#  On two-dimensional Dirac operators with $\delta$-shell interactions
supported on unbounded curves with straight ends

Jussi Behrndt  Institut für Angewandte Mathematik
Technische Universität Graz
Steyrergasse 30, 8010 Graz, Austria
E-mail<EMAIL_ADDRESS>,  Pavel Exner  Doppler Institute for Mathematical
Physics and Applied Mathematics
Czech Technical University in Prague
Břehová 7, 11519 Prague, Czech Republic, and Department of Theoretical Physics
Nuclear Physics Institute, Czech Academy of Sciences, 25068 Řež, Czech
Republic
E-mail<EMAIL_ADDRESS>,  Markus Holzmann  Institut für Angewandte
Mathematik
Technische Universität Graz
Steyrergasse 30, 8010 Graz, Austria
E-mail<EMAIL_ADDRESS>and  Matěj Tušek  Department of Mathematics,
Faculty of Nuclear Sciences and Physical Engineering
Czech Technical University in Prague
Trojanova 13, 120 00, Prague
E-mail<EMAIL_ADDRESS>
###### Abstract.

In this paper we study the self-adjointness and spectral properties of two-
dimensional Dirac operators with electrostatic, Lorentz scalar, and anomalous
magnetic $\delta$-shell interactions with constant weights that are supported
on a smooth unbounded curve that is straight outside a compact set and whose
ends are rays that are not parallel to each other. For all possible
combinations of interaction strengths we describe the self-adjoint
realizations and compute their essential spectra. Moreover, we prove in
different situations the existence of geometrically induced discrete
eigenvalues.

###### Key words and phrases:

Dirac operator, singular interaction, essential spectrum, geometrically
induced bound states

###### 2020 Mathematics Subject Classification:

Primary 81Q10; Secondary 35Q40

##  1\. Introduction

Operators with contact interactions are frequently used as quantum mechanical
Hamiltonians. True, as models of real physical systems they are idealized but
often they provide a description which preserves the salient features of the
system one wants to describe while being mathematically more accessible than
their ‘regular’ counterparts. In nonrelativistic quantum mechanics there is
wealth of such models; for a survey and bibliography we refer to the monograph
[2]. The motivation to investigate Dirac operators with contact interactions
was weaker for a long time; the only system of that type appeared in
connection with the so-called MIT bag model [23]. This changed with the advent
of graphene and similar materials, in which the effective dynamics is
described by the Dirac equation, see, e.g., [5], and the systems of interest
may have many geometric features depending on their experimental design.

The goal in this paper is to study the self-adjointness and the spectrum of a
Dirac operator in $\mathbb{R}^{2}$ with a combination of electrostatic,
Lorentz scalar, and anomalous magnetic $\delta$-shell potentials supported on
an unbounded curve $\Gamma$ that is straight outside a compact set and such
that its ends are not parallel to each other. Such an operator is associated
with the formal differential expression

$\mathcal{A}_{\eta,\tau,\lambda}^{m,c}:=-ic(\sigma\cdot\nabla)+mc^{2}\sigma_{3}+(\eta\sigma_{0}+\tau\sigma_{3}+\lambda
i(\sigma_{1}\nu_{1}+\sigma_{2}\nu_{2})\sigma_{3})\delta_{\Gamma},$ (1.1)

where $c>0$ is the speed of light, $m\geq 0$ is the mass of the particle (note
that we will later consider $m\in\mathbb{R}$), $\delta_{\Gamma}$ stands for
the $\delta$\- or single layer distribution supported on $\Gamma$ and
$\nu=(\nu_{1},\nu_{2})$ is a unit normal vector field along $\Gamma$; finally,
$\eta,\tau,\lambda\in\mathbb{R}$ are coupling constants for the electrostatic,
Lorentz scalar, and anomalous magnetic interaction, respectively. It is known
that with these three parameters one can define a general symmetric
$\delta$-interaction with constant coefficients for the two-dimensional Dirac
operator; a fourth possible parameter in the $2\times 2$-matrix-valued
coefficient of $\delta_{\Gamma}$ can be gauged away by a unitary
transformation, see [16, 22].

Operators of the form $\mathcal{A}_{\eta,\tau,\lambda}^{m,c}$ were first
studied in the one-dimensional case [32], in which the singular interaction is
supported on a point in $\mathbb{R}$. By using this analysis and a
decomposition to spherical harmonics, the three-dimensional counterpart of the
operator in (1.1) was investigated in the end of the 1980s with the
interaction support being a sphere [27]. It took then 25 years, until more
general bounded $C^{2}$-surfaces in $\mathbb{R}^{3}$ were considered as
supports of purely electrostatic interactions in [3]. After that the interest
on the operators induced by (1.1) grew, the more general interactions
including Lorentz scalar and anomalous magnetic terms were introduced, and the
self-adjointness, the spectral and scattering properties were studied for
bounded interaction supports in dimensions two and three, see [4, 7, 8, 11,
22], to mention just a few of references, and the recent review [13] for more
references. The case of unbounded interaction supports was not studied that
extensively, despite the fact that for the case of the straight line in
$\mathbb{R}^{2}$ interesting spectral transitions were observed in [15, 16].
We also mention the paper [18], where the self-adjointness and spectral
properties for combinations of electrostatic and Lorentz scalar interactions
supported on surfaces in $\mathbb{R}^{3}$, which are compact perturbations of
the plane, were investigated, the works [31, 39, 42, 43] about the self-
adjointness of the operator in (1.1) under different assumptions on the
interaction support including non-smooth broken lines, and the recent paper
[30], where the spectrum of the operator $\mathcal{A}_{0,\tau,0}^{m,c}$ was
studied, when $\Gamma$ is a broken line.

It is the main goal in this paper to treat the two-dimensional Dirac operator
in (1.1) with a singular potential that is supported on a smooth unbounded
curve in $\mathbb{R}^{2}$ that is straight outside a compact set in a
systematic way. As for bounded interaction supports, the operator associated
with $\mathcal{A}_{\eta,\tau,\lambda}^{m,c}$ lacks semi-boundedness and hence,
form methods are not suitable to introduce it as a self-adjoint operator.
Therefore, a different technique from extension theory of symmetric operators
will be applied. We follow closely the approach in [13], where two- and three-
dimensional Dirac operators with singular potentials supported on bounded
interaction supports were studied with the help of boundary triples and their
Weyl functions. More precisely, we construct a suitable ordinary boundary
triple, which allows us to introduce $\mathcal{A}_{\eta,\tau,\lambda}^{m,c}$
as a self-adjoint operator for all combinations of the interaction strengths
$\eta,\tau,\lambda\in\mathbb{R}$ and to identify the essential spectrum. In a
similar way as it is known for bounded interaction supports, it turns out that
there are special combinations of the interaction strengths
$\eta,\tau,\lambda$ for which there is a lack of Sobolev regularity in the
operator domain. Moreover, as in the case of the straight line [15, 16],
unexpected transitions in the essential spectra appear in several situations.

Finally, we are going to show for particular geometries of the interaction
support and/or parameter values the existence of isolated eigenvalues.
Relations between spectral properties of differential operators and the
geometry belong to the most traditional questions, and one of the intriguing
examples is the fact that geometrically induced bound states may exist even in
regions which are spatially infinitely extended. A classical example is the
Dirichlet Laplacian in strips around a bent reference curve, for which we
refer to the monograph [29] and the bibliography therein. The same effect was
observed for two-dimensional Schrödinger operators with attractive
$\delta$-potentials supported on unbounded curves [28, 38], and recently also
for Dirac operators with Lorentz scalar $\delta$-shell interactions on broken
lines [30]. We will prove that also for smooth unbounded interaction supports
$\Gamma$ that are straight outside a compact set geometrically induced
eigenvalues may exist. We do this first in the case, where the electrostatic
and the Lorentz scalar interaction strengths have the same negative value, no
magnetic term is present, and the speed of light $c$ is large by using the
nonrelativistic limit of the Dirac operator in combination with the known
result about the corresponding Schrödinger operator from [28]. Secondly, we
prove the existence of geometrically induced bound states for attractive
purely Lorentz scalar interactions that are supported on a compact
perturbation of a broken line with a sufficiently small opening angle by using
the claim proved in the paper [30]. While such existence results are of
obvious interest, they lack the universal character of their nonrelativistic
counterparts mentioned above, and their extension to a wider class of
operators represents an important challenge.

The paper is organized as follows: In Section 2 we state and discuss the main
results of this paper in detail. Then, in Section 3 we provide some
preliminary material that is necessary to prove the main results in Sections 4
and 5. Finally, in Appendix A we recall some useful abstract results about
ordinary boundary triples, and in Appendix B we prove a technical statement
about compactness properties of integral operators.

### Notations

Let

$\sigma_{1}:=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\qquad\sigma_{2}:=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\qquad\sigma_{3}:=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},$
(1.2)

be the Pauli spin matrices and denote by $\sigma_{0}$ the $2\times 2$-identity
matrix. Note that the Pauli matrices satisfy

$\sigma_{j}\sigma_{k}+\sigma_{k}\sigma_{j}=2\delta_{jk}\sigma_{0},\quad
j,k\in\\{1,2,3\\}.$ (1.3)

For $x=(x_{1},x_{2})$ we will often write $\sigma\cdot
x=\sigma_{1}x_{1}+\sigma_{2}x_{2}$ and
$\sigma\cdot\nabla=\sigma_{1}\partial_{1}+\sigma_{2}\partial_{2}$.

Let $\Omega$ be an open set. The $L^{2}$-based Sobolev spaces of $k$ times
weakly differentiable functions are denoted by $H^{k}(\Omega;\mathbb{C})$ and
we write
$H^{k}(\Omega;\mathbb{C}^{2}):=H^{k}(\Omega;\mathbb{C})\otimes\mathbb{C}^{2}$.
Moreover, if $\Gamma$ is an infinite curve that is straight outside a compact
set as in (2.1) with arc-length parametrization
$\gamma:\mathbb{R}\rightarrow\mathbb{R}^{2}$, then we define the mapping

$U_{\gamma}:\mathcal{S}^{\prime}(\Gamma;\mathbb{C})\rightarrow\mathcal{S}^{\prime}(\mathbb{R};\mathbb{C}),\quad(U_{\gamma}f)(\varphi)=f(\varphi(\gamma^{-1}(\cdot))).$
(1.4)

Clearly, if $f\in L^{1}(\Gamma;\mathbb{C})$, then the distribution
$U_{\gamma}f\in L^{1}(\mathbb{R};\mathbb{C})$ is generated by $f\circ\gamma$.
With the help of $U_{\gamma}$ we introduce the Sobolev space of order
$s\in\mathbb{R}$ on $\Gamma$ by

$H^{s}(\Gamma):=\big{\\{}f\in\mathcal{S}^{\prime}(\Gamma;\mathbb{C}):U_{\gamma}f\in
H^{s}(\mathbb{R};\mathbb{C})\big{\\}}.$

In the main part of the paper we will often use the mapping $\Lambda$ defined
by

$(\mathcal{F}U_{\gamma}\Lambda
f)(p)=\sqrt[4]{1+p^{2}}\,(\mathcal{F}U_{\gamma}f)(p),\quad p\in\mathbb{R},$
(1.5)

where $\mathcal{F}$ is the Fourier transform in
$\mathcal{S}^{\prime}(\mathbb{R};\mathbb{C})$. Evidently, for any
$s\in\mathbb{R}$ the operator $\Lambda:H^{s}(\Gamma;\mathbb{C})\rightarrow
H^{s-1/2}(\Gamma;\mathbb{C})$ is well defined and bijective and for
$s=\frac{1}{2}$ it can be viewed as an unbounded self-adjoint operator in
$L^{2}(\Gamma;\mathbb{C})$ with dense domain $H^{1/2}(\Gamma;\mathbb{C})$.

If $\mathcal{H}$ is a Hilbert space and $A$ is a closed operator in
$\mathcal{H}$, then we denote its domain of definition, its range, and its
kernel by $\mathrm{dom}\,A$, $\operatorname{ran}A$, and $\ker A$,
respectively. Furthermore, we use the symbols $\sigma(A)$,
$\sigma_{\textup{p}}(A)$, and $\rho(A)$ for the spectrum, the point spectrum,
and the resolvent set of $A$, respectively. If $A$ is self-adjoint, then
$\sigma_{\textup{disc}}(A)$ and $\sigma_{\textup{ess}}(A)$ denote the discrete
and the essential spectrum of $A$, respectively.

##  2\. Main results

###  2.1. Definition of $A_{\eta,\tau,\lambda}^{m,c}$ and confinement

Throughout the paper, we assume that the support of the singular interaction
is a smooth curve that does not intersect with itself and that is straight
outside a compact set. To make this mathematically more precise, we make the
following assumption:

######  Hypothesis 2.1.

Assume that $\gamma:\mathbb{R}\rightarrow\mathbb{R}^{2}$ is a
$C^{\infty}$-smooth and injective map that satisfies $|\dot{\gamma}(s)|=1$ for
all $s\in\mathbb{R}$ and

$\gamma(s)=\begin{cases}(a_{-}s+b_{-},c_{-}s+d_{-})&\text{ for }s<-M,\\\
(a_{+}s+b_{+},c_{+}s+d_{+})&\text{ for }s>M,\end{cases}$ (2.1)

for suitable constants $M>0$ and
$a_{\pm},b_{\pm},c_{\pm},d_{\pm}\in\mathbb{R}$ with
$a_{\pm}^{2}+c_{\pm}^{2}=1$ and $(a_{-},c_{-})\neq-(a_{+},c_{+})$. We set
$\Gamma:=\operatorname{ran}\gamma$ and denote by $\nu(x)$ the unit normal
vector at $x=\gamma(s)\in\Gamma$, $s\in\mathbb{R}$, that is given by
$\nu(\gamma(s))=(\dot{\gamma}_{2}(s),-\dot{\gamma}_{1}(s))$. The set
$\Omega_{+}$ is the domain with boundary $\Gamma$ such that $\nu$ is pointing
outwards of $\Omega_{+}$ and we define
$\Omega_{-}:=\mathbb{R}^{2}\setminus\overline{\Omega_{+}}$.

If not mentioned differently, we always assume that $\Gamma$ satisfies the
above hypothesis. The condition $(a_{-},c_{-})\neq-(a_{+},c_{+})$ in
Hypothesis 2.1 means that the two asymptotes of $\Gamma$ are not parallel rays
(of the same orientation) as $t\to\pm\infty$. Note that for $\gamma$ in
Hypothesis 2.1 there exists $C_{1}>0$ such that for all $s,t\in\mathbb{R}$

$C_{1}|s-t|\leq|\gamma(s)-\gamma(t)|\leq|s-t|$ (2.2)

holds. The second inequality in (2.2) is clear. To see the first one consider
the function $(s,t)\mapsto r(s,t):=\frac{|\gamma(s)-\gamma(t)|}{|s-t|}$, which
is positive and continuous, being extended to $r(s,s)=1$ as $\Gamma$ is smooth
by assumption. In the first and the third quadrant of the $(s,t)$ plane we
have $r(s,t)=1$ for all $s,t$ large enough. Moreover, for any
$a\in[0,+\infty)$,

$\lim_{s\to\pm\infty}\inf_{\pm
t\in[0,a]}r(s,t)=\lim_{s\to\pm\infty}\frac{|\gamma(s)|}{|s|}=1,$

and similarly with the roles of $s$ and $t$ interchanged. In the other two
quadrants we have $r(s,t)\leq 1$ and it is not difficult to check that

$\lim_{|s|\to\infty}r(s,-\mu s)=\frac{\sqrt{1+\mu^{2}-2\mu\cos\theta}}{1+\mu}$

holds for $\mu>0$, where $\theta$ is the angle between the asymptotes; the
right-hand side reaches its minimum $\sin\frac{\theta}{2}$ at the axis of the
quadrants, $\mu=1$. We conclude that there is a compact subset $M$ of the
plane outside of which we have $r(s,t)>\frac{1}{2}\sin\frac{\theta}{2}$. In
view of the compactness, the minimum is attained within $M$ and
$C_{2}:=\min_{M}r(s,t)>0$, because the curve does not intersect itself; it is
then enough to choose $C_{1}=\min\\{C_{2},\frac{1}{2}\sin\frac{\theta}{2}\\}$.

