 2}$-valued function

$r(x,y)=-\frac{\sigma\cdot(x-y)}{\pi|x-y|^{3}}(\sigma\cdot\nu(y)),\qquad x\neq
y.$

The map $\mathcal{R}$ is closely related to the Riesz transform on $\Sigma$.
We define also the formal adjoint of $\mathcal{R}$ with respect to the inner
product in $L^{2}(\Sigma;\mathbb{C}^{2})$ by

(4.4) $\mathcal{R}^{*}\varphi(x)=\lim_{\varepsilon\searrow
0}\int_{\Sigma\setminus
B(0,\varepsilon)}r(y,x)^{*}\varphi(y)\textup{d}\sigma(y).$

We remark that the definitions of $\mathcal{R}$ and $\mathcal{R}^{*}$ imply
the simple, but useful relations

(4.5)
$(\sigma\cdot\nu)\mathcal{R}=-\mathcal{R}^{*}(\sigma\cdot\nu)\quad\text{and}\quad(\sigma\cdot\nu)\mathcal{R}^{*}=-\mathcal{R}(\sigma\cdot\nu).$

In the following proposition we provide a link of $\mathcal{R}$ and
$\mathcal{R}^{*}$ and the operator $\mathcal{C}_{z}$ given in (2.4). In
particular, this implies the mapping properties of $\mathcal{R}$ and
$\mathcal{R}^{*}$.

######  Proposition 4.6.

Let $\mathcal{R}$ and $\mathcal{R}^{*}$ be defined by (4.3) and (4.4),
respectively, let $\Lambda$ be as in (4.2), let $\mathcal{C}_{z}$,
$z\in(-m,m)$, be given by (2.4), and let $s\in\mathbb{R}$. Then $\mathcal{R}$
and $\mathcal{R}^{*}$ give rise to bounded operators in
$H^{s}(\Sigma;\mathbb{C}^{2})$ and there exists an operator $\mathcal{K}$ that
is compact from $H^{s}(\Sigma;\mathbb{C}^{4})$ to
$H^{s+1}(\Sigma;\mathbb{C}^{4})$ such that

(4.6)
$\mathcal{C}_{z}=\begin{pmatrix}(z+m)\Lambda^{-2}I_{2}&-\frac{i}{4}\mathcal{R}(\sigma\cdot\nu)\\\
\frac{i}{4}(\sigma\cdot\nu)\mathcal{R}^{*}&(z-m)\Lambda^{-2}I_{2}\end{pmatrix}+\mathcal{K}.$

In particular, the realization of $\mathcal{K}$ in
$L^{2}(\Sigma;\mathbb{C}^{4})$ is self-adjoint.

###### Proof.

Denote by $\mathcal{T}$ the integral operator that is formally acting on a
sufficiently smooth function $\varphi:\Sigma\rightarrow\mathbb{C}^{2}$ as

$\mathcal{T}\varphi(x):=\lim_{\varepsilon\searrow 0}\int_{\Sigma\setminus
B(x,\varepsilon)}t(x-y)\varphi(y)\textup{d}\sigma(y),\quad x\in\Sigma,$

where the integral kernel
$t:\mathbb{R}^{3}\setminus\\{0\\}\rightarrow\mathbb{C}^{2\times 2}$ is given
by

(4.7) $t(x):=\left(1-i\sqrt{z^{2}-m^{2}}|x|\right)\frac{i(\sigma\cdot
x)}{4\pi|x|^{3}}e^{i\sqrt{z^{2}-m^{2}}|x|}.$

Recall that $\mathcal{S}(z^{2}-m^{2})$ denotes the single layer boundary
integral operator for $-\Delta+m^{2}-z^{2}$. With these notations we conclude
from (2.3) that

(4.8)
$\mathcal{C}_{z}=\begin{pmatrix}(z+m)\mathcal{S}(z^{2}-m^{2})I_{2}&\mathcal{T}\\\
\mathcal{T}&(z-m)\mathcal{S}(z^{2}-m^{2})I_{2}\end{pmatrix}.$

Note that (4.8) implies that $\mathcal{T}$ gives rise to a bounded operator in
$H^{s}(\Sigma;\mathbb{C}^{2})$, as $\mathcal{C}_{z}$ has this mapping property
by Proposition 2.1.

First, we analyze the diagonal terms and show that

(4.9)
$\begin{split}\mathcal{S}(z^{2}-m^{2})&=\Lambda^{-2}+\mathcal{K}_{1}\end{split}$

with a compact operator $\mathcal{K}_{1}:H^{s}(\Sigma)\rightarrow
H^{s+1}(\Sigma)$. Indeed (4.2) implies that

$\begin{split}\mathcal{S}(z^{2}-m^{2})&=\Lambda^{-2}+\mathcal{K}_{2}+\mathcal{K}_{3},\end{split}$

where

$\mathcal{K}_{2}:=\mathcal{S}(z^{2}-m^{2})-\mathcal{S}(-1)\quad\text{and}\quad\mathcal{K}_{3}:=\mathcal{S}(-1)-(\mathcal{S}(-1)^{-1}+c_{\Lambda})^{-1}.$

With (4.1) we get that $\mathcal{K}_{2}$ is an integral operator with kernel
$k_{2}(x-y)$, where

$\begin{split}k_{2}(x)&=\frac{e^{-\sqrt{m^{2}-z^{2}}|x|}}{4\pi|x|}-\frac{e^{-|x|}}{4\pi|x|},\quad
x\in\mathbb{R}^{3}\setminus\\{0\\}.\end{split}$

By making a power series expansion we see that the first term in the sums
cancels,

$\begin{split}k_{2}(x)&=\sum_{k=0}^{\infty}c_{k}|x|^{2k}+\sum_{k=0}^{\infty}d_{k}|x|^{2k+1}\end{split}$

with some suitable coefficients $c_{k},d_{k}\in\mathbb{R}$, and that both sums
converge absolutely, as the power series expansion of the exponential function
has this property as well. Hence, the first sum is an analytic function, while
the second sum is a pseudo-homogeneous kernel of class $-3$ in the sense of
Definition 4.3, cf. Lemma 4.2. Thus, we conclude with Proposition 4.4 that
$\mathcal{K}_{2}:H^{s}(\Sigma)\rightarrow H^{s+3}(\Sigma)$ is bounded for any
$s\in\mathbb{R}$ and in particular, $\mathcal{K}_{2}$ is compact as an
operator from $H^{s}(\Sigma)$ to $H^{s+1}(\Sigma)$.

Next, using the resolvent identity we see that

(4.10)
$\mathcal{K}_{3}=c_{\Lambda}\mathcal{S}(-1)(\mathcal{S}(-1)^{-1}+c_{\Lambda})^{-1}=c_{\Lambda}\mathcal{S}(-1)\Lambda^{-2},$

which is due to the mapping properties of $\mathcal{S}(-1)$ and $\Lambda$
bounded from $H^{s}(\Sigma)$ to $H^{s+2}(\Sigma)$ and hence compact from
$H^{s}(\Sigma)$ to $H^{s+1}(\Sigma)$. Thus, we have shown (4.9).