In the following we use the notation $f_{\pm}:=f\upharpoonright\Omega_{\pm}$
for a function $f$ that is defined on $\mathbb{R}^{2}$. Recall that
$\sigma_{1},\sigma_{2},\sigma_{3}$ are the $\mathbb{C}^{2\times 2}$-valued
Pauli spin matrices defined in (1.2) and $\sigma_{0}$ is the $2\times 2$
identity matrix. We define the spaces

$H(\sigma,\Omega_{\pm}):=\big{\\{}f_{\pm}\in
L^{2}(\Omega_{\pm};\mathbb{C}^{2}):(\sigma\cdot\nabla)f_{\pm}\in
L^{2}(\Omega_{\pm};\mathbb{C}^{2})\big{\\}}$

and note that for $f_{\pm}\in H(\sigma;\Omega_{\pm})$ the trace
$T^{D}_{\pm}f_{\pm}$ exists in the Sobolev space
$H^{-1/2}(\Gamma;\mathbb{C}^{2})$; cf. Section 3.1 for details. This allows us
to define for $m,\eta,\tau,\lambda\in\mathbb{R}$ and $c>0$ in
$L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})$ the operator

$\begin{split}&\,\,\,\,\,\,\,\,\,A_{\eta,\tau,\lambda}^{m,c}f:=\big{(}-ic(\sigma\cdot\nabla)+mc^{2}\sigma_{3}\big{)}f_{+}\oplus\big{(}-ic(\sigma\cdot\nabla)+mc^{2}\sigma_{3}\big{)}f_{-},\\\
&\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}:=\bigg{\\{}f=f_{+}\oplus f_{-}\in
H(\sigma,\Omega_{+})\oplus H(\sigma,\Omega_{-}):\\\ &\quad-
ic(\sigma\cdot\nu)(T^{D}_{+}f_{+}-T^{D}_{-}f_{-})=\frac{1}{2}(\eta\sigma_{0}+\tau\sigma_{3}+\lambda
i(\sigma\cdot\nu)\sigma_{3})(T^{D}_{+}f_{+}+T^{D}_{-}f_{-})\bigg{\\}}.\end{split}$
(2.3)

Using integration by parts, it can be seen that the operator in (2.3) is the
rigorous mathematical definition of the formal expression in (1.1), see, e.g.,
[19].

A remarkable property of $A_{\eta,\tau,\lambda}^{m,c}$ is that it decomposes
for special combinations of $\eta,\tau,\lambda$ to the orthogonal sum of two
Dirac operators in $L^{2}(\Omega_{\pm};\mathbb{C}^{2})$ with boundary
conditions. This phenomenon, which was observed originally in the three-
dimensional setting and later in the current two-dimensional case (but always
with bounded interaction supports) in [4, 8, 11, 13, 22, 27], is known as
confinement and means that the $\delta$-potential is, in this case,
impenetrable for the quantum particle. The mentioned boundary conditions are
known as quantum dot boundary conditions and play an important role in the
mathematical analysis of graphene [17]. Note that in all studied cases the
confinement occurs if and only if the parameter

$d:=\eta^{2}-\tau^{2}-\lambda^{2}$ (2.4)

satisfies $d=-4c^{2}$ and the same is also true in the current setting.

######  Proposition 2.2.

Let $\Gamma$ be as in Hypothesis 2.1, $m,\eta,\tau,\lambda\in\mathbb{R}$ and
$c>0$. Then, the following is true:

  * (i)

If $d\neq-4c^{2}$, then there exists a map
$Q_{\eta,\tau,\lambda}:\Gamma\rightarrow\mathbb{C}^{2\times 2}$ depending on
the parameters $\eta,\tau,\lambda$ such that $Q_{\eta,\tau,\lambda}$ is
pointwise invertible and $f\in\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}$ if
and only if $f\in H(\sigma;\Omega_{+})\oplus H(\sigma;\Omega_{-})$ and

$T^{D}_{+}f_{+}=Q_{\eta,\tau,\lambda}T^{D}_{-}f_{-}.$

  * (ii)

If $d=-4c^{2}$, then
$A_{\eta,\tau,\lambda}^{m,c}=A_{\eta,\tau,\lambda}^{+}\oplus
A_{\eta,\tau,\lambda}^{-}$, where $A_{\eta,\tau,\lambda}^{\pm}$ are the
operators acting in $L^{2}(\Omega_{\pm};\mathbb{C}^{2})$ given by

$\begin{split}A_{\eta,\tau,\lambda}^{\pm}f&=\big{(}-ic(\sigma\cdot\nabla)+mc^{2}\sigma_{3}\big{)}f,\\\
\mathrm{dom}\,A_{\eta,\tau,\lambda}^{\pm}&=\big{\\{}f\in
H(\sigma;\Omega_{\pm}):\big{(}2c\sigma_{0}\mp
i(\sigma\cdot\nu)(\eta\sigma_{0}+\tau\sigma_{3}+\lambda
i(\sigma\cdot\nu)\sigma_{3})\big{)}T^{D}_{\pm}f=0\big{\\}}.\end{split}$

The proof of this result can be done in exactly the same way as, e.g., in [4,
Theorem 5.5], [11, Lemma 4.1], [13, Lemma 5.11], or [22, Theorem 2.3], and
hence, we omit it in this paper.

###  2.2. Self-adjointness and essential spectrum of
$A_{\eta,\tau,\lambda}^{m,c}$

In the following theorem the self-adjointness and the essential spectrum of
$A_{\eta,\tau,\lambda}^{m,c}$ are discussed. Note that, in a similar way as it
was observed in [10, 11, 13, 22, 37] for compact interaction supports, there
is a loss of Sobolev regularity in the operator domain in the case of critical
interaction strengths $(\tfrac{d}{4}-c^{2})^{2}-\lambda^{2}c^{2}=0$. Moreover,
in the particular case of $\Gamma$ being a straight line the spectrum of
$A_{\eta,\tau,\lambda}^{m,c}$ was computed in [16]; Theorem 2.3 says that the
essential spectrum is preserved under the geometric perturbations as long as
$\Gamma$ is straight outside a compact set. We point out that to the best of
our knowledge item (i) of the following theorem on the Sobolev regularity in
$\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}$ was not known in this generality
before even for the straight line.

######  Theorem 2.3.

Let $m,\eta,\tau,\lambda\in\mathbb{R}$, $c>0$, $d$ be given in (2.4), and
$\Gamma$ be as in Hypothesis 2.1. Then, the operator
$A_{\eta,\tau,\lambda}^{m,c}$ is self-adjoint in
$L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})$ and the following holds:

  * (i)

For the domain of $A_{\eta,\tau,\lambda}^{m,c}$ one has:

    * (a)

If $(\tfrac{d}{4}-c^{2})^{2}-\lambda^{2}c^{2}\neq 0$, then
$\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}\subset
H^{1}(\mathbb{R}^{2}\setminus\Gamma;\mathbb{C}^{2})$.

    * (b)

If $(\tfrac{d}{4}-c^{2})^{2}-\lambda^{2}c^{2}=0$, then
$\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}\not\subset
H^{s}(\mathbb{R}^{2}\setminus\Gamma;\mathbb{C}^{2})$ for all $s>0$.

  * (ii)

For the essential spectrum of $A_{\eta,\tau,\lambda}^{m,c}$ one has:

    * (a)

If $d=4c^{2}$ and $\lambda\neq 0$, then

$\sigma_{\textup{ess}}(A_{\eta,\tau,\lambda}^{m,c})=\sigma(A_{\eta,\tau,\lambda}^{m,c})=\mathbb{R}.$

    * (b)

If $d=4c^{2}$ and $\lambda=0$, then

$\sigma_{\textup{ess}}(A_{\eta,\tau,\lambda}^{m,c})=(-\infty,-|m|c^{2}]\cup\left\\{-\frac{\tau}{\eta}mc^{2}\right\\}\cup[|m|c^{2},+\infty).$

    * (c)

If $d\neq 4c^{2}$, then

$(-\infty,-|m|c^{2}]\cup[|m|c^{2},+\infty)\subset\sigma_{\textup{ess}}(A_{\eta,\tau,\lambda}^{m,c})$

and
$\sigma_{\textup{ess}}(A_{\eta,\tau,\lambda}^{m,c})\cap(-|m|c^{2},|m|c^{2})$
coincides with the set

$\overline{\left\\{z_{\pm}(k):(d-4c^{2})\left(\frac{\eta}{c^{2}}z_{\pm}(k)+\lambda
k+\tau m\right)>0\right\\}}\cap(-|m|c^{2},|m|c^{2}),$

where $z_{\pm}(k)$, $k\in\mathbb{R}$, is given by

$\begin{split}z_{\pm}(k)&=\frac{1}{\tfrac{\eta^{2}}{c^{2}}+\big{(}\tfrac{d}{4c^{2}}-1\big{)}^{2}}\bigg{(}-\eta(\lambda
k+\tau m)\\\
&\,\,\pm\left|\frac{d}{4c^{2}}-1\right|\sqrt{\big{(}\tau^{2}c^{2}+\big{(}\tfrac{d}{4}+c^{2}\big{)}^{2}\big{)}k^{2}-2\lambda\tau
mkc^{2}+\big{(}\lambda^{2}c^{2}+\big{(}\tfrac{d}{4}+c^{2}\big{)}^{2}\big{)}m^{2}}\bigg{)}.\end{split}$

Moreover, if $\Gamma$ is the straight line, then
$\sigma(A_{\eta,\tau,\lambda}^{m,c})=\sigma_{\textup{ess}}(A_{\eta,\tau,\lambda}^{m,c})$,
i.e. the spectrum of $A_{\eta,\tau,\lambda}^{m,c}$ is fully characterized by
(ii) in this case.

We point out that an interesting spectral transition appears in the case when
$d=4c^{2}$: If $\lambda\neq 0$, then always
$\sigma(A_{\eta,\tau,\lambda}^{m,c})=\mathbb{R}$, but for $\lambda=0$ the
essential spectrum within the interval $(-|m|c^{2},|m|c^{2})$ breaks down to
the single point $-\frac{\tau}{\eta}mc^{2}$.

Theorem 2.3 will be proved in Section 4 with the help of a suitable boundary
triple. In Subsection 4.4 we will also show a variant of the Birman-Schwinger
principle, a Krein type resolvent formula, and an isospectral relation that
are useful in the analysis of $A_{\eta,\tau,\lambda}^{m,c}$.

###  2.3. Special cases

The next results follow immediately from Theorem 2.3 (and thus, no proofs of
these corollaries are given), but we prefer to state them here as they are
more explicit than Theorem 2.3 and of particular interest in applications.
First, we discuss the case of purely electrostatic interactions, i.e. when
$\eta\in\mathbb{R}$ and $\tau=\lambda=0$. Note that, as in the case of the
straight line [15], there appears a spectral transition at $\eta=\pm 2c$,
where an interval of continuous spectrum of $A_{\eta,0,0}^{m,c}$ collapses to
a single point in the essential spectrum.

######  Corollary 2.4.

Let $\eta\in\mathbb{R}$. Then, one has

$\sigma_{\textup{ess}}(A_{\eta,0,0}^{m,c})=\begin{cases}(-\infty,\frac{4c^{2}-\eta^{2}}{4c^{2}+\eta^{2}}|m|c^{2}]\cup[|m|c^{2},+\infty)&\text{
for }\eta\in(-\infty,-2c),\\\
(-\infty,-|m|c^{2}]\cup\\{0\\}\cup[|m|c^{2},+\infty)&\text{ for }\eta=-2c,\\\
(-\infty,-|m|c^{2}]\cup[\frac{4c^{2}-\eta^{2}}{4c^{2}+\eta^{2}}|m|c^{2},+\infty)&\text{
for }\eta\in(-2c,0),\\\ (-\infty,-|m|c^{2}]\cup[|m|c^{2},+\infty)&\text{ for
}\eta=0,\\\
(-\infty,\frac{\eta^{2}-4c^{2}}{\eta^{2}+4c^{2}}|m|c^{2}]\cup[|m|c^{2},+\infty)&\text{
for }\eta\in(0,2c),\\\
(-\infty,-|m|c^{2}]\cup\\{0\\}\cup[|m|c^{2},+\infty)&\text{ for }\eta=2c,\\\
(-\infty,-|m|c^{2}]\cup[\frac{\eta^{2}-4c^{2}}{\eta^{2}+4c^{2}}|m|c^{2},+\infty)&\text{
for }\eta\in(2c,+\infty).\end{cases}$

For purely Lorentz scalar interactions, i.e. for $\tau\in\mathbb{R}$ and
$\eta=\lambda=0$, the following result holds. Note that for $\tau m\geq 0$ the
essential spectrum of $A_{0,\tau,0}^{m,c}$ coincides with the spectrum of the
free Dirac operator.

######  Corollary 2.5.

Let $\tau\in\mathbb{R}$. Then, one has

$\sigma_{\textup{ess}}(A_{0,\tau,0}^{m,c})=\begin{cases}(-\infty,-|m|c^{2}]\cup[|m|c^{2},+\infty)&\text{
for }\tau m\geq 0,\\\
(-\infty,-\frac{|4c^{2}-\tau^{2}|}{4c^{2}+\tau^{2}}|m|c^{2}]\cup[\frac{|4c^{2}-\tau^{2}|}{4c^{2}+\tau^{2}}|m|c^{2},+\infty)&\text{
for }\tau m<0.\end{cases}$

Finally, we consider the case of purely anomalous magnetic interactions, i.e.
when $\lambda\in\mathbb{R}$ and $\eta=\tau=0$. Note that in this configuration
the essential spectrum of $A_{0,0,\lambda}^{m,c}$ always coincides with the
spectrum of the free Dirac operator.

######  Corollary 2.6.

Let $\lambda\in\mathbb{R}$. Then, one has

$\sigma_{\textup{ess}}(A_{0,0,\lambda}^{m,c})=(-\infty,-|m|c^{2}]\cup[|m|c^{2},+\infty).$

###  2.4. Geometrically induced bound states

In the present paper we prove the existence of geometrically induced bound
states in two situations: Firstly, for $A_{\eta/2,\eta/2,0}^{m,c}$ for large
values of $c$ with the help of the nonrelativistic limit, and secondly for
$A_{0,\tau,0}^{m,c}$ by using the test functions from [30].

The nonrelativistic limit is one way to relate a relativistic Dirac operator
$A^{m,c}$ with the Schrödinger operator $H$ that describes the same physical
system [45, Chapter 6]. To compute the nonrelativistic limit, one has to
subtract the particle’s rest energy $mc^{2}$, which is a purely relativistic
quantity, from the total energy and compute the limit in the operator norm of
the resolvent of the resulting operator $A^{m,c}-mc^{2}$ for
$c\rightarrow+\infty$. The expected limit is then the resolvent of $H$ times a
projection onto the upper component. This result is of interest by its own, as
it gives a physical interpretation of the model, but taking properties of
convergence of the resolvents into account, this allows to relate the spectral
properties of $H$ to those of $A^{m,c}$ for large values of $c$.

In this paper we consider a simple case and compute the nonrelativistic limit
of $A_{\eta/2,\eta/2,0}^{m,c}$, see also Remark 5.3 for a more general
consideration. In the following, we assume that $\Gamma$ is given as in
Hypothesis 2.1 and that $m>0$. First, we have to introduce the expected limit
operator, a Schrödinger operator with a $\delta$-potential supported on
$\Gamma$. Define for $\eta\in\mathbb{R}$ in $L^{2}(\mathbb{R}^{2};\mathbb{C})$
the sesquilinear form

$\mathfrak{h}_{\eta}[f,g]:=\frac{1}{2m}(\nabla f,\nabla
g)_{L^{2}(\mathbb{R}^{2};\mathbb{C})}+\eta(T^{D}f,T^{D}g)_{L^{2}(\Gamma;\mathbb{C})},\quad\mathrm{dom}\,\mathfrak{h}_{\eta}=H^{1}(\mathbb{R}^{2};\mathbb{C}),$
(2.5)

where $T^{D}:H^{1}(\mathbb{R};\mathbb{C})\rightarrow
H^{1/2}(\Gamma;\mathbb{C})$ is the Dirichlet trace operator. It is well-known
that $\mathfrak{h}_{\eta}$ is densely defined, symmetric, and closed [20, 28].
Hence, by the first representation theorem [36, Chapter VI, Theorem 2.1] there
exists a unique self-adjoint operator $H_{\eta}$ that is associated with
$\mathfrak{h}_{\eta}$. The operator $H_{\eta}$ can be regarded as the rigorous
way to interpret the formal expression
$-\frac{1}{2m}\Delta+\eta\delta_{\Gamma}$. With this notation we can formulate
the result about the nonrelativistic limit of $A_{\eta/2,\eta/2,0}^{m,c}$,
which yields, combined with a result from [28], a statement about the
existence of geometrically induced bound states:

######  Theorem 2.7.

Let $\eta\in\mathbb{R}$, $m>0$, and $\Gamma$ be as in Hypothesis 2.1. Then,
there exists $z_{0}<0$ such that every $z\in(-\infty,z_{0})$ belongs to
$\rho(A_{\eta/2,\eta/2,0}^{m,c}-mc^{2})\cap\rho(H_{\eta})$, whenever $c$ is
above a $z$-dependent threshold value $c_{z}$. Moreover, there exists a
constant $K>0$ such that

$\left\|\big{(}A_{\eta/2,\eta/2,0}^{m,c}-(z+mc^{2})\big{)}^{-1}-(H_{\eta}-z)^{-1}\begin{pmatrix}1&0\\\
0&0\end{pmatrix}\right\|_{L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})\rightarrow
L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})}\leq\frac{K}{c}$

for all $c>c_{z}$. In particular, if $\Gamma$ is not the straight line,
$\eta<0$, and $c$ is sufficiently large, then
$\sigma_{\textup{disc}}(A_{\eta/2,\eta/2,0}^{m,c})\neq\emptyset$.

Theorem 2.7 will be proved in Section 5 with the help of the Krein type
resolvent formula for $A_{\eta/2,\eta/2,0}^{m,c}$ provided in Section 4. In
this resolvent formula, convergence of each term appearing there can be
proved, when $c\rightarrow+\infty$, which yields the claim about the
nonrelativistic limit. The statement about the discrete spectrum is then a
simple consequence of the nonrelativistic limit, the non-emptiness of
$\sigma_{\textup{disc}}(H_{\eta})$ under the stated assumptions shown in [28],
and general results that are known for operators that converge in the norm
resolvent sense.

Another way to show the existence of geometrically induced bound states is to
transfer a recent result from [30]. Define for $\omega\in(0,\frac{\pi}{2})$
the broken line $\widetilde{\Gamma}_{\omega}$ of opening angle $2\omega$ by

$\widetilde{\Gamma}_{\omega}:=\big{\\{}(r\cos(\omega),r\sin(\omega)):r>0\big{\\}}\cup\big{\\{}(r\cos(\omega),-r\sin(\omega)):r>0\big{\\}}.$
(2.6)

It was shown in [30] that a Dirac operator with a Lorentz scalar
$\delta$-shell interaction supported on $\widetilde{\Gamma}_{\omega}$ of
strength $\tau\in(-\infty,0)\setminus\\{-2\\}$ always has non-empty discrete
spectrum, if $\omega$ is sufficiently small. This can be transferred to obtain
an associated result for $A_{0,\tau,0}^{m,c}$, if the interaction support
$\Gamma$ is a compact smooth perturbation of $\widetilde{\Gamma}_{\omega}$. We
remark that any curve $\Gamma$ satisfying Hypothesis 2.1, that is not a
compact perturbation of the straight line, is, up to a translation and
rotation, of this form and that the following result, in contrast to the one
in Theorem 2.7, can be shown for arbitrary $c>0$.

######  Theorem 2.8.

Let $\tau\in(-\infty,0)\setminus\\{-2\\}$ and $\Gamma$ be as in Hypothesis 2.1
such that $\Gamma$ coincides with $\widetilde{\Gamma}_{\omega}$ away from a
compact set in $\mathbb{R}^{2}$, and $m,c>0$. Then, for any $N\in\mathbb{N}$
there exists an angle $\omega_{\star}\in(0,\frac{\pi}{2})$ depending on
$N,\tau$ such that for all $\omega\in(0,\omega_{\star}]$ the operator
$A_{0,\tau,0}^{m,c}$ has at least $N$ discrete eigenvalues with multiplicities
taken into account.

Theorem 2.8 is also shown in Section 5 with the help of the min-max principle
applied to the non-negative and self-adjoint operator
$(A_{0,\tau,0}^{m,c})^{2}$ and a family of test functions constructed in [30]
that are supported in a region where $\Gamma$ and
$\widetilde{\Gamma}_{\omega}$ coincide, which allows to transfer the result
from [30] to the situation studied in this paper.

##  3\. Preliminaries

In this section we collect several results, that are needed to prove the main
claims made on $A_{\eta,\tau,\lambda}^{m,c}$ in Section 2. First, in Section
3.1 we summarize some well-known properties of the free Dirac operator
$A_{0}^{m,c}$ and in Section 3.2 we state some properties of integral
operators that are associated with the integral kernel of the resolvent of
$A_{0}^{m,c}$. In Section 3.3 we construct a boundary triple that is useful to
study $A_{\eta,\tau,\lambda}^{m,c}$. Throughout this section, we always assume
that $\Gamma\subset\mathbb{R}^{2}$ is as in Hypothesis 2.1.

###  3.1. The free Dirac operator and function spaces for Dirac operators

In this section we recall the definition of the free Dirac operator in
$\mathbb{R}^{2}$ and state several related results that are needed in the main
part of the paper. Let $\sigma_{1},\sigma_{2},\sigma_{3}$ be the Pauli spin
matrices defined in (1.2). For $m\in\mathbb{R}$ and $c>0$ the free Dirac
operator in $\mathbb{R}^{2}$ is

$A_{0}^{m,c}f:=-ic(\sigma\cdot\nabla)f+mc^{2}\sigma_{3}f,\quad\mathrm{dom}\,A_{0}^{m,c}=H^{1}(\mathbb{R}^{2};\mathbb{C}^{2}).$
(3.1)

We will sometimes make use of the relation $A_{0}^{m,c}=cA_{0}^{mc,1}$. With
the help of the Fourier transform it is not difficult to show that
$A_{0}^{m,c}$ is self-adjoint in $L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})$ and
that

$\sigma(A_{0}^{m,c})=\sigma_{\text{ess}}(A_{0}^{m,c})=(-\infty,-|m|c^{2}]\cup[|m|c^{2},+\infty).$

Moreover, for $z\in\rho(A_{0}^{m,c})$ the resolvent of $A_{0}^{m,c}$ is given
by

$(A_{0}^{m,c}-z)^{-1}f(x)=\int_{\mathbb{R}^{2}}G_{z}^{m,c}(x-y)f(y)dy,\quad
f\in L^{2}(\mathbb{R}^{2};\mathbb{C}^{2}),~{}x\in\mathbb{R}^{2},$

where

$\begin{split}G_{z}^{m,c}(x)&=\frac{1}{2\pi
c}\sqrt{\tfrac{z^{2}}{c^{2}}-(mc)^{2}}K_{1}\left(-i\sqrt{\tfrac{z^{2}}{c^{2}}-(mc)^{2}}|x|\right)\frac{(\sigma\cdot
x)}{|x|}\\\ &\qquad\qquad+\frac{1}{2\pi
c}K_{0}\left(-i\sqrt{\tfrac{z^{2}}{c^{2}}-(mc)^{2}}|x|\right)\left(\frac{z}{c}\sigma_{0}+mc\sigma_{3}\right)\end{split}$
(3.2)

and $K_{j}$ is the modified Bessel function of the second kind and order $j$;
cf. [1]. We remark that we always choose the complex square root such that for
all $w\in\mathbb{C}\setminus[0,+\infty)$ one has $\textup{Im}(\sqrt{w})>0$ and
hence $\overline{\sqrt{w}}=-\sqrt{\overline{w}}$ for
$w\in\mathbb{C}\setminus[0,+\infty)$. Note also that
$A_{0}^{m,c}=cA_{0}^{mc,1}$ yields

$(A_{0}^{m,c}-z)^{-1}=\frac{1}{c}\left(A_{0}^{mc,1}-\frac{z}{c}\right)^{-1}\quad\text{and}\quad
G_{z}^{m,c}=\frac{1}{c}G_{z/c}^{mc,1},\quad z\in\rho(A_{0}^{m,c}).$ (3.3)

Recall that $\Omega_{\pm}\subset\mathbb{R}^{2}$ are the two open sets with
joint boundary $\partial\Omega_{\pm}=\Gamma$, see Hypothesis 2.1, and that we
denote elements in $L^{2}(\Omega_{\pm};\mathbb{C}^{2})$ with a subscript
$\pm$. We consider the spaces

$H(\sigma,\Omega_{\pm}):=\big{\\{}f_{\pm}\in
L^{2}(\Omega_{\pm};\mathbb{C}^{2}):(\sigma\cdot\nabla)f_{\pm}\in
L^{2}(\Omega_{\pm};\mathbb{C}^{2})\big{\\}},$ (3.4)

which are, endowed with the norms

$\|f_{\pm}\|_{H(\sigma,\Omega_{\pm})}^{2}:=\|f_{\pm}\|_{L^{2}(\Omega_{\pm};\mathbb{C}^{2})}^{2}+\|(\sigma\cdot\nabla)f_{\pm}\|_{L^{2}(\Omega_{\pm};\mathbb{C}^{2})}^{2},$

Hilbert spaces; cf. [17, Lemma 2.1] for similar arguments in the case when
$\Gamma$ is bounded. An important property of $H(\sigma,\Omega_{\pm})$ is the
fact that the trace operator can be extended to these spaces. The proof of the
next lemma for our geometric setting follows very similar ideas as the ones in
[17, Lemma 2.3] or [39, Lemma 2.3].

######  Lemma 3.1.

The map $T^{D}_{\pm,0}:H^{1}(\Omega_{\pm};\mathbb{C}^{2})\rightarrow
H^{1/2}(\Gamma;\mathbb{C}^{2}),T^{D}_{\pm,0}f=f|_{\Gamma}$, extends uniquely
by continuity to a bounded linear operator
$T^{D}_{\pm}:H(\sigma,\Omega_{\pm})\rightarrow
H^{-1/2}(\Gamma;\mathbb{C}^{2})$.

In the main part of the paper we will work with the densely defined closed
symmetric restriction $S$ of $A_{0}^{m,c}$ onto
$H^{1}_{0}(\mathbb{R}^{2}\setminus\Gamma;\mathbb{C}^{2})$, that is,

$Sf=-ic(\sigma\cdot\nabla)f+mc^{2}\sigma_{3}f,\quad\mathrm{dom}\,S=H^{1}_{0}(\mathbb{R}^{2}\setminus\Gamma;\mathbb{C}^{2}),$
(3.5)

and its adjoint

$\begin{split}S^{*}f&=\big{(}-ic(\sigma\cdot\nabla)+mc^{2}\sigma_{3}\big{)}f_{+}\oplus(-ic(\sigma\cdot\nabla)+mc^{2}\sigma_{3}\big{)}f_{-},\\\
\mathrm{dom}\,S^{*}&=H(\sigma,\Omega_{+})\oplus
H(\sigma,\Omega_{-}).\end{split}$ (3.6)

###  3.2. Integral operators for the Dirac operator

In this section, we introduce two families of integral operators that will
play an important role in the analysis of $A_{\eta,\tau,\lambda}^{m,c}$. For
this, recall the fundamental solution $G_{z}^{m,c}$ in (3.2) and note that
$\Gamma$ given in Hypothesis 2.1 belongs to the class of curves considered in
[12].