It remains to consider the anti-diagonal blocks. The integral kernel $t$ of
the operator $\mathcal{T}$ in (4.7) has the series expansion

(4.11) $t(x)=\frac{i(\sigma\cdot
x)}{4\pi|x|^{3}}+\sum_{k=0}^{\infty}\tilde{c}_{k}(\sigma\cdot
x)|x|^{2k}+\sum_{k=0}^{\infty}\tilde{d}_{k}(\sigma\cdot x)|x|^{2k-1}$

with some suitable coefficients $\tilde{c}_{k},\tilde{d}_{k}\in\mathbb{C}$, as
the terms with $|x|^{-2}$ cancel. Since the power series for the exponential
function is absolutely converging, both sums in (4.11) are absolutely
converging as well for any $x\neq 0$. In particular, the first sum defines an
analytic function. Moreover, by Lemma 4.2 the second sum is a pseudo-
homogeneous kernel of class $-2$ in the sense of Definition 4.3. Hence, we
conclude with Proposition 4.4 that

(4.12) $\mathcal{T}=-\frac{i}{4}\mathcal{R}(\sigma\cdot\nu)+\mathcal{K}_{4},$

where $\mathcal{K}_{4}$ is bounded from $H^{s}(\Sigma;\mathbb{C}^{2})$ to
$H^{s+2}(\Sigma;\mathbb{C}^{2})$ and, in particular, compact from
$H^{s}(\Sigma;\mathbb{C}^{2})$ to $H^{s+1}(\Sigma;\mathbb{C}^{2})$. Combining
(4.12) with (4.5), we can also write

(4.13)
$\mathcal{T}=\frac{i}{4}(\sigma\cdot\nu)\mathcal{R}^{*}+\mathcal{K}_{4}.$

In particular, as $\mathcal{T}$ is bounded in $H^{s}(\Sigma;\mathbb{C}^{2})$
and $\sigma\cdot\nu$ is a bijection in $H^{s}(\Sigma;\mathbb{C}^{2})$, we
conclude that $\mathcal{R}$ and $\mathcal{R}^{*}$ give rise to bounded
operators in $H^{s}(\Sigma;\mathbb{C}^{2})$. Finally, equations (4.8), (4.9),
(4.12), and (4.13) imply (4.6). Furthermore, as the realization of
$\mathcal{C}_{z}$ in $L^{2}(\Sigma;\mathbb{C}^{4})$ is self-adjoint for
$z\in(-m,m)$ by Proposition 2.1 (ii), also the operator $\mathcal{K}$ in (4.6)
must be self-adjoint in $L^{2}(\Sigma;\mathbb{C}^{4})$. ∎

Let us mention that, since $\mathcal{R}$ and $\mathcal{R}^{*}$ are formally
adjoint to each other with respect to the inner product in
$L^{2}(\Sigma;\mathbb{C}^{2})$, a continuity argument shows for all
$\varphi\in H^{s}(\Sigma;\mathbb{C}^{2})$ and $\psi\in
H^{-s}(\Sigma;\mathbb{C}^{2})$, $s\in\mathbb{R}$, that

(4.14) $(\mathcal{R}\varphi,\psi)_{H^{s}(\Sigma;\mathbb{C}^{2})\times
H^{-s}(\Sigma;\mathbb{C}^{2})}=(\varphi,\mathcal{R}^{*}\psi)_{H^{s}(\Sigma;\mathbb{C}^{2})\times
H^{-s}(\Sigma;\mathbb{C}^{2})},$

where $(\cdot,\cdot)_{H^{s}(\Sigma;\mathbb{C}^{2})\times
H^{-s}(\Sigma;\mathbb{C}^{2})}$ denotes the sesquilinear duality product in
$H^{s}(\Sigma;\mathbb{C}^{2})\times H^{-s}(\Sigma;\mathbb{C}^{2})$.

In the following lemma we state a variant of (4.6) that is particularly useful
in the application in the boundary triple framework, as it is done, e.g., in
[10]. Define the matrix $V\in\mathbb{C}^{4\times 4}$ by

(4.15) $V=\begin{pmatrix}I_{2}&0\\\ 0&-i(\sigma\cdot\nu)\end{pmatrix}.$

Then we have the following result:

######  Lemma 4.7.

Let $\mathcal{R}$ and $\mathcal{R}^{*}$ be defined by (4.3) and (4.4),
respectively, let $\Lambda$ be as in (4.2), and let $\mathcal{C}_{z}$,
$z\in(-m,m)$, be given by (2.4). Then, for any $s\in\mathbb{R}$ there exists
an operator $\mathcal{K}$ that is compact in $H^{s}(\Sigma;\mathbb{C}^{4})$
such that the bounded and everywhere defined operator

$4\Lambda
V(\alpha\cdot\nu)\mathcal{C}_{z}(\alpha\cdot\nu)V^{*}\Lambda:H^{s}(\Sigma;\mathbb{C}^{4})\rightarrow
H^{s-1}(\Sigma;\mathbb{C}^{4})$

can be written as

(4.16) $4\Lambda
V(\alpha\cdot\nu)\mathcal{C}_{z}(\alpha\cdot\nu)V^{*}\Lambda=-\Lambda\begin{pmatrix}0&\mathcal{R}^{*}\\\
\mathcal{R}&0\end{pmatrix}\Lambda+4\begin{pmatrix}z-m&0\\\
0&z+m\end{pmatrix}+\mathcal{K}.$

In particular, the realization of $\mathcal{K}$ in
$L^{2}(\Sigma;\mathbb{C}^{4})$ is self-adjoint.

###### Proof.

First, equation (4.6) and a simple calculation imply

(4.17) $\begin{split}4\Lambda
V(\alpha\cdot\nu)&\mathcal{C}_{z}(\alpha\cdot\nu)V^{*}\Lambda\\\
&=\Lambda\begin{pmatrix}4(z-m)(\sigma\cdot\nu)\Lambda^{-2}(\sigma\cdot\nu)&-\mathcal{R}^{*}\\\
-\mathcal{R}&4(z+m)\Lambda^{-2}\end{pmatrix}\Lambda+\mathcal{K}_{1},\end{split}$

where $\mathcal{K}_{1}$ is a compact operator in
$H^{s}(\Sigma;\mathbb{C}^{4})$ due to the mapping properties of $\Lambda$.