Following [12, Appendix C], we introduce for

$z\in\rho(A_{0}^{m,c})=\mathbb{C}\setminus\bigl{(}(-\infty,-|m|c^{2}]\cup[|m|c^{2},+\infty)\bigr{)}$

the operator

$(\Phi_{z}^{m,c})^{*}:=T^{D}(A_{0}^{m,c}-\overline{z})^{-1}:L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})\rightarrow
H^{1/2}(\Gamma;\mathbb{C}^{2}),$ (3.7)

which is well-defined and bounded due to the mapping properties of the
Dirichlet trace $T^{D}:H^{1}(\mathbb{R}^{2})\rightarrow H^{1/2}(\Gamma)$ and
the resolvent of $A_{0}^{m,c}$. Hence, also its anti-dual

$\Phi_{z}^{m,c}:=\big{(}(\Phi_{z}^{m,c})^{*}\big{)}^{\prime}:H^{-1/2}(\Gamma;\mathbb{C}^{2})\rightarrow
L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})$ (3.8)

is well-defined and bounded. Using (3.7) and
$G_{z}^{m,c}(x)^{*}=G_{\overline{z}}^{m,c}(-x)$ one finds that the action of
$\Phi_{z}^{m,c}$ on $\varphi\in L^{2}(\Gamma;\mathbb{C}^{2})$ is

$\Phi_{z}^{m,c}\varphi(x)=\int_{\Gamma}G_{z}^{m,c}(x-y)\varphi(y)d\sigma(y),\quad
x\in\mathbb{R}^{2}\setminus\Gamma.$

Moreover, as $\ker(\Phi_{z}^{m,c})^{*}=\operatorname{ran}(S-\overline{z})$,
where $S$ is the symmetric operator in (3.5), and
$\operatorname{ran}(\Phi_{z}^{m,c})^{*}$ is closed, we get that

$\operatorname{ran}\Phi_{z}^{m,c}=\ker(S^{*}-z)\subset
H(\sigma,\Omega_{+})\oplus H(\sigma,\Omega_{-}).$ (3.9)

Since by (3.3) one has $\Phi_{z}^{m,c}=\frac{1}{c}\Phi_{z/c}^{mc,1}$ and
$\Phi_{z/c}^{mc,1}=\Phi_{z/c}$ in the notation of [12] (when $m$ is replaced
by $mc$), we conclude from [12, Proposition 2.8] that $\Phi_{z}^{m,c}$ has the
bounded restriction

$\widetilde{\Phi}_{z}^{m,c}:H^{1/2}(\Gamma;\mathbb{C}^{2})\rightarrow
H^{1}(\mathbb{R}^{2}\setminus\Gamma;\mathbb{C}^{2}).$ (3.10)

The boundary integral operator formally given by

$\mathcal{C}_{z}^{m,c}\varphi(x):=\text{p.v.}\int_{\Gamma}G_{z}^{m,c}(x-y)\varphi(y)d\sigma(y),\quad
x\in\Gamma,~{}\varphi\in L^{2}(\Gamma;\mathbb{C}^{2}),$ (3.11)

where $z\in\rho(A_{0}^{m,c})$ and the integral is understood as the principal
value, will be used throughout this paper. Since
$\mathcal{C}_{z}^{m,c}=\frac{1}{c}\mathcal{C}_{z/c}^{mc,1}$ holds by (3.3) and
$\mathcal{C}_{z/c}^{mc,1}=\mathcal{C}_{z/c}$ in the notation of [12] (when $m$
is replaced by $mc$), we get with [12, Proposition 2.9] that
$\mathcal{C}_{z}^{m,c}$ gives rise to a bounded operator in
$H^{r}(\Gamma;\mathbb{C}^{2})$ for $r\in[-\frac{1}{2},\frac{1}{2}]$. Moreover,
as $G_{z}^{m,c}(x)^{*}=G_{\overline{z}}^{m,c}(-x)$,
$x\in\mathbb{R}^{2}\setminus\\{0\\}$, it can be shown that the adjoint of
$\mathcal{C}_{z}^{m,c}$, viewed as an operator in
$L^{2}(\Gamma;\mathbb{C}^{2})$, is
$(\mathcal{C}_{z}^{m,c})^{*}=\mathcal{C}_{\overline{z}}^{m,c}$.

Next, we explain that in a suitable sense $\mathcal{C}_{z}^{m,c}$ can be
regarded as a compact perturbation of the same mapping
$\mathcal{C}_{z}^{m,c,\Sigma}$ on the straight line
$\Sigma=\mathbb{R}\times\\{0\\}$, i.e. of the operator that acts on
$\varphi\in L^{2}(\Sigma;\mathbb{C}^{2})$ as

$\mathcal{C}_{z}^{m,c,\Sigma}\varphi(s,0):=\text{p.v.}\int_{-\infty}^{\infty}G_{z}^{m,c}(s-t,0)\varphi(t,0)dt,\quad
s\in\mathbb{R}.$ (3.12)

In the formulation of the next proposition, the multiplication operator with
the matrix-valued function

$V=\begin{pmatrix}1&0\\\ 0&\overline{\boldsymbol{t}}\end{pmatrix},$ (3.13)

where $\boldsymbol{t}=t_{1}+it_{2}$ with
$(t_{1},t_{2})=\dot{\gamma}\circ\gamma^{-1}$ being the unit tangent vector at
$\Gamma$, will play an important role. Since $\boldsymbol{t}$ is
$C^{\infty}$-smooth and $|\boldsymbol{t}|=1$, one clearly has that for any
$r\in\mathbb{R}$ the multiplication with $V$ gives rise to a bounded and
bijective operator in $H^{r}(\Gamma;\mathbb{C}^{2})$. Moreover, recall the
definition of the map $U_{g}$ from (1.4) for any arc-length parametrization
$g$ of an unbounded curve. The technical proof of Proposition 3.2 is given in
Appendix B.

######  Proposition 3.2.

Let
$z\in\rho(A_{0}^{m,c})=\mathbb{C}\setminus((-\infty,-|m|c^{2}]\cup[|m|c^{2},+\infty))$,
$\Gamma$ be as in Hypothesis 2.1, and $x(t)=(t,0)$ be a parametrization of
$\Sigma$. Then the operator

$U_{\gamma}V(\sigma\cdot\nu)\mathcal{C}_{z}^{m,c}(\sigma\cdot\nu)V^{*}U_{\gamma}^{-1}-U_{x}\sigma_{2}\mathcal{C}_{z}^{m,c,\Sigma}\sigma_{2}U_{x}^{-1}:H^{-1/2}(\mathbb{R};\mathbb{C}^{2})\rightarrow
H^{1/2}(\mathbb{R};\mathbb{C}^{2})$

is compact.

######  Remark 3.3.

In fact, we will show in the proof of Proposition 3.2 that the map
$U_{\gamma}V(\sigma\cdot\nu)\mathcal{C}_{z}^{m,c}(\sigma\cdot\nu)V^{*}U_{\gamma}^{-1}-U_{x}\sigma_{2}\mathcal{C}_{z}^{m,c,\Sigma}\sigma_{2}U_{x}^{-1}$
can be extended, for any $r\in[-1,0]$, to a compact operator from
$H^{r}(\mathbb{R};\mathbb{C}^{2})$ to $H^{r+1}(\mathbb{R};\mathbb{C}^{2})$.

Finally, we state a useful variant of the Plemelj-Sokhotskii formula, which
relates the trace of $\Phi_{z}^{m,c}\varphi$ to
$\mathcal{C}_{z}^{m,c}\varphi$; cf. [12, Proposition 2.9 (ii)]. For a direct
proof of this formula one can also follow line by line the proof of [11,
Proposition 3.4] using the classical Plemelj-Sokhotskii formula for the Cauchy
transform on unbounded curves, see, e.g., [33, Proposition 4.6.1].

######  Lemma 3.4.

Let
$z\in\rho(A_{0}^{m,c})=\mathbb{C}\setminus((-\infty,-|m|c^{2}]\cup[|m|c^{2},+\infty))$.
Then one has for all $\varphi\in H^{-1/2}(\Gamma;\mathbb{C}^{2})$

$T^{D}_{\pm}\Phi_{z}^{m,c}\varphi=\mp\frac{i}{2c}(\sigma\cdot\nu)\varphi+\mathcal{C}_{z}^{m,c}\varphi.$

Eventually, we note that Lemma 3.4 allows to show, in the same way as in [3,
Lemma 3.3 (ii)], the identity

$-4c^{2}\big{(}(\sigma\cdot\nu)\mathcal{C}_{z}^{m,c}\big{)}^{2}=-4c^{2}\big{(}\mathcal{C}_{z}^{m,c}(\sigma\cdot\nu)\big{)}^{2}=I,\quad
z\in\rho(A_{0}^{m,c}).$ (3.14)

###  3.3. A boundary triple for $A_{\eta,\tau,\lambda}^{m,c}$

In this section we show how the operator $A_{\eta,\tau,\lambda}^{m,c}$ defined
in (2.3) can be described with the help of a boundary triple. The general
definition and some useful properties of boundary triples are recalled in
Appendix A. As before, $\Gamma$, $\Omega_{\pm}$, and $\nu$ are as in
Hypothesis 2.1, and we use for $f:\mathbb{R}^{2}\rightarrow\mathbb{C}^{2}$ the
notation $f_{\pm}:=f\upharpoonright\Omega_{\pm}$. First, we introduce an
auxiliary Dirac operator and discuss its properties. Recall that $A_{0}^{m,c}$
is the free Dirac operator given by (3.1), define the unitary operator

$U:L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})\rightarrow
L^{2}(\mathbb{R}^{2};\mathbb{C}^{2}),\quad U(f_{+}\oplus
f_{-})=f_{+}\oplus(-f_{-}),$

and introduce in $L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})$ the operator

$B:=U^{*}A_{0}^{m,c}U.$ (3.15)

In a more explicit form the operator $B$ is given by

$\begin{split}Bf&=\big{(}-ic(\sigma\cdot\nabla)+mc^{2}\sigma_{3}\big{)}f_{+}\oplus\big{(}-ic(\sigma\cdot\nabla)+mc^{2}\sigma_{3}\big{)}f_{-},\\\
\mathrm{dom}\,B&=\big{\\{}f=f_{+}\oplus f_{-}\in
H^{1}(\Omega_{+};\mathbb{C}^{2})\oplus
H^{1}(\Omega_{-};\mathbb{C}^{2}):T^{D}_{+}f_{+}+T^{D}_{-}f_{-}=0\big{\\}}.\end{split}$

The basic properties of $B$ are stated in the following lemma. Recall that
$\widetilde{\Phi}_{z}^{m,c}$ and $\mathcal{C}_{z}^{m,c}$ are the operators
defined by (3.10) and (3.11), respectively.

######  Lemma 3.5.

The operator $B$ is self-adjoint in $L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})$,
its spectrum is given by
$\sigma(B)=\sigma_{\textup{ess}}(B)=\sigma(A_{0}^{m,c})=(-\infty,-|m|c^{2}]\cup[|m|c^{2},+\infty)$,
and for $z\in\rho(B)$ one has

$(B-z)^{-1}=(A_{0}^{m,c}-z)^{-1}+4c^{2}\widetilde{\Phi}_{z}^{m,c}(\sigma\cdot\nu)\mathcal{C}_{z}^{m,c}(\sigma\cdot\nu)(\Phi_{\overline{z}}^{m,c})^{*}.$

###### Proof.

Since the operator $U$ is unitary, it follows from the definition of $B$ in
(3.15) and the properties of $A_{0}^{m,c}$ in (3.1) that $B$ is self-adjoint
in $L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})$ and that
$\sigma(B)=\sigma(A_{0}^{m,c})$. It remains to show the claimed resolvent
formula. For this, fix $z\in\rho(B)=\rho(A_{0}^{m,c})$ and $f\in
L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})$, and define

$g:=(A_{0}^{m,c}-z)^{-1}f+4c^{2}\widetilde{\Phi}_{z}^{m,c}(\sigma\cdot\nu)\mathcal{C}_{z}^{m,c}(\sigma\cdot\nu)(\Phi_{\overline{z}}^{m,c})^{*}f.$

Then, by the mapping properties of $(A_{0}^{m,c}-z)^{-1}$,
$\widetilde{\Phi}_{z}^{m,c}$, $\mathcal{C}_{z}^{m,c}$, and
$(\Phi_{\overline{z}}^{m,c})^{*}$ in (3.10), (3.11), and (3.7), respectively,
we get $g\in H^{1}(\Omega_{+};\mathbb{C}^{2})\oplus
H^{1}(\Omega_{-};\mathbb{C}^{2})$. Moreover, using Lemma 3.4 and (3.14), we
find

$\begin{split}T^{D}_{+}g_{+}+T^{D}_{-}g_{-}&=2T^{D}(A_{0}^{m,c}-z)^{-1}f+8c^{2}\mathcal{C}_{z}^{m,c}(\sigma\cdot\nu)\mathcal{C}_{z}^{m,c}(\sigma\cdot\nu)(\Phi_{\overline{z}}^{m,c})^{*}f\\\
&=2T^{D}(A_{0}^{m,c}-z)^{-1}f-2(\Phi_{\overline{z}}^{m,c})^{*}f=0\end{split}$

and hence, $g\in\mathrm{dom}\,B$. Finally, equation (3.9) yields

$((B-z)g)_{\pm}=(-ic(\sigma\cdot\nabla)+mc^{2}\sigma_{3}-z\sigma_{0})\big{(}(A_{0}^{m,c}-z)^{-1}f\big{)}_{\pm}=f_{\pm}.$

Taking the injectivity of $B-z$ and the definition of $g$ into account, we see
that also the claimed resolvent formula is true. ∎

Next, we show how a boundary triple that is useful to study
$A_{\eta,\tau,\lambda}^{m,c}$ is given, see also [13, Theorem 4.8] and [11,
22] for similar constructions. In order to formulate the result, recall the
definition of the map $\Lambda$ in (1.5), of the free Dirac operator
$A_{0}^{m,c}$ in (3.1), of its restriction $S$ and extension $S^{*}$ in (3.5)
and (3.6), respectively, of the boundary integral operator
$\mathcal{C}_{z}^{m,c}$ in (3.11), and of the unitary map $V$ in (3.13).

######  Proposition 3.6.

Let $\zeta\in\rho(A_{0}^{m,c})$. Define the maps
$\Gamma_{0},\Gamma_{1}:\mathrm{dom}\,S^{*}\rightarrow
L^{2}(\Gamma;\mathbb{C}^{2})$ acting on $f=f_{+}\oplus
f_{-}\in\mathrm{dom}\,S^{*}$ by

$\begin{split}\Gamma_{0}f&:=\frac{1}{2}\Lambda^{-1}V\big{(}T^{D}_{+}f_{+}+T^{D}_{-}f_{-}\big{)},\\\
\Gamma_{1}f&:=-\Lambda\big{(}icV(\sigma\cdot\nu)(T^{D}_{+}f_{+}-T^{D}_{-}f_{-})\\\
&\qquad\qquad+c^{2}V(\sigma\cdot\nu)(\mathcal{C}_{\zeta}^{m,c}+\mathcal{C}_{\overline{\zeta}}^{m,c})(\sigma\cdot\nu)(T^{D}_{+}f_{+}+T^{D}_{-}f_{-})\big{)}.\end{split}$

Then $\\{L^{2}(\Gamma;\mathbb{C}^{2}),\Gamma_{0},\Gamma_{1}\\}$ is a boundary
triple for $S^{*}$. Moreover, the associated $\gamma$-field and Weyl function
are given by

$\rho(A_{0}^{m,c})\ni
z\mapsto\gamma(z)=-4c^{2}\Phi_{z}^{m,c}(\sigma\cdot\nu)\mathcal{C}_{z}^{m,c}(\sigma\cdot\nu)V^{*}\Lambda$

and

$\rho(A_{0}^{m,c})\ni z\mapsto M(z)=4c^{2}\Lambda
V(\sigma\cdot\nu)\left(\mathcal{C}_{z}^{m,c}-\frac{1}{2}(\mathcal{C}_{\zeta}^{m,c}+\mathcal{C}_{\overline{\zeta}}^{m,c})\right)(\sigma\cdot\nu)V^{*}\Lambda,$

respectively.