By (4.2) we can write

$(\sigma\cdot\nu)\Lambda^{-2}-\Lambda^{-2}(\sigma\cdot\nu)=\mathcal{T}+\mathcal{K}_{2},$

where $\mathcal{T}$ is the integral operator acting as

(4.18)
$\mathcal{T}\varphi(x)=\big{(}(\sigma\cdot\nu)\mathcal{S}(-1)-\mathcal{S}(-1)(\sigma\cdot\nu)\big{)}\varphi(x)=\int_{\Sigma}t(x,y)\varphi\textup{d}\sigma(y)$

with

(4.19)
$t(x,y)=\big{(}\sigma\cdot(\nu(x)-\nu(y))\big{)}\frac{e^{-|x-y|}}{4\pi|x-y|}$

and $\mathcal{K}_{2}$ is given by

$\mathcal{K}_{2}=(\sigma\cdot\nu)\big{(}(\mathcal{S}(-1)+c_{\Lambda})^{-1}-\mathcal{S}(-1)\big{)}+\big{(}\mathcal{S}(-1)-(\mathcal{S}(-1)+c_{\Lambda})^{-1}\big{)}(\sigma\cdot\nu).$

A similar argument as in (4.10) shows that $\mathcal{K}_{2}$ is compact as a
mapping from $H^{s-1/2}(\Sigma;\mathbb{C}^{2})$ to
$H^{s+1/2}(\Sigma;\mathbb{C}^{2})$.

Next, we show that $\mathcal{T}$ in (4.18) is compact as an operator from
$H^{s-1/2}(\Sigma;\mathbb{C}^{2})$ to $H^{s+1/2}(\Sigma;\mathbb{C}^{2})$.
Denote by $\tilde{\nu}=(\tilde{\nu}_{1},\tilde{\nu}_{2},\tilde{\nu}_{3})$ a
smooth and compactly supported extension of $\nu$ that is defined on
$\mathbb{R}^{3}$. Then, this function has a Taylor series expansion of the
form

$\tilde{\nu}(x)=\tilde{\nu}(y)+D\tilde{\nu}(y)(x-y)+\sum_{j=1}^{3}(x-y)^{\top}H\tilde{\nu}_{j}(y)(x-y)e_{j}+\dots,$

where $D\tilde{\nu}$ and $H\tilde{\nu}_{j}$ are the Jacobi matrix and the
Hessian of $\tilde{\nu}$ and $\tilde{\nu}_{j}$, respectively, and $e_{j}$ are
the canonical basis vectors in $\mathbb{R}^{3}$. Using this and the power
series expansion of the exponential function we find that the function $t$
defined in (4.19) can be written as

$\begin{split}t(x,y)&=\sigma\cdot\bigg{(}D\tilde{\nu}(y)(x-y)+\sum_{j=1}^{3}(x-y)^{\top}H\tilde{\nu}_{j}(y)(x-y)e_{j}+\dots\bigg{)}\sum_{k=0}^{\infty}\frac{|x-y|^{2k-1}}{4\pi(2k)!}\\\
&\qquad-\sigma\cdot(\tilde{\nu}(x)-\tilde{\nu}(y))\sum_{k=0}^{\infty}\frac{|x-y|^{2k}}{4\pi(2k+1)!}.\end{split}$

Since the second term is a smooth function, we find with Lemma 4.2 that
$t(x,y)$ is a pseudo-homogeneous kernel of class $-2$. Therefore, we get with
Proposition 4.4 that $\mathcal{T}$ is bounded from
$H^{s-1/2}(\Sigma;\mathbb{C}^{2})$ to $H^{s+3/2}(\Sigma;\mathbb{C}^{2})$ and,
in particular, compact from $H^{s-1/2}(\Sigma;\mathbb{C}^{2})$ to
$H^{s+1/2}(\Sigma;\mathbb{C}^{2})$. Hence, we conclude that

$\Lambda(\sigma\cdot\nu)\Lambda^{-2}(\sigma\cdot\nu)\Lambda=I_{2}+\mathcal{K}_{3}$

holds, where $\mathcal{K}_{3}$ is compact in $H^{s}(\Sigma;\mathbb{C}^{2})$.
Together with (4.17) this implies (4.16).

Finally, equation (4.16) and the fact that the realization of
$\mathcal{C}_{z}$ in $L^{2}(\Sigma;\mathbb{C}^{4})$ is self-adjoint for
$z\in(-m,m)$ by Proposition 2.1 yield that also $\mathcal{K}$ must be self-
adjoint in $L^{2}(\Sigma;\mathbb{C}^{4})$. ∎

In the following proposition some further properties of $\mathcal{R}$ and
$\mathcal{R}^{*}$ that follow from known properties of $\mathcal{C}_{z}$ are
stated:

######  Proposition 4.8.

Let $\mathcal{R}$ and $\mathcal{R}^{*}$ be defined by (4.3) and (4.4),
respectively, and let $s\in\mathbb{R}$. Then, the following holds:

  * (i)

The operator
$\mathcal{R}-\mathcal{R}^{*}:H^{s-1}(\Sigma;\mathbb{C}^{2})\rightarrow
H^{s}(\Sigma;\mathbb{C}^{2})$ gives rise to a bounded operator.

  * (ii)

One has $\mathcal{R}^{2}=4I_{2}$ and $(\mathcal{R}^{*})^{2}=4I_{2}$. In
particular, this implies that

(4.20)
$\mathcal{R}\mathcal{R}^{*}-4I_{2}:H^{s-1}(\Sigma;\mathbb{C}^{2})\rightarrow
H^{s}(\Sigma;\mathbb{C}^{2})$

gives rise to a bounded operator.

  * (iii)

The operator
$\Lambda^{-2}\mathcal{R}-\mathcal{R}\Lambda^{-2}:H^{s-1}(\Sigma;\mathbb{C}^{2})\rightarrow
H^{s}(\Sigma;\mathbb{C}^{2})$ is compact.

###### Proof.

(i) Taking the definitions of $\mathcal{R}$ and $\mathcal{R}^{*}$ into
account, we see that their difference is a singular integral operator with
kernel

(4.21)
$\begin{split}r(x,y)&-r(y,x)^{*}=-\frac{\sigma\cdot(x-y)}{\pi|x-y|^{3}}(\sigma\cdot\nu(y))-(\sigma\cdot\nu(x))\frac{\sigma\cdot(x-y)}{\pi|x-y|^{3}}\\\
&=\frac{\sigma\cdot(x-y)}{\pi|x-y|^{3}}\sigma\cdot(\nu(x)-\nu(y))-2\frac{\nu(x)\cdot(x-y)}{\pi|x-y|^{3}}=:s_{1}(x,y)+s_{2}(x,y).\end{split}$

Following ideas from the proof of [21, Proposition 2.8], we remark first that
the kernel $s_{2}(x,y):=-2\frac{\nu(x)\cdot(x-y)}{\pi|x-y|^{3}}$ is pseudo-
homogeneous of class $-1$, as $\nu(x)\cdot(x-y)\sim|x-y|^{2}$ for
$x\rightarrow y$ (for details see the considerations on the kernel $K_{1}$ in
the proof of [21, Proposition 2.8]). Concerning the first term in the sum in
(4.21), one gets in a similar way as in the study of (4.19) with Lemma 4.2
that $s_{1}(x,y)$ is a pseudo-homogeneous kernel of class $-1$ in the sense of
Definition 4.3. Summing up, we find that $r(x,y)-r(y,x)^{*}$ is a kernel of
class $-1$ and hence, by Proposition 4.4 the operator
$\mathcal{R}-\mathcal{R}^{*}:H^{s-1}(\Sigma;\mathbb{C}^{2})\rightarrow
H^{s}(\Sigma;\mathbb{C}^{2})$ is bounded, which is the claim of item (i).

(ii) First, we show that $\mathcal{R}^{2}=4I_{2}$ holds. It is known from [5,
Proposition 3.5 (i)] that $\mathcal{C}_{m}:=\lim_{z\rightarrow
m}\mathcal{C}_{z}$ exists with respect to the norm in the space of bounded
operators in $L^{2}(\Sigma;\mathbb{C}^{4})$ and the limit is given by

$\mathcal{C}_{m}=\frac{1}{4}\begin{pmatrix}2m\widehat{S}&-i\mathcal{R}(\sigma\cdot\nu)\\\
-i\mathcal{R}(\sigma\cdot\nu)&0\end{pmatrix}$

with

$\widehat{S}\varphi(x)=\int_{\Sigma}\frac{1}{\pi|x-y|}\varphi(y)\textup{d}\sigma(y),\quad\varphi\in
L^{2}(\Sigma;\mathbb{C}^{2}).$

Combining this with Proposition 2.1 (iii) yields

$\begin{split}I_{4}&=4\big{(}\mathcal{C}_{m}(i\alpha\cdot\nu)\big{)}^{2}=4\left(\frac{1}{4}\begin{pmatrix}2m\widehat{S}&-i\mathcal{R}(\sigma\cdot\nu)\\\
-i\mathcal{R}(\sigma\cdot\nu)&0\end{pmatrix}\begin{pmatrix}0&i(\sigma\cdot\nu)\\\
i(\sigma\cdot\nu)&0\end{pmatrix}\right)^{2}\\\
&=\frac{1}{4}\begin{pmatrix}\mathcal{R}&2im\widehat{S}(\sigma\cdot\nu)\\\
0&\mathcal{R}\end{pmatrix}^{2}=\frac{1}{4}\begin{pmatrix}\mathcal{R}^{2}&2im\mathcal{R}\widehat{S}(\sigma\cdot\nu)+2im\widehat{S}(\sigma\cdot\nu)\mathcal{R}\\\
0&\mathcal{R}^{2}\end{pmatrix},\end{split}$

which shows $\mathcal{R}^{2}=4I_{2}$. By taking the adjoint this implies
$(\mathcal{R}^{*})^{2}=4I_{2}$. Using this we also find that

$\mathcal{R}\mathcal{R}^{*}-4I_{2}=\mathcal{R}\big{(}\mathcal{R}^{*}-\mathcal{R}\big{)}$

holds. This and the result from item (i) yield the mapping properties in
(4.20).

(iii) First we note that Proposition 4.6 implies that

(4.22)
$\mathcal{C}_{z}(\alpha\cdot\nu)=\frac{1}{4}\begin{pmatrix}-i\mathcal{R}&4(z+m)\Lambda^{-2}(\sigma\cdot\nu)\\\
4(z-m)\Lambda^{-2}(\sigma\cdot\nu)&i(\sigma\cdot\nu)\mathcal{R}^{*}(\sigma\cdot\nu)\end{pmatrix}+\mathcal{K}_{1}$

holds, where $\mathcal{K}_{1}:H^{s-1}(\Sigma;\mathbb{C}^{4})\rightarrow
H^{s}(\Sigma;\mathbb{C}^{4})$ is compact. Using this representation in
$-4(\mathcal{C}_{z}(\alpha\cdot\nu))^{2}=I_{4}$, see Proposition 2.1 (iii), we
find that the upper right block of this equation has the form

$\begin{split}0&=i(z+m)\mathcal{R}\Lambda^{-2}(\sigma\cdot\nu)-i(z+m)\Lambda^{-2}\mathcal{R}^{*}(\sigma\cdot\nu)+\mathcal{K}_{2}\\\
&=i(z+m)\big{(}\mathcal{R}\Lambda^{-2}-\Lambda^{-2}\mathcal{R}\big{)}(\sigma\cdot\nu)+i(z+m)\Lambda^{-2}(\mathcal{R}-\mathcal{R}^{*})(\sigma\cdot\nu)+\mathcal{K}_{2}\end{split}$

with a compact operator
$\mathcal{K}_{2}:H^{s-1}(\Sigma;\mathbb{C}^{2})\rightarrow
H^{s}(\Sigma;\mathbb{C}^{2})$. Since
$\Lambda^{-2}(\mathcal{R}-\mathcal{R}^{*})$ is bounded from
$H^{s-1}(\Sigma;\mathbb{C}^{2})$ to $H^{s+1}(\Sigma;\mathbb{C}^{2})$ by item
(i) and the mapping properties of $\Lambda$, the last equation and Rellich’s
embedding theorem yield the claim. ∎

Next, we state a result on the sum $\mathcal{R}+\mathcal{R}^{*}$, that will
play a crucial role in [10] in the proof of the self-adjointness and the
spectral analysis of Dirac operators with singular potentials.

######  Theorem 4.9.

Let $\mathcal{R}$ and $\mathcal{R}^{*}$ be defined as above and let
$s\in[-1,1]$. Then, there exist a bounded operator
$\mathcal{K}:H^{s}(\Sigma;\mathbb{C}^{2})\rightarrow
H^{1}(\Sigma;\mathbb{C}^{2})$ and closed subspaces $\mathcal{H}_{\pm}$ of
$H^{s}(\Sigma;\mathbb{C}^{2})$ satisfying ${\rm dim\,}\mathcal{H}_{+}={\rm
dim\,}\mathcal{H}_{-}$ and
$H^{s}(\Sigma;\mathbb{C}^{2})=\mathcal{H}_{+}\dot{+}\mathcal{H}_{-}$ such that
the realization of $\mathcal{R}+\mathcal{R}^{*}$ in the space
$H^{s}(\Sigma;\mathbb{C}^{2})$ can be written as

$\mathcal{R}+\mathcal{R}^{*}=4P_{+}-4P_{-}+\mathcal{K},$

where $P_{\pm}$ is the projection onto $\mathcal{H}_{\pm}$.

###### Proof.

The proof of this result is separated in four steps. In Step 1 we consider the
realization of $\mathcal{R}+\mathcal{R}^{*}$ in the space
$L^{2}(\Sigma;\mathbb{C}^{2})$ and prove a spectral representation that is
suitable to define the map $\mathcal{K}$. In Step 2 we show an auxiliary
embedding result, which is then used in Step 3 to conclude the mapping
properties of $\mathcal{K}$. Finally, in Step 4 we show the claims about
$\mathcal{H}_{\pm}$ and $P_{\pm}$. Throughout the proof, we will denote by
$(\cdot,\cdot)$ the inner product in $L^{2}(\Sigma;\mathbb{C}^{2})$.