###### Proof.

We apply the construction from the end of Appendix A leading to Proposition
A.3 with the self-adjoint operator $B$ in (3.15) and the boundary mapping
$\mathcal{T}:\mathrm{dom}\,B\rightarrow L^{2}(\Gamma;\mathbb{C}^{2})$ given by

$\mathcal{T}f=-ic\Lambda
V(\sigma\cdot\nu)(T^{D}_{+}f_{+}-T^{D}_{-}f_{-}),\quad f\in\mathrm{dom}\,B.$
(3.16)

Since $T^{D}_{\pm}:H^{1}(\Omega_{\pm};\mathbb{C}^{2})\rightarrow
H^{1/2}(\Gamma;\mathbb{C}^{2})$ is surjective, $V$ and $\sigma\cdot\nu$ give
rise to bijective maps in $H^{1/2}(\Gamma;\mathbb{C}^{2})$ (these functions
are pointwise invertible and $C^{\infty}$-smooth), and
$\Lambda:H^{1/2}(\Gamma;\mathbb{C}^{2})\rightarrow
L^{2}(\Gamma;\mathbb{C}^{2})$ is bijective by construction (1.5), the operator
$\mathcal{T}$ is surjective. Moreover, as $\mathrm{dom}\,B$ is a closed
subspace of $H^{1}(\mathbb{R}^{2}\setminus\Gamma;\mathbb{C}^{2})$, the
embedding of $\mathrm{dom}\,B$ equipped with the norm inherited from
$H^{1}(\mathbb{R}^{2}\setminus\Gamma;\mathbb{C}^{2})$ into $\mathrm{dom}\,B$
equipped with the graph norm is bounded and bijective. Therefore, the graph
norm induced by $B$ and the norm in
$H^{1}(\mathbb{R}^{2}\setminus\Gamma;\mathbb{C}^{2})$ are equivalent on
$\mathrm{dom}\,B$. Hence, by the properties of $T^{D}_{\pm}$,
$\sigma\cdot\nu$, $V$, and $\Lambda$ the map $\mathcal{T}$ is continuous.
Furthermore,
$\ker\mathcal{T}=H^{1}_{0}(\mathbb{R}^{2}\setminus\Sigma;\mathbb{C}^{2})$ is
dense in $L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})$. Next, with Lemma 3.4 and
Lemma 3.5, one finds

$\mathcal{G}_{z}=(\mathcal{T}(B-\overline{z})^{-1})^{*}=-4c^{2}\Phi_{z}^{m,c}(\sigma\cdot\nu)\mathcal{C}_{z}^{m,c}(\sigma\cdot\nu)V^{*}\Lambda$
(3.17)

for $z\in\rho(A_{0}^{m,c})=\rho(B)$. Recall that $\zeta\in\rho(B)$. For
$f=f_{\zeta}+\mathcal{G}_{\zeta}\xi$ with $f_{\zeta}\in\mathrm{dom}\,B$ and
$\xi\in L^{2}(\Gamma;\mathbb{C}^{2})$ it follows from (3.17), Lemma 3.4, and
(3.14) that

$\begin{split}&\frac{1}{2}\Lambda^{-1}V\big{(}T^{D}_{+}+T^{D}_{-}\big{)}(f_{\zeta}+\mathcal{G}_{\zeta}\xi)\\\
&\qquad\qquad=-2c^{2}\Lambda^{-1}V\big{(}T^{D}_{+}+T^{D}_{-}\big{)}\Phi_{\zeta}^{m,c}(\sigma\cdot\nu)\mathcal{C}_{\zeta}^{m,c}(\sigma\cdot\nu)V^{*}\Lambda\xi\\\
&\qquad\qquad=-4c^{2}\Lambda^{-1}V\big{(}\mathcal{C}_{\zeta}^{m,c}(\sigma\cdot\nu)\big{)}^{2}V^{*}\Lambda\xi\\\
&\qquad\qquad=\xi,\end{split}$ (3.18)

and hence we obtain the representation of $\Gamma_{0}$ from Proposition A.3.
In order to obtain the formula for $\Gamma_{1}$ we note first that the mapping
$\mathcal{T}$ in (3.16) can be extended onto $\mathrm{dom}\,S^{*}$. Using
(3.16), (3.17), and Lemma 3.4 we then compute

$\begin{split}\mathcal{T}\mathcal{G}_{z}\xi&=4ic^{3}\Lambda
V(\sigma\cdot\nu)(T^{D}_{+}-T^{D}_{-})\Phi_{z}^{m,c}(\sigma\cdot\nu)\mathcal{C}_{z}^{m,c}(\sigma\cdot\nu)V^{*}\Lambda\xi\\\
&=4c^{2}\Lambda
V(\sigma\cdot\nu)\mathcal{C}_{z}^{m,c}(\sigma\cdot\nu)V^{*}\Lambda\xi\end{split}$
(3.19)

for $z\in\rho(A_{0}^{m,c})=\rho(B)$ and $\xi\in L^{2}(\Gamma;\mathbb{C}^{2})$.
Now consider
$f=f_{\zeta}+\mathcal{G}_{\zeta}\xi=f_{\overline{\zeta}}+\mathcal{G}_{\overline{\zeta}}\xi$
and observe that by Proposition A.3

$\Gamma_{1}f=\frac{1}{2}\mathcal{T}(f_{\zeta}+f_{\overline{\zeta}})=\frac{1}{2}\mathcal{T}(f-\mathcal{G}_{\zeta}\xi+f-\mathcal{G}_{\overline{\zeta}}\xi)=\mathcal{T}f-\frac{1}{2}\mathcal{T}\big{(}\mathcal{G}_{\zeta}+\mathcal{G}_{\overline{\zeta}}\big{)}\xi.$
(3.20)

Inserting $\mathcal{T}f$ and the expressions
$\mathcal{T}\mathcal{G}_{\zeta}\xi$,
$\mathcal{T}\mathcal{G}_{\overline{\zeta}}\xi$ from (3.19) in (3.20), and
making use of
$\xi=\Gamma_{0}f=\frac{1}{2}\Lambda^{-1}V(T^{D}_{+}f_{+}+T^{D}_{-}f_{-})$ we
obtain the representation of $\Gamma_{1}$. Finally, the expressions for the
$\gamma$-field and Weyl function follow immediately from
$\gamma(z)=\mathcal{G}_{z}$ and
$M(z)=\mathcal{T}(\mathcal{G}_{z}-\tfrac{1}{2}(\mathcal{G}_{\zeta}+\mathcal{G}_{\overline{\zeta}}))$
in Proposition A.3. ∎

In the following proposition we describe the operator
$A_{\eta,\tau,\lambda}^{m,c}$ with the help of the boundary triple from the
previous proposition. For similar arguments see [13, Section 6] and also [22,
Proposition 6.3] or [11, Proposition 4.3]. Recall that $V$ is given by (3.13)
and define for $\eta,\tau,\lambda\in\mathbb{R}$ the matrix

$F:=\begin{pmatrix}\eta+\tau&\lambda\\\ \lambda&\eta-\tau\end{pmatrix},$
(3.21)

implying

$V^{*}FV=\eta\sigma_{0}+\tau\sigma_{3}+i\lambda(\sigma\cdot\nu)\sigma_{3}.$
(3.22)

######  Proposition 3.7.

Let $\eta,\tau,\lambda\in\mathbb{R}$ and define in
$L^{2}(\Gamma;\mathbb{C}^{2})$ the operator

$\begin{split}\Theta\varphi&:=\Lambda\big{[}F-2c^{2}V(\sigma\cdot\nu)(\mathcal{C}_{\zeta}^{m,c}+\mathcal{C}_{\overline{\zeta}}^{m,c})(\sigma\cdot\nu)V^{*}\big{]}\Lambda\varphi,\\\
\mathrm{dom}\,\Theta&=\big{\\{}\varphi\in L^{2}(\Gamma;\mathbb{C}^{2}):\\\
&\qquad\quad\big{[}F-2c^{2}V(\sigma\cdot\nu)(\mathcal{C}_{\zeta}^{m,c}+\mathcal{C}_{\overline{\zeta}}^{m,c})(\sigma\cdot\nu)V^{*}\big{]}\Lambda\varphi\in
H^{1/2}(\Gamma;\mathbb{C}^{2})\big{\\}}.\end{split}$ (3.23)

Then
$A_{\eta,\tau,\lambda}^{m,c}=S^{*}\upharpoonright\ker(\Gamma_{1}-\Theta\Gamma_{0})$.

###### Proof.

Recall that $f\in\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}$ if and only if
$f_{\pm}\in H(\sigma,\Omega_{\pm})$ and

$-ic(\sigma\cdot\nu)(T^{D}_{+}f_{+}-T^{D}_{-}f_{-})=\frac{1}{2}(\eta\sigma_{0}+\tau\sigma_{3}+i\lambda(\sigma\cdot\nu)\sigma_{3})(T^{D}_{+}f_{+}+T^{D}_{-}f_{-}).$

Using (3.22) and

$\begin{split}\frac{1}{2}(T^{D}_{+}f_{+}+T^{D}_{-}f_{-})&=V^{*}\Lambda\Gamma_{0}f,\\\
-ic(\sigma\cdot\nu)(T^{D}_{+}f_{+}-T^{D}_{-}f_{-})&=V^{*}\Lambda^{-1}\Gamma_{1}f+2c^{2}(\sigma\cdot\nu)(\mathcal{C}_{\zeta}^{m,c}+\mathcal{C}_{\overline{\zeta}}^{m,c})(\sigma\cdot\nu)V^{*}\Lambda\Gamma_{0}f,\end{split}$

one finds that $f\in\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}$ if and only if
$f\in\mathrm{dom}\,S^{*}$ and $\Gamma_{1}f-\Theta\Gamma_{0}f=0$. ∎

##  4\. Proof of Theorem 2.3 and further properties of
$A_{\eta,\tau,\lambda}^{m,c}$

The claims in Theorem 2.3 are shown in various steps: Subsection 4.1 is
devoted to the self-adjointness of $A_{\eta,\tau,\lambda}^{m,c}$, the claim in
Theorem 2.3 (i) about the Sobolev regularity of
$\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}$ is then shown in Subsection 4.2.
Afterwards, $\sigma_{\textup{ess}}(A_{\eta,\tau,\lambda}^{m,c})$ is computed
in Subsection 4.3. Finally, in Subsection 4.4 we state some further properties
of $A_{\eta,\tau,\lambda}^{m,c}$ like a useful Krein type resolvent formula
and a Birman-Schwinger principle.

We will make use of the following two main ingredients: We use that
$A_{\eta,\tau,\lambda}^{m,c}$ can be described with the boundary triple in
Proposition 3.6 as explained in Proposition 3.7, and the abstract results in
Appendix A. And we will use that the claimed properties can be obtained
explicitly when the interaction support is the straight line $\Sigma$; cf.
[16]. To distinguish the appearing objects on the straight line from the case
of the more general interaction support $\Gamma$, we will denote them with a
superindex $\Sigma$. Eventually, we will make often use of the arc-length
parametrization

$x:\mathbb{R}\rightarrow\mathbb{R}^{2},\quad x(t)=(t,0),$ (4.1)

of the straight line $\Sigma$.

###  4.1. Self-adjointness of $A_{\eta,\tau,\lambda}^{m,c}$

Let $\\{L^{2}(\Gamma;\mathbb{C}^{2}),\Gamma_{0},\Gamma_{1}\\}$ be the boundary
triple for $S^{*}$ from Proposition 3.6 and $\Theta$ be defined by (3.23).
Recall that

$A_{\eta,\tau,\lambda}^{m,c}=S^{*}\upharpoonright\ker(\Gamma_{1}-\Theta\Gamma_{0}).$

Hence, by Theorem A.2 the operator $A_{\eta,\tau,\lambda}^{m,c}$ is self-
adjoint in $L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})$ if and only if $\Theta$ is
self-adjoint in $L^{2}(\Gamma;\mathbb{C}^{2})$. We show that the latter is the
case.

Denote by $\Lambda^{\Sigma}$ the map in (1.5) on $\Sigma$, by
$\mathcal{C}_{z}^{m,c,\Sigma}$ the operator in (3.11) on the straight line
$\Sigma$, and we write $\Theta^{\Sigma}$ for the parameter defined as in
(3.23) on $\Sigma$. Since $\sigma\cdot\nu^{\Sigma}=-\sigma_{2}$ and
$V^{\Sigma}=\sigma_{0}$ on $\Sigma$, one has the more explicit representation

$\Theta^{\Sigma}=\Lambda^{\Sigma}\big{[}F-2c^{2}\sigma_{2}(\mathcal{C}_{\zeta}^{m,c,\Sigma}+\mathcal{C}_{\overline{\zeta}}^{m,c,\Sigma})\sigma_{2}\big{]}\Lambda^{\Sigma}$
(4.2)

and as in (3.23) this map is defined on the maximal domain. This operator is
self-adjoint in $L^{2}(\Sigma;\mathbb{C}^{2})$ by Theorem A.2 and Proposition
3.7 (applied for $\Gamma=\Sigma$), as $A_{\eta,\tau,\lambda}^{m,c,\Sigma}$ is
self-adjoint in $L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})$; this follows for
$d\neq-4c^{2}$ from [16, Section 6] noting that
$A_{\eta,\tau,\lambda}^{m,c,\Sigma}=cA_{\eta/c,\tau/c,\lambda/c}^{mc,1,\Sigma}$
and $A_{\eta/c,\tau/c,\lambda/c}^{mc,1,\Sigma}=A_{\eta/c,\tau/c,\lambda/c}$ is
the operator studied in [16], when $m$ is replaced by $mc$, and for
$d=-4c^{2}$ from [34]. Next, recall that $\gamma$ is the arc-length
parametrization of $\Gamma$ given by (2.1) and that $x$ is the arc-length
parametrization of $\Sigma$ in (4.1). By definition, one has
$\Lambda^{\Sigma}=U_{x}^{-1}U_{\gamma}\Lambda U_{\gamma}^{-1}U_{x}$. Then, by
Proposition 3.2

$\begin{split}U_{\gamma}&\Theta
U_{\gamma}^{-1}-U_{x}\Theta^{\Sigma}U_{x}^{-1}\\\ &=-2c^{2}U_{\gamma}\Lambda
U_{\gamma}^{-1}\big{[}U_{\gamma}V(\sigma\cdot\nu)\mathcal{C}_{\zeta}^{m,c}(\sigma\cdot\nu)V^{*}U_{\gamma}^{-1}-U_{x}\sigma_{2}\mathcal{C}_{\zeta}^{m,c,\Sigma}\sigma_{2}U_{x}^{-1}\\\
&\qquad\qquad\quad+U_{\gamma}V(\sigma\cdot\nu)\mathcal{C}_{\overline{\zeta}}^{m,c}(\sigma\cdot\nu)V^{*}U_{\gamma}^{-1}-U_{x}\sigma_{2}\mathcal{C}_{\overline{\zeta}}^{m,c,\Sigma}\sigma_{2}U_{x}^{-1}\big{]}U_{\gamma}\Lambda
U_{\gamma}^{-1},\end{split}$ (4.3)

defined on its natural maximal domain, can be extended to a compact operator
in $L^{2}(\mathbb{R};\mathbb{C}^{2})$. Since the adjoint of
$\mathcal{C}_{\zeta}^{m,c}$ in $L^{2}(\Gamma;\mathbb{C}^{2})$ is given by
$\mathcal{C}_{\overline{\zeta}}^{m,c}$, the right hand side of the latter
equation is symmetric and can be extended to a self-adjoint operator. As
$\Theta^{\Sigma}$ is self-adjoint in $L^{2}(\Sigma;\mathbb{C}^{2})$ and
$U_{\gamma}:L^{2}(\Gamma;\mathbb{C}^{2})\rightarrow
L^{2}(\mathbb{R};\mathbb{C}^{2})$ and
$U_{x}:L^{2}(\Sigma;\mathbb{C}^{2})\rightarrow
L^{2}(\mathbb{R};\mathbb{C}^{2})$ are unitary, this implies that $\Theta$ is
self-adjoint.