Step 1: Consider $\mathcal{R}$ and $\mathcal{R}^{*}$ as bounded operators in
$L^{2}(\Sigma;\mathbb{C}^{2})$. Then, Proposition 4.8 (ii) implies that

(4.23)
$(\mathcal{R}+\mathcal{R}^{*})^{2}=\mathcal{R}^{2}+(\mathcal{R}^{*})^{2}+\mathcal{R}^{*}\mathcal{R}+\mathcal{R}\mathcal{R}^{*}=16I_{2}+\mathcal{A}^{2},$

where we have set

$\mathcal{A}:=i\mathcal{R}-i\mathcal{R}^{*}.$

By definition, $\mathcal{A}$ is a bounded and self-adjoint operator in
$L^{2}(\Sigma;\mathbb{C}^{2})$ and by Proposition 4.8 (i) $\mathcal{A}$ gives
rise to a bounded operator from $L^{2}(\Sigma;\mathbb{C}^{2})$ to
$H^{1}(\Sigma;\mathbb{C}^{2})$ and hence, by Rellich’s embedding theorem,
$\mathcal{A}$ is compact in $L^{2}(\Sigma;\mathbb{C}^{2})$. Thus, the spectral
theorem implies that

(4.24)
$\sigma(\mathcal{R}+\mathcal{R}^{*})\subset\left\\{\pm\sqrt{16+\lambda^{2}}:\lambda\in\sigma(\mathcal{A})\right\\}$

and $\sigma(\mathcal{R}+\mathcal{R}^{*})\setminus\\{\pm 4\\}$ is purely
discrete. We claim that the inclusion in (4.24) is actually an equality. To
see this, it suffices to show that
$\sigma_{\textup{p}}(\mathcal{R}+\mathcal{R}^{*})$ is symmetric with respect
to the origin. Let $\mu\in\sigma_{\textup{p}}(\mathcal{R}+\mathcal{R}^{*})$
and $e\in L^{2}(\Sigma;\mathbb{C}^{2})$ such that
$(\mathcal{R}+\mathcal{R}^{*})e=\mu e$. With (4.5) we see that

(4.25)
$(\mathcal{R}+\mathcal{R}^{*})(\sigma\cdot\nu)e=-(\sigma\cdot\nu)(\mathcal{R}+\mathcal{R}^{*})e=-\mu(\sigma\cdot\nu)e,$

i.e. also $-\mu\in\sigma_{\textup{p}}(\mathcal{R}+\mathcal{R}^{*})$ and hence

$\sigma(\mathcal{R}+\mathcal{R}^{*})=\left\\{\pm\sqrt{16+\lambda^{2}}:\lambda\in\sigma(\mathcal{A})\right\\}.$

Thus, if we denote by
$\mu_{k}=\text{sign}\,\mu_{k}\cdot\sqrt{16+\lambda_{k}^{2}}$ all eigenvalues
of $\mathcal{R}+\mathcal{R}^{*}$ with some
$\lambda_{k}\in\sigma_{\textup{p}}(\mathcal{A})$ and by $e_{k}^{\pm}$ the
corresponding eigenfunctions (where the superscript corresponds to the sign of
the associated eigenvalue and which are by (4.23) also eigenfunctions of
$\mathcal{A}$), and that are by the spectral theorem an orthonormal basis in
$L^{2}(\Sigma;\mathbb{C}^{2})$, we can write

(4.26)
$\begin{split}(\mathcal{R}+\mathcal{R}^{*})\varphi=\sum_{k}\mu_{k}(\varphi,e_{k}^{\pm})e_{k}^{\pm}=4\widetilde{P}_{+}\varphi-4\widetilde{P}_{-}\varphi+\widetilde{\mathcal{K}}\varphi\end{split}$

with

(4.27)
$\begin{split}\widetilde{P}_{+}\varphi&:=\sum_{\mu_{k}>0}(\varphi,e_{k}^{+})e_{k}^{+},\qquad\widetilde{P}_{-}\varphi:=\sum_{\mu_{k}<0}(\varphi,e_{k}^{-})e_{k}^{-},\\\
\widetilde{\mathcal{K}}\varphi&:=\sum_{\mu_{k}>0}\Big{(}\sqrt{16+\lambda_{k}^{2}}-4\Big{)}(\varphi,e_{k}^{+})e_{k}^{+}+\sum_{\mu_{k}<0}\Big{(}4-\sqrt{16+\lambda_{k}^{2}}\Big{)}(\varphi,e_{k}^{-})e_{k}^{-}.\end{split}$

It remains to show that $\widetilde{P}_{\pm}$ give rise to bounded operators
$P_{\pm}$ in $H^{s}(\Sigma;\mathbb{C}^{2})$, that $\mathcal{H}_{\pm}:={\rm
ran\,}P_{\pm}$ fulfil
$H^{s}(\Sigma;\mathbb{C}^{2})=\mathcal{H}_{+}\dot{+}\mathcal{H}_{-}$, and that
$\widetilde{\mathcal{K}}$ can be extended to a bounded operator from
$H^{s}(\Sigma;\mathbb{C}^{2})$ to $H^{1}(\Sigma;\mathbb{C}^{2})$.

Step 2: Define the space $\mathcal{G}:={\rm ran\,}\mathcal{A}$ and endow it
with the norm

$\begin{split}\|\varphi\|_{\mathcal{G}}^{2}&=\big{\|}(\mathcal{A}\upharpoonright(L^{2}(\Sigma;\mathbb{C}^{2})\ominus{{\rm
ker\,}\,}\mathcal{A}))^{-1}\varphi\big{\|}_{L^{2}(\Sigma;\mathbb{C}^{2})}^{2}+\|\varphi\|_{L^{2}(\Sigma;\mathbb{C}^{2})}^{2}\\\
&=\sum_{\lambda_{k}\in\sigma(\mathcal{A})\setminus\\{0\\}}(\lambda_{k}^{-2}+1)|(\varphi,e_{k}^{\pm})|^{2}.\end{split}$

Remark that
$(\mathcal{A}\upharpoonright(L^{2}(\Sigma;\mathbb{C}^{2})\ominus{{\rm
ker\,}\,}\mathcal{A}))^{-1}$ is self-adjoint in the Hilbert space
$\overline{{\rm ran\,}\mathcal{A}}$ and hence, it is closed and $\mathcal{G}$
is a Hilbert space. Clearly, those eigenfunctions $e_{k}^{\pm}$ of
$\mathcal{A}$ that correspond to eigenvalues $\lambda_{k}\neq 0$ are an
orthogonal basis in $\mathcal{G}$. Note that Proposition 4.8 (i) implies that
$\mathcal{G}={\rm ran\,}(\mathcal{R}-\mathcal{R}^{*})\subset
H^{1}(\Sigma;\mathbb{C}^{2})$. In this step, we show that the embedding
$\iota:\mathcal{G}\rightarrow H^{1}(\Sigma;\mathbb{C}^{2})$ is continuous.