###  4.2. Sobolev regularity in $\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}$

In this subsection we show item (i) of Theorem 2.3. For this we prove an
auxiliary lemma in which we analyze $\mathrm{dom}\,\Theta$. Afterwards the
Krein type resolvent formula for $A_{\eta,\tau,\lambda}^{m,c}$ from Theorem
A.2 allows us to show the claims on
$\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}$. Since $mc^{2}\sigma_{3}$ is a
bounded perturbation of the self-adjoint operator
$A_{\eta,\tau,\lambda}^{m,c}$, it does not affect
$\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}$, and hence, it is no restriction
to assume in this subsection that $m\neq 0$ and to choose $\zeta=0$ in
Proposition 3.6.

######  Lemma 4.1.

Assume that $m\neq 0$. Then, for the domain of the operator $\Theta$ defined
in (3.23) with $\zeta=0$ one has:

  * (a)

If $(\tfrac{d}{4}-c^{2})^{2}-\lambda^{2}c^{2}\neq 0$, then
$\mathrm{dom}\,\Theta=H^{1}(\Gamma;\mathbb{C}^{2})$.

  * (b)

If $(\tfrac{d}{4}-c^{2})^{2}-\lambda^{2}c^{2}=0$, then
$\mathrm{dom}\,\Theta\not\subset H^{s}(\Gamma;\mathbb{C}^{2})$ for all $s>0$.

###### Proof.

The proof of this lemma is split into two steps: We analyze
$\mathrm{dom}\,\Theta^{\Sigma}$ first and with the help of Proposition 3.2 we
transfer these results to the case of more general curves $\Gamma$.

Step 1. We use the Fourier transform $\mathcal{F}$ in
$L^{2}(\mathbb{R};\mathbb{C}^{2})=U_{x}L^{2}(\Sigma;\mathbb{C}^{2})$ and note
first that
$\mathcal{F}U_{x}\mathcal{C}_{z}^{m,c,\Sigma}U_{x}^{-1}\mathcal{F}^{-1}$ is
the maximal multiplication operator associated with the function

$c_{z}(p)=\frac{1}{2\sqrt{p^{2}c^{2}+(mc^{2})^{2}-z^{2}}}\begin{pmatrix}\tfrac{z}{c}+mc&p\\\
p&\tfrac{z}{c}-mc\end{pmatrix};$

to see this, one can apply [15, Lemma 2.1] and note with (3.3) that
$\mathcal{C}_{z}^{m,c,\Sigma}=\tfrac{1}{c}\mathcal{C}_{z/c}^{mc,1,\Sigma}$ and
$\mathcal{C}_{z}=\mathcal{C}_{z}^{mc,1,\Sigma}$ in the notation of [15].
Recall that we chose $\zeta=0$ in the definition of $\Theta^{\Sigma}$. Hence,
with (4.2) one sees that the operator
$\mathcal{F}U_{x}\Theta^{\Sigma}U_{x}^{-1}\mathcal{F}^{-1}$ is the maximal
multiplication operator associated with the function

$\theta(p)=\sqrt{p^{2}+1}\left[F-\sigma_{2}\frac{2c^{2}}{\sqrt{p^{2}c^{2}+(mc^{2})^{2}}}\big{(}mc\sigma_{3}+p\sigma_{1}\big{)}\sigma_{2}\right]=\theta_{1}(p)+\theta_{2}(p)$

with

$\begin{split}\theta_{1}(p)&=\sqrt{p^{2}+1}\begin{pmatrix}\eta+\tau&\lambda+\frac{2pc}{\sqrt{p^{2}+(mc)^{2}}}\\\
\lambda+\frac{2pc}{\sqrt{p^{2}+(mc)^{2}}}&\eta-\tau\end{pmatrix},\\\
\theta_{2}(p)&=\frac{2\sqrt{p^{2}+1}}{\sqrt{p^{2}+(mc)^{2}}}mc^{2}\sigma_{3}.\end{split}$

Since $\theta_{2}$ is a bounded function, its associated multiplication
operator is bounded and everywhere defined in
$L^{2}(\mathbb{R};\mathbb{C}^{2})$. The eigenvalues of $\theta_{1}(p)$ are
given by

$\begin{split}\mu_{\pm}(p)&=\sqrt{p^{2}+1}\left(\eta\pm\sqrt{\eta^{2}-d+\frac{4p^{2}c^{2}}{p^{2}+(mc)^{2}}+\frac{4pc\lambda}{\sqrt{p^{2}+(mc)^{2}}}}\right)\\\
&=\sqrt{p^{2}+1}\left(\eta\pm\sqrt{\eta^{2}-d+4c^{2}+4\lambda
c\,\mathrm{sign}\,(p)+f(p)}\right),\end{split}$

where $d$ is given by (2.4) and $f$ is a continuous function such that
$f(p)=\mathcal{O}\big{(}\frac{1}{p^{2}}\big{)}$ for $p\rightarrow\pm\infty$.
Hence, if $(\tfrac{d}{4}-c^{2})^{2}-\lambda^{2}c^{2}\neq 0$, then
$\mu_{\pm}(p)\sim\sqrt{p^{2}+1}$ and thus,
$\mathrm{dom}\,\Theta^{\Sigma}=H^{1}(\Sigma;\mathbb{C}^{2})$. On the other
hand, if $(\tfrac{d}{4}-c^{2})^{2}-\lambda^{2}c^{2}=0$, then at least one of
the functions $\mu_{\pm}$ is bounded on $\mathbb{R}_{+}$ or on
$\mathbb{R}_{-}$ (or bounded everywhere on $\mathbb{R}$, if
$0=\lambda=d-4c^{2}$), implying that $\mathrm{dom}\,\Theta^{\Sigma}\not\subset
H^{s}(\Sigma;\mathbb{C}^{2})$ for all $s>0$. Thus, the claim of the lemma is
true for the straight line $\Gamma=\Sigma$.

Step 2. Recall that $U_{\gamma}\Theta
U_{\gamma}^{-1}-U_{x}\Theta^{\Sigma}U_{x}^{-1}$ can be extended to a compact
operator in $L^{2}(\mathbb{R};\mathbb{C})$, see Proposition 3.2 and (4.3).
Hence, the maximal domain of $\Theta$ can be described in the form
$\mathrm{dom}\,\Theta=\mathrm{dom}\,(\Theta^{\Sigma}U_{x}^{-1}U_{\gamma})$ and
now the claim follows from our observation in Step 1. ∎

###### Proof of Theorem 2.3 (i).

As explained above we can assume, without loss of generality, that $m\neq 0$,
and choose $\zeta=0$ in (3.23). To prove item (a) we use
$\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}=\operatorname{ran}(A_{\eta,\tau,\lambda}^{m,c}-z)^{-1}$
for $z\in\mathbb{C}\setminus\mathbb{R}$ and the resolvent formula

$(A_{\eta,\tau,\lambda}^{m,c}-z)^{-1}=(A_{0}^{m,c}-z)^{-1}+\gamma(z)\big{(}\Theta-M(z)\big{)}^{-1}\gamma(\overline{z})^{*}$
(4.4)

from Theorem A.2 (iv). If $(\tfrac{d}{4}-c^{2})^{2}-\lambda^{2}c^{2}\neq 0$,
then by Lemma 4.1 one has

$\mathrm{dom}\,(\Theta-M(z))=\mathrm{dom}\,\Theta=H^{1}(\Gamma;\mathbb{C}^{2}).$

Hence,
$\operatorname{ran}(A_{0}^{m,c}-z)^{-1}=\mathrm{dom}\,A_{0}^{m,c}=H^{1}(\mathbb{R}^{2};\mathbb{C}^{2})$,
the explicit form
$\gamma(z)=-4c^{2}\Phi_{z}^{m,c}(\sigma\cdot\nu)\mathcal{C}_{z}^{m,c}(\sigma\cdot\nu)V^{*}\Lambda$
from Proposition 3.6, and the mapping properties in (1.5), (3.10), and (3.11)
imply $\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}\subset
H^{1}(\mathbb{R}^{2}\setminus\Gamma;\mathbb{C}^{2})$, that is, the assertion
in Theorem 2.3 (i) (a) holds.

Next, we show assertion (b), i.e. that
$\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}\not\subset
H^{s}(\mathbb{R}^{2}\setminus\Gamma;\mathbb{C}^{2})$ for all $s>0$, if
$(\tfrac{d}{4}-c^{2})^{2}-\lambda^{2}c^{2}=0$. Note that, due to the
surjectivity of $(\Gamma_{0},\Gamma_{1})$, for all
$\varphi\in\mathrm{dom}\,\Theta$ there exists $f\in\mathrm{dom}\,S^{*}$ such
that $\Gamma_{0}f=\varphi$ and $\Gamma_{1}f=\Theta\varphi$. Hence, for each
$\varphi\in\mathrm{dom}\,\Theta$ there exists
$f\in\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}$ such that
$\Gamma_{0}f=\varphi$. If we assume now
$\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}\subset
H^{s}(\mathbb{R}^{2}\setminus\Gamma;\mathbb{C}^{2})$ for an $s>0$, then by the
definition of $\Gamma_{0}$ we would conclude that any
$\varphi\in\mathrm{dom}\,\Theta$ also belongs to
$H^{s}(\Gamma;\mathbb{C}^{2})$. However, this contradicts Lemma 4.1, showing
that also Theorem 2.3 (i) (b) is true. ∎

###  4.3. Essential spectrum of $A_{\eta,\tau,\lambda}^{m,c}$

First, we claim that for all combinations $\eta,\tau,\lambda\in\mathbb{R}$ we
have

$(-\infty,-|m|c^{2}]\cup[|m|c^{2},+\infty)\subset\sigma_{\textup{ess}}(A_{\eta,\tau,\lambda}^{m,c}),$
(4.5)

which can be proved via a suitable singular sequence. To show (4.5), we can
assume that $\Gamma\subset[0,+\infty)\times\mathbb{R}$. In fact, if this is
not the case, then via a suitable rotation and translation one can transform
$A_{\eta,\tau,\lambda}^{m,c}$ to a unitary equivalent Dirac operator, which
may have different Dirac matrices
$\alpha_{1},\alpha_{2},\beta\in\mathbb{C}^{2\times 2}$ instead of
$\sigma_{1},\sigma_{2},\sigma_{3}$, see, e.g., [35, p. 150] or [43,
Proposition 4], with a singular potential that is supported on a curve
contained in $[0,+\infty)\times\mathbb{R}$, so this assumption is not a
restriction.

Let now $z\in(-\infty,-|m|c^{2})\cup(|m|c^{2},+\infty)$ be fixed and consider
for $n\in\mathbb{N}$ the function

$f_{n}(x_{1},x_{2}):=\frac{1}{n}\chi\left(\frac{1}{n}|x-y_{n}|\right)e^{i\sqrt{z^{2}-m^{2}c^{4}}x_{1}/c}\big{(}\sqrt{z^{2}-m^{2}c^{4}}\sigma_{1}+mc^{2}\sigma_{3}+z\sigma_{0}\big{)}\xi,$

where $\chi:\mathbb{R}\to[0,1]$ is a $C^{\infty}$-function such that
$\chi(t)=1$ for $|t|\leq\frac{1}{2}$ and $\chi(t)=0$ for $|t|\geq 1$, the
vector $\xi\in\mathbb{C}^{2}$ is chosen such that
$(\sqrt{z^{2}-m^{2}c^{4}}\sigma_{1}+mc^{2}\sigma_{3}+z\sigma_{0})\xi\neq 0$,
and we set $y_{n}:=(0,-n^{2})$, $n\in\mathbb{N}$. Then
$f_{n}\in\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}$ and one can show as in
[10, Theorem 5.7 (i)] that
$\|(A_{\eta,\tau,\lambda}^{m,c}-z)f_{n}\|/\|f_{n}\|\rightarrow 0$ for
$n\to\infty$. Since $z\in(-\infty,-|m|c^{2})\cup(|m|c^{2},+\infty)$ was
arbitrary, the inclusion (4.5) follows.

For $m=0$ the inclusion (4.5) implies
$\sigma_{\textup{ess}}(A_{\eta,\tau,\lambda}^{0,c})=\mathbb{R}$, so there is
nothing left to show. Hence we can assume $m\neq 0$ in the following. In order
to analyze
$\sigma_{\textup{ess}}(A_{\eta,\tau,\lambda}^{m,c})\cap(-|m|c^{2},|m|c^{2})$,
we use that by Proposition 3.7 and Theorem A.2 (iii)

$z\in\sigma_{\textup{ess}}(A_{\eta,\tau,\lambda}^{m,c})\cap(-|m|c^{2},|m|c^{2})\quad\text{if
and only if}\quad 0\in\sigma_{\textup{ess}}(\Theta-M(z)),$ (4.6)

where $M$ is the Weyl function of the boundary triple in Proposition 3.6.
Using the definition of $\Theta$ in (3.23) one verifies for
$z\in(-|m|c^{2},|m|c^{2})$

$\Theta-M(z)=\Lambda[F-4c^{2}V(\sigma\cdot\nu)\mathcal{C}_{z}^{m,c}(\sigma\cdot\nu)V^{*}]\Lambda$

and in the same way for the straight line

$\Theta^{\Sigma}-M^{\Sigma}(z)=\Lambda^{\Sigma}[F-4c^{2}\sigma_{2}\mathcal{C}_{z}^{m,c,\Sigma}\sigma_{2})]\Lambda^{\Sigma}.$

Since $\Lambda^{\Sigma}=U_{x}^{-1}U_{\gamma}\Lambda U_{\gamma}^{-1}U_{x}$ we
obtain

$\begin{split}U_{\gamma}&(\Theta-M(z))U_{\gamma}^{-1}-U_{x}(\Theta^{\Sigma}-M^{\Sigma}(z))U_{x}^{-1}\\\
&=-4c^{2}U_{\gamma}\Lambda
U_{\gamma}^{-1}\big{[}U_{\gamma}V(\sigma\cdot\nu)\mathcal{C}_{z}^{m,c}(\sigma\cdot\nu)V^{*}U_{\gamma}^{-1}-U_{x}\sigma_{2}\mathcal{C}_{z}^{m,c,\Sigma}\sigma_{2}U_{x}^{-1}\big{]}U_{\gamma}\Lambda
U_{\gamma}^{-1}\end{split}$

and it follows from Proposition 3.2 that this operator can be extended to a
compact operator in $L^{2}(\mathbb{R};\mathbb{C}^{2})$. Since $U_{\gamma}$ and
$U_{x}$ are unitary, we conclude that

$\sigma_{\textup{ess}}(\Theta-M(z))=\sigma_{\textup{ess}}(\Theta^{\Sigma}-M^{\Sigma}(z)),$

and hence, by (4.6),

$z\in\sigma_{\textup{ess}}(A_{\eta,\tau,\lambda}^{m,c})\cap(-|m|c^{2},|m|c^{2})\quad\text{if
and only if}\quad 0\in\sigma_{\textup{ess}}(\Theta^{\Sigma}-M^{\Sigma}(z)).$

Applying Theorem A.2 (iii) and Proposition 3.7 to the Dirac operator
$A_{\eta,\tau,\lambda}^{m,c,\Sigma}$ with the singular interaction supported
on the straight line $\Sigma$, we find
$\sigma_{\textup{ess}}(A_{\eta,\tau,\lambda}^{m,c})=\sigma_{\textup{ess}}(A_{\eta,\tau,\lambda}^{m,c,\Sigma})$,
which finally yields the statement in Theorem 2.3 (ii); this follows in the
case $\eta^{2}-\tau^{2}-\lambda^{2}\neq-4c^{2}$ from [16, Theorem 6.2] noting
that
$A_{\eta,\tau,\lambda}^{m,c,\Sigma}=cA_{\eta/c,\tau/c,\lambda/c}^{mc,1,\Sigma}$
and $A_{\eta/c,\tau/c,\lambda/c}^{mc,1,\Sigma}=A_{\eta/c,\tau/c,\lambda/c}$ is
the operator studied in [16], when $m$ is replaced by $mc$; to obtain the
result, it is convenient to replace the term $k$ in [16, Theorem 6.2] by $kc$.
The claim is also true for $\eta^{2}-\tau^{2}-\lambda^{2}=-4c^{2}$, which is
excluded in [16], see [34].