For that, we show that $\iota$ is closed. So let
$(\varphi_{n})\subset\mathcal{G}$ such that $\varphi_{n}\rightarrow\varphi$ in
$\mathcal{G}$ and $\iota\varphi_{n}=\varphi_{n}\rightarrow\psi$ in
$H^{1}(\Sigma;\mathbb{C}^{2})$. Then, as $\mathcal{G}$ is complete
$\varphi\in\mathcal{G}$ and by the same argument $\psi\in
H^{1}(\Sigma;\mathbb{C}^{2})$. Since both $\mathcal{G}$ and
$H^{1}(\Sigma;\mathbb{C}^{2})$ are boundedly embedded in
$L^{2}(\Sigma;\mathbb{C}^{2})$, we have that $(\varphi_{n})$ converges to
$\varphi$ and $\psi$ in $L^{2}(\Sigma;\mathbb{C}^{2})$. This can only be true
if $\varphi=\psi$. This finishes the proof that $\iota$ is closed and thus,
continuous.

Step 3: In this step we prove that the map $\widetilde{\mathcal{K}}$
introduced in (4.27) can be extended to a bounded operator from
$H^{-1}(\Sigma;\mathbb{C}^{2})$ to $H^{1}(\Sigma;\mathbb{C}^{2})$, which gives
then rise to a bounded mapping
$\mathcal{K}:H^{s}(\Sigma;\mathbb{C}^{2})\rightarrow
H^{1}(\Sigma;\mathbb{C}^{2})$.

To show the stated boundedness property, let $\varphi\in
L^{2}(\Sigma;\mathbb{C}^{2})$ be fixed. We claim that
$\widetilde{\mathcal{K}}\varphi\in\mathcal{G}$. To see this, we use that the
definition of $\widetilde{\mathcal{K}}$ in (4.27) yields
$|(\widetilde{\mathcal{K}}\varphi,e_{k}^{\pm})|=(\sqrt{16+\lambda_{k}^{2}}-4)|(\varphi,e_{k}^{\pm})|$
and thus,

(4.28)
$\begin{split}\|\widetilde{\mathcal{K}}\varphi\|_{\mathcal{G}}^{2}&=\sum_{\lambda_{k}\in\sigma(\mathcal{A})\setminus\\{0\\}}(\lambda_{k}^{-2}+1)|(\widetilde{\mathcal{K}}\varphi,e_{k}^{\pm})|^{2}\\\
&=\sum_{\lambda_{k}\in\sigma(\mathcal{A})\setminus\\{0\\}}(\lambda_{k}^{-2}+1)\left(\frac{\lambda_{k}^{2}}{\sqrt{16+\lambda_{k}^{2}}+4}\right)^{2}|(\varphi,e_{k}^{\pm})|^{2}\\\
&\leq
c_{1}\sum_{\lambda_{k}\in\sigma(\mathcal{A})\setminus\\{0\\}}\lambda_{k}^{2}|(\varphi,e_{k}^{\pm})|^{2}=c_{1}\|\mathcal{A}\varphi\|_{L^{2}(\Sigma;\mathbb{C}^{2})}^{2}<\infty.\end{split}$

Therefore, we have shown $\widetilde{\mathcal{K}}\varphi\in\mathcal{G}$.
Combining now (4.28) with Proposition 4.8 (i) (applied with $s=0$) and the
result in Step 2 we conclude

$\begin{split}\|\widetilde{\mathcal{K}}\varphi\|_{H^{1}(\Sigma;\mathbb{C}^{2})}^{2}&\leq
c_{2}\|\widetilde{\mathcal{K}}\varphi\|_{\mathcal{G}}^{2}\leq
c_{3}\|\mathcal{A}\varphi\|_{L^{2}(\Sigma;\mathbb{C}^{2})}^{2}\leq
c_{4}\|\varphi\|_{H^{-1}(\Sigma;\mathbb{C}^{2})}.\end{split}$

Since $L^{2}(\Sigma;\mathbb{C}^{2})$ is a dense subspace of
$H^{-1}(\Sigma;\mathbb{C}^{2})$, the latter estimate shows that
$\widetilde{\mathcal{K}}$ can indeed be extended to a bounded map from
$H^{-1}(\Sigma;\mathbb{C}^{2})$ to $H^{1}(\Sigma;\mathbb{C}^{2})$.

Step 4: It remains to show that $\widetilde{P}_{\pm}$ give rise to bounded
operators $P_{\pm}$ in $H^{s}(\Sigma;\mathbb{C}^{2})$ and that
$\mathcal{H}_{\pm}:={\rm ran\,}P_{\pm}$ fulfil
$H^{s}(\Sigma;\mathbb{C}^{2})=\mathcal{H}_{+}\dot{+}\mathcal{H}_{-}$. To see
the first claim, remark that the eigenfunctions $e_{k}^{\pm}$ of
$\mathcal{R}+\mathcal{R}^{*}$ form an orthonormal basis of
$L^{2}(\Sigma;\mathbb{C}^{2})$ and that the definition of
$\widetilde{P}_{\pm}$ in (4.27) implies
$\widetilde{P}_{+}+\widetilde{P}_{-}=I_{2}$. Hence, formula (4.26) yields

(4.29)
$\widetilde{P}_{\pm}=\pm\frac{1}{8}\big{(}\mathcal{R}+\mathcal{R}^{*}\pm
4I_{2}-\widetilde{\mathcal{K}}\big{)}.$

Since the operator on the right hand side gives rise to a continuous map in
$H^{s}(\Sigma;\mathbb{C}^{2})$, the same is true for the left hand side and we
denote this realization by $P_{\pm}$.

It remains to show that the direct sum decomposition
$H^{s}(\Sigma;\mathbb{C}^{2})=\mathcal{H}_{+}\dot{+}\mathcal{H}_{-}$ holds.
First, we note that (4.29) implies $P_{+}+P_{-}=I_{2}$ and hence,
$H^{s}(\Sigma;\mathbb{C}^{2})=\mathcal{H}_{+}+\mathcal{H}_{-}$. To see that
the sum is direct, assume that $\varphi\in\mathcal{H}_{+}\cap\mathcal{H}_{-}$.
Then, there exist $\varphi_{-},\varphi_{+}\in H^{s}(\Sigma;\mathbb{C}^{2})$
such that

(4.30) $\varphi=P_{+}\varphi_{+}=P_{-}\varphi_{-}.$

Note that the definition of $\widetilde{P}_{\pm}$ in (4.27) implies that
$\widetilde{P}_{\pm}\widetilde{P}_{\mp}=0$. Thus, also $P_{\pm}P_{\mp}=0$. An
Application of $P_{\pm}$ to (4.30) shows then $P_{\pm}\varphi=0$, i.e.
$\varphi=P_{+}\varphi+P_{-}\varphi=0$. Therefore,
$H^{s}(\Sigma;\mathbb{C}^{2})=\mathcal{H}_{+}\dot{+}\mathcal{H}_{-}$ as a
direct sum and all claims are shown. ∎

######  Remark 4.10.

It follows from the proof of Theorem 4.9 that $\sigma\cdot\nu$ is an
isomorphism from $\mathcal{H}_{\pm}$ to $\mathcal{H}_{\mp}$. Indeed, it is
shown in (4.25) that $(\sigma\cdot\nu){{\rm
ker\,}\,}(\mathcal{R}+\mathcal{R}^{*}-\mu)={{\rm
ker\,}\,}(\mathcal{R}+\mathcal{R}^{*}+\mu)$ holds for any $\mu\neq 0$. Since
$\mathcal{H}_{\pm}$ is defined for $s\in[-1,0]$ as the completion of

$\begin{split}{\rm ran\,}\widetilde{P}_{\pm}=\textup{span}\big{\\{}u\in
L^{2}(\Sigma;\mathbb{C}^{2}):u\in{{\rm
ker\,}\,}(\mathcal{R}+\mathcal{R}^{*}-\mu)\text{ for
}\pm\mu>0\big{\\}}\end{split}$

in $H^{s}(\Sigma;\mathbb{C}^{2})$ and for $s\in[0,1]$ as the completion of

$\begin{split}\textup{span}\big{\\{}u\in
L^{2}(\Sigma;\mathbb{C}^{2}):u\in{{\rm
ker\,}\,}(\mathcal{R}+\mathcal{R}^{*}-\mu)\text{ for }\pm\mu>4\big{\\}}&\\\
+\big{(}{{\rm ker\,}\,}(\mathcal{R}+\mathcal{R}^{*}\mp 4)\cap
H^{s}(\Sigma;\mathbb{C}^{2})&\big{)}\end{split}$

in $H^{s}(\Sigma;\mathbb{C}^{2})$, cf. (4.27), and $\sigma\cdot\nu$ gives rise
to a bounded map in $H^{s}(\Sigma;\mathbb{C}^{2})$, this map is indeed an
isomorphism between $\mathcal{H}_{\pm}$ and $\mathcal{H}_{\mp}$.

Eventually, we state a result about a special operator in
$L^{2}(\Sigma;\mathbb{C}^{2})$ involving the realizations
$\widetilde{\mathcal{R}},\widetilde{\mathcal{R}}^{*}$ of
$\mathcal{R},\mathcal{R}^{*}$ in $H^{-1/2}(\Sigma;\mathbb{C}^{2})$. In the
following proposition we show that zero belongs to the essential spectrum of
$\Lambda(\widetilde{\mathcal{R}}\widetilde{\mathcal{R}}^{*}-4I_{2})\Lambda$
under some suitable assumptions on the surface $\Sigma$. Remark that in the
recent paper [15], that was developed independently from this note, a related
result was shown with pseudodifferential techniques.

######  Proposition 4.11.

Assume that there exists an open subset $\Sigma_{0}\subset\Sigma$ such that
$\Sigma_{0}$ is contained in a plane. Then, there exists a sequence
$(\varphi_{n})\subset L^{2}(\Sigma;\mathbb{C}^{2})$ such that
$\|\varphi_{n}\|_{L^{2}(\Sigma;\mathbb{C}^{2})}=1$, $\varphi_{n}$ converges
weakly to zero, and
$\Lambda(\widetilde{\mathcal{R}}\widetilde{\mathcal{R}}^{*}-4I_{2})\Lambda\varphi_{n}\rightarrow
0$ in $L^{2}(\Sigma;\mathbb{C}^{2})$. It is possible to choose this sequence
such that $(\Lambda\varphi_{n})\subset\mathcal{H}_{\pm}$.

###### Proof.

The proof of this proposition is split into two steps. In Step 1 we follow
closely arguments from [7, Theorem 5.9 (i)] to show that there exists a
singular sequence $(\varphi_{n})$ for
$\Lambda(\widetilde{\mathcal{R}}\widetilde{\mathcal{R}}^{*}-4I_{2})\Lambda$,
while in Step 2 we prove that this sequence $(\varphi_{n})$ can be chosen such
that $\Lambda\varphi_{n}\in\mathcal{H}_{\pm}$.

Step 1: We show the existence of the singular sequence for
$\Lambda(\widetilde{\mathcal{R}}\widetilde{\mathcal{R}}^{*}-4I_{2})\Lambda$.
For this we note that by Proposition 4.8 (ii) the operator
$\Lambda(\widetilde{\mathcal{R}}\widetilde{\mathcal{R}}^{*}-4I_{2})\Lambda$ is
bounded and self-adjoint in $L^{2}(\Sigma;\mathbb{C}^{2})$ and hence, the
claim follows, if we can show that
$0\in\sigma_{\textup{ess}}(\Lambda(\widetilde{\mathcal{R}}\widetilde{\mathcal{R}}^{*}-4I_{2})\Lambda)$,
which is true, if

(4.31) $\nexists\mathcal{G}\subset L^{2}(\Sigma;\mathbb{C}^{2}):{\rm
dim\,}\mathcal{G}<\infty\text{ and }{\rm
ran\,}(\Lambda(\widetilde{\mathcal{R}}\widetilde{\mathcal{R}}^{*}-4I_{2})\Lambda)=L^{2}(\Sigma;\mathbb{C}^{2})\ominus\mathcal{G}.$

Let $\Sigma_{1}\subset\Sigma_{0}$ such that
$\overline{\Sigma_{1}}\subset\Sigma_{0}$, where the closure is understood in
the relative topology on $\Sigma_{0}$. Following closely arguments from Step 4
from the proof of [7, Theorem 5.9 (i)], we will show that
$((\widetilde{\mathcal{R}}^{*}-\widetilde{\mathcal{R}})\Lambda\varphi)\upharpoonright\Sigma_{1}\in
H^{1}(\Sigma_{1};\mathbb{C}^{2})$ for all $\varphi\in
L^{2}(\Sigma;\mathbb{C}^{2})$. Since Proposition 4.8 (ii) implies that
$\Lambda\widetilde{\mathcal{R}}$ is a bijection from
$H^{1/2}(\Sigma;\mathbb{C}^{2})$ to $L^{2}(\Sigma;\mathbb{C}^{2})$, this
yields (4.31).