Finally, the claim that
$\sigma(A_{\eta,\tau,\lambda}^{m,c})=\sigma_{\textup{ess}}(A_{\eta,\tau,\lambda}^{m,c})$,
if $\Gamma=\Sigma$ is the straight line, is a part of the results in [16, 34].

###  4.4. Krein type resolvent formula and further spectral properties of
$A_{\eta,\tau,\lambda}^{m,c}$

In the next proposition we formulate a Birman-Schwinger principle to
characterize discrete eigenvalues of $A_{\eta,\tau,\lambda}^{m,c}$. The proof
of the following result is the same as the proof of [13, Proposition 6.5 (ii)]
and will be omitted here.

######  Proposition 4.2.

Let $\eta,\tau,\lambda\in\mathbb{R}$ and
$z\notin\sigma_{\textup{ess}}(A_{\eta,\tau,\lambda}^{m,c})$. Then,

$z\in\sigma_{\textup{disc}}(A_{\eta,\tau,\lambda}^{m,c})\quad\text{if and only
if}\quad-1\in\sigma_{\textup{p}}((\eta\sigma_{0}+\tau\sigma_{3}+i\lambda(\sigma\cdot\nu)\sigma_{3})\mathcal{C}_{z}^{m,c}),$

where
$(\eta\sigma_{0}+\tau\sigma_{3}+i\lambda(\sigma\cdot\nu)\sigma_{3})\mathcal{C}_{z}^{m,c}$
is viewed as an operator in $H^{-1/2}(\Gamma;\mathbb{C}^{2})$.

Next, we state a Krein type resolvent formula for
$A_{\eta,\tau,\lambda}^{m,c}$. Recall that the operators $A_{0}^{m,c}$ and
$\Phi_{z}^{m,c}$ are given by (3.1) and (3.8), respectively.

######  Proposition 4.3.

Let $\eta,\tau,\lambda\in\mathbb{R}$. Then,
$z\in\rho(A_{\eta,\tau,\lambda}^{m,c})$ if and only if the operator
$\sigma_{0}+(\eta\sigma_{0}+\tau\sigma_{3}+i\lambda(\sigma\cdot\nu)\sigma_{3})\mathcal{C}_{z}^{m,c}$
admits a bounded inverse from $H^{1/2}(\Gamma;\mathbb{C}^{2})$ to
$H^{-1/2}(\Gamma;\mathbb{C}^{2})$ and in this case the resolvent formula

$\begin{split}(&A_{\eta,\tau,\lambda}^{m,c}-z)^{-1}=(A_{0}^{m,c}-z)^{-1}\\\
&~{}~{}-\Phi_{z}^{m,c}\big{(}\sigma_{0}+(\eta\sigma_{0}+\tau\sigma_{3}+i\lambda(\sigma\cdot\nu)\sigma_{3})\mathcal{C}_{z}^{m,c}\big{)}^{-1}(\eta\sigma_{0}+\tau\sigma_{3}+i\lambda(\sigma\cdot\nu)\sigma_{3})(\Phi_{\overline{z}}^{m,c})^{*}\end{split}$

holds.

###### Proof.

First, one shows with Lemma 3.5 in exactly the same way as in [13, Proposition
6.5 (i)] that for $z\in\rho(A_{\eta,\tau,\lambda}^{m,c})$ the map
$\sigma_{0}+(\eta\sigma_{0}+\tau\sigma_{3}+i\lambda(\sigma\cdot\nu)\sigma_{3})\mathcal{C}_{z}^{m,c}$
admits a bounded inverse from $H^{1/2}(\Gamma;\mathbb{C}^{2})$ to
$H^{-1/2}(\Gamma;\mathbb{C}^{2})$ and that the claimed resolvent formula is
true.

Conversely, assume that
$\sigma_{0}+(\eta\sigma_{0}+\tau\sigma_{3}+i\lambda(\sigma\cdot\nu)\sigma_{3})\mathcal{C}_{z}^{m,c}$
admits a bounded inverse from $H^{1/2}(\Gamma;\mathbb{C}^{2})$ to
$H^{-1/2}(\Gamma;\mathbb{C}^{2})$. Then, by Proposition 4.2 the operator
$A_{\eta,\tau,\lambda}^{m,c}-z$ is injective. Furthermore, by the mapping
properties of $\Phi_{z}^{m,c}$ and $(\Phi_{z}^{m,c})^{*}$ in (3.8), (3.9), and
(3.7), respectively, the map

$\begin{split}&R_{z}:=(A_{0}^{m,c}-z)^{-1}\\\
&~{}~{}-\Phi_{z}^{m,c}\big{(}\sigma_{0}+(\eta\sigma_{0}+\tau\sigma_{3}+i\lambda(\sigma\cdot\nu)\sigma_{3})\mathcal{C}_{z}^{m,c}\big{)}^{-1}(\eta\sigma_{0}+\tau\sigma_{3}+i\lambda(\sigma\cdot\nu)\sigma_{3})(\Phi_{\overline{z}}^{m,c})^{*}\end{split}$

is bounded in $L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})$ and
$\operatorname{ran}R_{z}\subset H(\sigma,\mathbb{R}^{2}\setminus\Gamma)$.
Moreover, with the help of Lemma 3.1 one finds that for any $f\in
L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})$ the function $R_{z}f$ satisfies the
transmission condition in $\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}$ and
thus, $R_{z}f\in\mathrm{dom}\,A_{\eta,\tau,\lambda}^{m,c}$. Finally, one
obtains with (3.9) that

$((A_{\eta,\tau,\lambda}^{m,c}-z)R_{z}f)_{\pm}=(-ic(\sigma\cdot\nabla)+mc^{2}\sigma_{3}-z\sigma_{0})\big{(}(A_{0}^{m,c}-z)^{-1}f\big{)}_{\pm}=f_{\pm}.$

Since $A_{\eta,\tau,\lambda}^{m,c}-z$ is injective, we see that also the
claimed resolvent formula is true. ∎

Eventually, two symmetry relations about the eigenvalues of
$A_{\eta,\tau,\lambda}^{m,c}$, that are sometimes referred to as isospectral
relations, hold. For the proof of the following result, one can follow the one
of [22, Proposition 4.2].

######  Proposition 4.4.

Let $\eta,\tau,\lambda\in\mathbb{R}$ and $d$ be given by (2.4). Then, the
following holds:

  * (i)

If $d\neq 0$, then $z\in\sigma_{\textup{p}}(A_{\eta,\tau,\lambda}^{m,c})$ if
and only if
$z\in\sigma_{\textup{p}}(A_{-4\eta/d,-4\tau/d,-4\lambda/d}^{m,c})$.

  * (ii)

$z\in\sigma_{\textup{p}}(A_{\eta,\tau,\lambda}^{m,c})$ if and only if
$-z\in\sigma_{\textup{p}}(A_{-\eta,\tau,-\lambda}^{m,c})$.

##  5\. Proof of Theorems 2.7 and 2.8

Throughout this section we assume that $m>0$. First, we follow ideas from [7,
Section 5] and show Theorem 2.7 about the nonrelativistic limit of
$A_{\eta/2,\eta/2,0}^{m,c}$ and the geometrically induced bound states for
large $c$, for which some preparations are necessary. Recall that $H_{\eta}$,
$\eta\in\mathbb{R}$, is the self-adjoint operator associated with the
quadratic form $\mathfrak{h}_{\eta}$ defined by (2.5). To state the resolvent
formula for $H_{\eta}$, define for $z\in\mathbb{C}\setminus[0,+\infty)$ the
single layer potential $SL(z):L^{2}(\Gamma;\mathbb{C})\rightarrow
L^{2}(\mathbb{R}^{2};\mathbb{C})$ acting as

$SL(z)\varphi(x)=\frac{m}{\pi}\int_{\Gamma}K_{0}\left(-i\sqrt{2mz}|x-y|\right)\varphi(y)d\sigma(y),\quad
x\in\mathbb{R}^{2}\setminus\Gamma,~{}\varphi\in L^{2}(\Gamma;\mathbb{C}),$
(5.1)

and the single layer boundary integral operator
$\mathcal{S}(z):L^{2}(\Gamma;\mathbb{C})\rightarrow L^{2}(\Gamma;\mathbb{C})$
given by

$\mathcal{S}(z)\varphi(x)=\frac{m}{\pi}\int_{\Gamma}K_{0}\left(-i\sqrt{2mz}|x-y|\right)\varphi(y)d\sigma(y),\quad
x\in\Gamma,~{}\varphi\in L^{2}(\Gamma;\mathbb{C}).$ (5.2)

It is well-known that both operators $SL(z)$ and $\mathcal{S}(z)$ are well-
defined and bounded, see, e.g., [6, Theorem 2.8 (a)] applied for $d=2$ for a
reference under the present assumptions. In the following, we denote by
$-\Delta$ the free Laplacian defined on $H^{2}(\mathbb{R}^{2};\mathbb{C})$.
Then, one has the following relation of $SL(z)$ and $\mathcal{S}(z)$ and the
resolvent of $H_{\eta}$, see [20, Lemma 2.3] or [6, Theorem 2.8]:

######  Lemma 5.1.

Let $\eta\in\mathbb{R}$ and $H_{\eta}$ be the self-adjoint operator associated
with the quadratic form $\mathfrak{h}_{\eta}$ in (2.5). If
$z\in\mathbb{C}\setminus[0,\infty)$ and $0\in\rho(I+\eta\mathcal{S}(z))$, then
$z\in\rho(H_{\eta})$ and

$(H_{\eta}-z)^{-1}=\left(-\frac{1}{2m}\Delta-z\right)^{-1}-SL(z)\big{(}I+\eta\mathcal{S}(z)\big{)}^{-1}\eta
SL(\overline{z})^{*}.$

In particular, there exists $z_{0}<0$ such that for any $z\in(-\infty,z_{0})$
one has $0\in\rho(I+\eta\mathcal{S}(z))$.

In the following proposition we state some preliminary estimates for the
convergence of the operators $\Phi_{z+mc^{2}}^{m,c}$ and
$\mathcal{C}_{z+mc^{2}}^{m,c}$ given by (3.8) and (3.11), as
$c\rightarrow\infty$, which are necessary to compute the desired
nonrelativistic limit. In the following we make use of

$P_{+}:=\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}\in\mathbb{C}^{2\times
2}\quad\text{and}\quad e:=\begin{pmatrix}1\\\
0\end{pmatrix}\in\mathbb{C}^{2}.$ (5.3)

Note that $P_{+}=ee^{\top}$.

######  Proposition 5.2.

Let $z\in\mathbb{C}\setminus[0,\infty)$ and $c>\sqrt{\frac{|z|}{|m|}}$. Then,
there exists a constant $K$ depending on $m,z,\Gamma$ such that

$\displaystyle\left\|\left(A_{0}^{m,c}-(z+mc^{2})\right)^{-1}-\left(-\frac{1}{2m}\Delta-z\right)^{-1}P_{+}\right\|_{L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})\rightarrow
L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})}$ $\displaystyle\leq\frac{K}{c},$ (5.4a)
$\displaystyle\big{\|}\Phi_{z+mc^{2}}^{m,c}e-SL(z)e\big{\|}_{L^{2}(\Gamma;\mathbb{C})\rightarrow
L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})}$ $\displaystyle\leq\frac{K}{c},$ (5.4b)
$\displaystyle\big{\|}e^{\top}(\Phi_{z+mc^{2}}^{m,c})^{*}-e^{\top}SL(z)^{*}\big{\|}_{L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})\rightarrow
L^{2}(\Gamma;\mathbb{C})}$ $\displaystyle\leq\frac{K}{c},$ (5.4c)
$\displaystyle\big{\|}e^{\top}\mathcal{C}_{z+mc^{2}}^{m,c}e-\mathcal{S}(z)\big{\|}_{L^{2}(\Gamma;\mathbb{C})\rightarrow
L^{2}(\Gamma;\mathbb{C})}$ $\displaystyle\leq\frac{K}{c}.$ (5.4d)

###### Proof.

First, the estimate in (5.4a) is well-known, see, e.g., [14, Lemma 3.1] for a
proof in the case $m=\frac{1}{2}$. To prove (5.4b), we note first that
$\Phi_{z+mc^{2}}^{m,c}e-SL(z)e$ acts on $\varphi\in L^{2}(\Gamma;\mathbb{C})$
as

$(\Phi_{z+mc^{2}}^{m,c}e-SL(z)e)\varphi(x)=\int_{\Gamma}\big{(}k_{1}(x-y)+k_{2}(x-y)\big{)}\varphi(y)d\sigma(y),\quad
x\in\mathbb{R}^{2},$

with

$\begin{split}k_{1}(x)&:=\frac{1}{2\pi}\left(\left(2m+\frac{z}{c^{2}}\right)K_{0}\left(-i\sqrt{\tfrac{z^{2}}{c^{2}}+2mz}|x|\right)-2mK_{0}\left(-i\sqrt{2mz}|x|\right)\right)e,\\\
k_{2}(x)&:=\frac{1}{2\pi
c}\sqrt{\tfrac{z^{2}}{c^{2}}+2mz}K_{1}\left(-i\sqrt{\tfrac{z^{2}}{c^{2}}+2mz}|x|\right)\frac{x_{1}+ix_{2}}{|x|}\begin{pmatrix}0\\\
1\end{pmatrix}.\end{split}$ (5.5)

In the same way as in the proof of [14, Lemma 3.1] (see the considerations on
the functions $t_{1},t_{2},t_{3}$ there) one finds that there exists constant
$M_{1},\kappa_{1},R>0$ such that