Recall that $\widetilde{\mathcal{R}}^{*}-\widetilde{\mathcal{R}}$ is an
integral operator with integral kernel

$t(x,y):=r(y,x)^{*}-r(x,y)=\frac{\sigma\cdot(x-y)}{\pi|x-y|^{3}}\sigma\cdot(\nu(y)-\nu(x))+2\frac{\nu(x)\cdot(x-y)}{\pi|x-y|^{3}},$

cf. (4.21). Clearly, $t(x,y)=0$ holds for all $x,y\in\Sigma_{0}$. Let
$U_{1}\subset\mathbb{R}^{2}$ and $\phi:U_{1}\rightarrow\mathbb{R}^{3}$ be a
map that parametrizes $\Sigma_{1}$, i.e. ${\rm ran\,}\phi=\Sigma_{1}$. Since
for any $y\in\Sigma$ the map $U_{1}\ni u\mapsto t(\phi(u),y)$ is
$C^{\infty}$-smooth and $\Sigma\ni y\mapsto\partial_{u_{j}}t(\phi(u),y)$ is
$C^{1}$-smooth for any fixed $j\in\\{1,2\\}$ and $u\in U_{1}$, we find for any
$\psi\in L^{2}(\Sigma;\mathbb{C}^{2})$ that

$\partial_{u_{j}}\big{(}(\widetilde{\mathcal{R}}^{*}-\widetilde{\mathcal{R}})\psi\big{)}(\phi(u))=\int_{\Sigma}\partial_{u_{j}}t(\phi(u),y)\psi(y)\text{d}\sigma(y)$

holds. With this, we get

$\begin{split}\big{\|}\partial_{u_{j}}\big{(}(\widetilde{\mathcal{R}}^{*}-\widetilde{\mathcal{R}})\psi\big{)}\big{\|}_{L^{2}(\Sigma_{1};\mathbb{C}^{2})}^{2}&=C_{1}\int_{U_{1}}\left|\int_{\Sigma}\partial_{u_{j}}t(\phi(u),y)\psi(y)\text{d}\sigma(y)\right|^{2}\text{d}u\\\
&=C_{1}\int_{U_{1}}\left|\big{(}\partial_{u_{j}}t(\phi(u),\cdot),\psi\big{)}_{H^{1/2}(\Sigma;\mathbb{C}^{2})\times
H^{-1/2}(\Sigma;\mathbb{C}^{2})}\right|^{2}\text{d}u\\\ &\leq
C_{1}\int_{U_{1}}\big{\|}\partial_{u_{j}}t(\phi(u),\cdot)\big{\|}_{H^{1/2}(\Sigma;\mathbb{C}^{2\times
2})}^{2}\cdot\|\psi\|_{H^{-1/2}(\Sigma;\mathbb{C}^{2})}^{2}\text{d}u\\\
&=C_{2}\|\psi\|_{H^{-1/2}(\Sigma;\mathbb{C}^{2})}^{2}.\end{split}$

By continuity, this can be extended for all $\psi\in
H^{-1/2}(\Sigma;\mathbb{C}^{2})$, which shows that
$(\widetilde{\mathcal{R}}^{*}-\widetilde{\mathcal{R}})\Lambda\varphi)\upharpoonright\Sigma_{1}\in
H^{1}(\Sigma_{1};\mathbb{C}^{2})$ for all $\varphi\in
L^{2}(\Sigma;\mathbb{C}^{2})$ and yields the claim of the first step.

Step 2: By the result of Step 1 there exists a sequence $(\varphi_{n})\subset
L^{2}(\Sigma;\mathbb{C}^{2})$ such that
$\|\varphi_{n}\|_{L^{2}(\Sigma;\mathbb{C}^{2})}=1$, $\varphi_{n}$ converges
weakly to zero, and
$\Lambda(\widetilde{\mathcal{R}}\widetilde{\mathcal{R}}^{*}-4I_{2})\Lambda\varphi_{n}\rightarrow
0$ in $L^{2}(\Sigma;\mathbb{C}^{2})$. It remains to verify that this sequence
can be chosen such that $\Lambda\varphi_{n}\in\mathcal{H}_{\pm}$.

Clearly, at least one of the sequences
$(\Lambda^{-1}P_{\pm}\Lambda\varphi_{n})$ has the claimed properties. So we
assume w.l.o.g. that $(\Lambda\varphi_{n})\subset\mathcal{H}_{+}$ such that
$\|\varphi_{n}\|_{L^{2}(\Sigma;\mathbb{C}^{2})}=1$, $\varphi_{n}$ converges
weakly to zero, and
$\Lambda(\widetilde{\mathcal{R}}\widetilde{\mathcal{R}}^{*}-4I_{2})\Lambda\varphi_{n}\rightarrow
0$ in $L^{2}(\Sigma;\mathbb{C}^{2})$. But then, in view of Remark 4.10, also
the sequence defined by
$\psi_{n}:=\Lambda^{-1}(\sigma\cdot\nu)\Lambda\varphi_{n}$ satisfies
$\|\psi_{n}\|_{L^{2}(\Sigma;\mathbb{C}^{2})}\geq c$ for some $c>0$, $\psi_{n}$
converges weakly to zero, and $\Lambda\psi_{n}\in\mathcal{H}_{-}$. Moreover,
the relations in (4.5) and Proposition 4.8 (ii) imply that

$\begin{split}\Lambda(\widetilde{\mathcal{R}}\widetilde{\mathcal{R}}^{*}-4I_{2})\Lambda\psi_{n}&=\Lambda(\widetilde{\mathcal{R}}\widetilde{\mathcal{R}}^{*}-4I_{2})(\sigma\cdot\nu)\Lambda\varphi_{n}=\Lambda(\sigma\cdot\nu)(\widetilde{\mathcal{R}}^{*}\widetilde{\mathcal{R}}-4I_{2})\Lambda\varphi_{n}\\\
&=\Lambda(\sigma\cdot\nu)\widetilde{\mathcal{R}}^{*}\widetilde{\mathcal{R}}\left(I_{2}-\frac{1}{4}\widetilde{\mathcal{R}}\widetilde{\mathcal{R}}^{*}\right)\Lambda\varphi_{n}\rightarrow
0\end{split}$

in $L^{2}(\Sigma;\mathbb{C}^{2})$. Thus, also $(\psi_{n})$ fulfils all claimed
properties and all claims are shown. ∎

Finally, combining the results from Lemma 4.7 and Theorem 4.9, we get the
following decomposition of $\mathcal{C}_{z}$ into a positive and a negative
part.

######  Corollary 4.12.

Let $\mathcal{C}_{z}$, $z\in(-m,m)$, be defined by (2.4), let $P_{\pm}$ be as
in Theorem 4.9, and let $V$ be given by (4.15). Then, for any $s\in[-1,1]$
there exists a bounded operator
$\mathcal{K}:H^{s}(\Sigma;\mathbb{C}^{4})\rightarrow
H^{\tau}(\Sigma;\mathbb{C}^{4})$ with $\tau={\rm min\,}\\{1,s+1\\}$ such that

$\mathcal{C}_{z}=\frac{1}{2}(\alpha\cdot\nu)V^{*}\begin{pmatrix}0&P_{-}-P_{+}\\\
P_{-}-P_{+}&0\end{pmatrix}V(\alpha\cdot\nu)+\mathcal{K}.$

In particular, the realization of $\mathcal{K}$ in
$L^{2}(\Sigma;\mathbb{C}^{4})$ is self-adjoint.

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