$\big{|}k_{1}(x)+k_{2}(x)\big{|}^{2}\leq\frac{M_{1}^{2}}{c^{2}}k_{3}(x)k_{4}(x),$
(5.6)

where

$k_{3}(x):=\begin{cases}|x|^{-3/2},&\text{if }|x|\leq R,\\\
e^{-\kappa_{1}|x|},&\text{if }|x|>R,\end{cases}$

and

$k_{4}(x):=\begin{cases}|x|^{-1/2},&\text{if }|x|\leq R,\\\
e^{-\kappa_{1}|x|},&\text{if }|x|>R.\end{cases}$

Clearly, by the translation invariance of the Lebesgue measure we have for any
$y\in\Gamma$ that

$\int_{\mathbb{R}^{2}}|k_{3}(x-y)|dx=\int_{\mathbb{R}^{2}}|k_{3}(x)|dx$

and thus,

$M_{2}:=\sup_{y\in\Gamma}\int_{\mathbb{R}^{2}}|k_{3}(x-y)|dx=\int_{\mathbb{R}^{2}}|k_{3}(x)|dx<\infty.$
(5.7)

Next, recall that $\gamma$ is the arc-length parametrization of $\Gamma$
defined in (2.1) and choose for a fixed $x\in\mathbb{R}^{2}$ a number
$t=t(x)\in\mathbb{R}$ such that
$|x-\gamma(t)|=\min_{s\in\mathbb{R}}|x-\gamma(s)|$. Then, taking (2.2) into
account, we see that

$\frac{C_{1}}{2}|t(x)-s|\leq\frac{1}{2}|\gamma(t(x))-\gamma(s)|\leq|x-\gamma(s)|.$

This implies
$B_{R}(x)\cap\Gamma\subset\\{\gamma(s):|t(x)-s|\leq\frac{2R}{C_{1}}\\}$, where
$B_{R}(x)$ is the ball of radius $R$ around $x$, and

$\begin{split}M_{3}&:=\sup_{x\in\mathbb{R}^{2}}\int_{\Gamma}|k_{4}(x-y)|d\sigma(y)=\sup_{x\in\mathbb{R}^{2}}\int_{\mathbb{R}}|k_{4}(x-\gamma(s))|ds\\\
&=\sup_{x\in\mathbb{R}^{2}}\left(\int_{\gamma^{-1}(B_{R}(x)\cap\Gamma)}|x-\gamma(s)|^{-1/2}ds+\int_{\gamma^{-1}(\Gamma\setminus
B_{R}(x))}e^{-\kappa_{1}|x-\gamma(s)|}ds\right)\\\
&\leq\sup_{x\in\mathbb{R}^{2}}\left(\int_{t(x)-2R/C_{1}}^{t(x)+2R/C_{1}}|x-\gamma(s)|^{-1/2}ds+\int_{\mathbb{R}}e^{-\kappa_{1}|x-\gamma(s)|}ds\right)\\\
&\leq\sup_{x\in\mathbb{R}^{2}}M_{4}\left(\int_{t(x)-2R/C_{1}}^{t(x)+2R/C_{1}}|t(x)-s|^{-1/2}ds+\int_{\mathbb{R}}e^{-\kappa_{2}|t(x)-s|}ds\right)<+\infty,\end{split}$
(5.8)

where $M_{4},\kappa_{2}>0$ are suitable constants. Using (5.6)–(5.8) in the
Schur test, see, e.g., [47, Theorem 6.24] or [36, Example III 2.4], we get

$\big{\|}\Phi_{z+mc^{2}}^{m,c}e-SL(z)e\big{\|}_{L^{2}(\Gamma;\mathbb{C})\rightarrow
L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})}\leq\frac{M_{1}}{c}\sqrt{M_{2}M_{3}},$

which is exactly (5.4b). By taking adjoints this also implies (5.4c).

It remains to prove (5.4d). Note that for $\varphi\in
L^{2}(\Gamma;\mathbb{C})$ one has

$\big{(}e^{\top}\mathcal{C}_{z+mc^{2}}^{m,c}e-\mathcal{S}(z)\big{)}\varphi(x)=\int_{\Gamma}e^{\top}k_{1}(x-y)\varphi(y)d\sigma(y),\quad
x\in\Gamma,$

where $k_{1}$ is given by (5.5). The function $e^{\top}k_{1}$ can be further
decomposed in $e^{\top}k_{1}=k_{5}+k_{6}$ with

$\begin{split}k_{5}(x)&:=\frac{1}{2\pi}\left(2m+\frac{z}{c^{2}}\right)\left(K_{0}\left(-i\sqrt{\tfrac{z^{2}}{c^{2}}+2mz}|x|\right)-K_{0}\left(-i\sqrt{2mz}|x|\right)\right),\\\
k_{6}(x)&:=\frac{z}{c^{2}}K_{0}\left(-i\sqrt{2mz}|x|\right).\end{split}$

With the same argument as in [14, equation (3.11)] one sees that

$|k_{5}(x)|\leq\frac{M_{5}}{c}\begin{cases}1,&\text{if }|x|\leq R,\\\
e^{-\kappa_{3}|x|},&\text{if }|x|>R,\end{cases}$

while for $k_{6}$ one gets with the asymptotics for the Bessel function
$K_{0}$ in [1, Sections 9.6 and 9.7] that

$|k_{6}(x)|\leq\frac{M_{5}}{c}\begin{cases}|\log|x||+1,&\text{if }|x|\leq
R,\\\ e^{-\kappa_{3}|x|},&\text{if }|x|>R,\end{cases}$

where $M_{5},\kappa_{3}>0$ are suitable constants. Note that for a fixed
$y=\gamma(s)\in\Gamma$, $s\in\mathbb{R}$, we have
$B_{R}(\gamma(s))\cap\Gamma\subset\\{\gamma(t):|t-s|\leq\frac{R}{C_{1}}\\}$ by
(2.2). Hence, one obtains as in (5.8) with constants
$M_{6},M_{7},M_{8},\kappa_{4}>0$ that

$\begin{split}&\sup_{x\in\Gamma}\int_{\Gamma}|k_{1}(x-y)|d\sigma(y)=\sup_{y\in\Gamma}\int_{\Gamma}|k_{1}(x-y)|d\sigma(x)=\sup_{y\in\Gamma}\int_{\mathbb{R}}|k_{1}(\gamma(s)-y)|ds\\\
~{}&\leq\sup_{y\in\Gamma}\frac{M_{5}}{c}\left(\int_{\gamma^{-1}(B_{R}(x)\cap\Gamma)}\big{(}\big{|}\log|\gamma(s)-y|\big{|}+1\big{)}ds+\int_{\gamma^{-1}(\Gamma\setminus
B_{R}(x))}e^{-\kappa_{3}|\gamma(s)-y|}ds\right)\\\
~{}&\leq\sup_{y=\gamma(t)\in\Gamma}\frac{M_{6}}{c}\left(\int_{t-R/C_{1}}^{t+R/C_{1}}\big{(}|\gamma(s)-\gamma(t)|^{-1/2}+1\big{)}ds+\int_{\mathbb{R}}e^{-\kappa_{3}|\gamma(s)-\gamma(t)|}ds\right)\\\
~{}&\leq\sup_{y=\gamma(t)\in\Gamma}\frac{M_{7}}{c}\left(\int_{t-R/C_{1}}^{t+R/C_{1}}\big{(}|s-t|^{-1/2}+1\big{)}ds+\int_{\mathbb{R}}e^{-\kappa_{4}|s-t|}ds\right)\\\
~{}&\leq\frac{M_{8}}{c}<+\infty.\end{split}$

This yields, together with the Schur test, the claim in (5.4d). ∎

Using the result from Proposition 5.2 it is not difficult to prove that the
nonrelativistic limit of $A_{\eta/2,\eta/2,0}^{m,c}$ is $H_{\eta}$. Recall
that the matrix $P_{+}$ is defined in (5.3).

###### Proof of Theorem 2.7.

First, we prove the result about the nonrelativistic limit. Let $z_{0}$ be as
in Lemma 5.1, $z<z_{0}$, and $c$ be sufficiently large so that
$z+mc^{2}\in\rho(A_{0}^{m,c})$. Note that (5.4d) implies that

$\big{\|}(I+\eta\mathcal{S}(z))-(I+\eta
e^{\top}\mathcal{C}_{z+mc^{2}}^{m,c}e)\big{\|}_{L^{2}(\Gamma;\mathbb{C})\rightarrow
L^{2}(\Gamma;\mathbb{C})}\leq\frac{K}{c}$

and hence, for sufficiently large $c>0$ the operator $I+\eta
e^{\top}\mathcal{C}_{z+mc^{2}}^{m,c}e$ is boundedly invertible in
$L^{2}(\Gamma;\mathbb{C})$, as $I+\eta\mathcal{S}(z)$ has this property by
Lemma 5.1. Applying [36, Theorem IV 1.16] and (5.4d) yields

$\big{\|}(I+\eta\mathcal{S}(z))^{-1}-(I+\eta
e^{\top}\mathcal{C}_{z+mc^{2}}^{m,c}e)^{-1}\big{\|}_{L^{2}(\Gamma;\mathbb{C})\rightarrow
L^{2}(\Gamma;\mathbb{C})}\leq\frac{K}{c}.$ (5.9)

Since $I+\eta e^{\top}\mathcal{C}_{z+mc^{2}}^{m,c}e$ is boundedly invertible,
also

$\sigma_{0}+\eta
ee^{\top}\mathcal{C}_{z+mc^{2}}^{m,c}=\sigma_{0}+\left(\frac{\eta}{2}\sigma_{0}+\frac{\eta}{2}\sigma_{3}\right)\mathcal{C}_{z+mc^{2}}^{m,c}$

is boundedly invertible in $L^{2}(\Gamma;\mathbb{C}^{2})$. Thus, it follows
from Proposition 4.3 that $z\in\rho(A_{\eta/2,\eta/2,0}^{m,c})$ and

$\begin{split}\big{(}A_{\eta/2,\eta/2,0}^{m,c}-(z+mc^{2})\big{)}^{-1}&=\big{(}A_{0}^{m,c}-(z+mc^{2})\big{)}^{-1}\\\
&\qquad-\Phi_{z+mc^{2}}^{m,c}\big{(}\sigma_{0}+\eta
ee^{\top}\mathcal{C}_{z+mc^{2}}^{m,c}\big{)}^{-1}\eta
ee^{\top}(\Phi_{\overline{z}+mc^{2}}^{m,c})^{*}\\\
&=\big{(}A_{0}^{m,c}-(z+mc^{2})\big{)}^{-1}\\\
&\qquad-\Phi_{z+mc^{2}}^{m,c}e\big{(}I+\eta
e^{\top}\mathcal{C}_{z+mc^{2}}^{m,c}e\big{)}^{-1}\eta
e^{\top}(\Phi_{\overline{z}+mc^{2}}^{m,c})^{*}.\end{split}$

Using Proposition 5.2 and (5.9) in this equation, one finds

$\begin{split}\lim_{c\rightarrow+\infty}\big{(}A_{\eta/2,\eta/2,0}^{m,c}-(z+mc^{2})\big{)}^{-1}&=\left(-\frac{1}{2m}\Delta-z\right)^{-1}P_{+}\\\
&\qquad-SL(z)e\big{(}I+\eta\mathcal{S}(z)\big{)}^{-1}\eta
e^{\top}SL(\overline{z})^{*}\end{split}$

and that the order of convergence is $\frac{1}{c}$. Since $ee^{\top}=P_{+}$,
we see with Lemma 5.1 that the last expression is $(H_{\eta}-z)^{-1}P_{+}$.

To show $\sigma_{\textup{disc}}(A_{\eta/2,\eta/2,0}^{m,c})\not=\emptyset$ in
the case $\eta<0$ and $\Gamma$ is not the straight line, we follow closely the
proof of [7, Proposition 5.5] and transfer the result from [28] about the
spectrum of $H_{\eta}$ to $A_{\eta/2,\eta/2,0}^{m,c}$. Note first that, due to
the assumption $\eta<0$ and $\Gamma$ is not the straight line,
$\sigma_{\textup{disc}}(H_{\eta}P_{+})\neq\emptyset$, see [28, Theorem 5.2].
Now fix some $z<z_{0}$. We remark that $(H_{\eta}-z)^{-1}P_{+}$ is the
resolvent of a self-adjoint relation (multivalued operator) in
$L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})$, whose operator part is
$H_{\eta}P_{+}$. Since $(A_{\eta/2,\eta/2,0}^{m,c}-(z-+mc^{2}))^{-1}$
converges in the operator norm to $(H_{\eta}-z)^{-1}P_{+}$, as
$c\rightarrow+\infty$, it follows in the same way as in [44, Theorem VIII.23
(b)] or [46, Satz 9.24 b)] that the spectral measures of
$A_{\eta/2,\eta/2,0}^{m,c}-mc^{2}$ converge to the spectral measures of
$H_{\eta}P_{+}$. Thus, one finds with [46, Satz 2.58 a)] that also
$\sigma_{\textup{disc}}(A_{\eta/2,\eta/2,0}^{m,c})\neq\emptyset$. ∎

######  Remark 5.3.

One can show, in a similar way as in Theorem 2.7, that
$A_{\eta,\tau,\lambda}^{m,c}$ converges in the nonrelativistic limit to
$H_{\eta+\tau}$. Hence, as long as $\Gamma$ is not the straight line and
$\eta+\tau<0$, one concludes as above that
$\sigma_{\textup{disc}}(A_{\eta,\tau,\lambda}^{m,c})\neq\emptyset$ in this
situation. However, since the coefficients of the Lorentz scalar and the
anomalous magnetic interactions should be scaled differently to obtain a
nonrelativistic limit modelling the same physics, we prefer to state the
simpler result on $A_{\eta/2,\eta/2,0}^{m,c}$ in this section.

Next, we prove Theorem 2.8. Recall that $\widetilde{\Gamma}_{\omega}$ is the
broken line defined in (2.6) and that the essential spectrum of
$A_{0,\tau,0}^{m,c}$ was identified in Corollary 2.5.

###### Proof of Theorem 2.8.

In the following we denote by $\widetilde{\Omega}_{\pm}$ the two disjoint
domains with joint boundary $\widetilde{\Gamma}_{\omega}$, which are chosen
such that $(1,0)\in\widetilde{\Omega}_{+}$, and we denote by $\widetilde{\nu}$
the unit normal vector field at $\widetilde{\Gamma}_{\omega}$ that is pointing
outwards of $\widetilde{\Omega}_{+}$. We will make use of the operator

$\begin{split}\widetilde{S}_{\tau}^{m,c}f&:=\big{(}-ic(\sigma\cdot\nabla)+mc^{2}\sigma_{3}\big{)}f_{+}\oplus\big{(}-ic(\sigma\cdot\nabla)+mc^{2}\sigma_{3}\big{)}f_{-},\\\
\mathrm{dom}\,\widetilde{S}_{\tau}^{m,c}&:=\bigg{\\{}f=f_{+}\oplus f_{-}\in
H^{1}(\widetilde{\Omega}_{+};\mathbb{C}^{2})\oplus
H(\widetilde{\Omega}_{-};\mathbb{C}^{2}):\\\ &\qquad\qquad\quad-
ic(\sigma\cdot\widetilde{\nu})(f_{+}|_{\widetilde{\Gamma}_{\omega}}-f_{-}|_{\widetilde{\Gamma}_{\omega}})=\frac{\tau}{2}\sigma_{3}(f_{+}|_{\widetilde{\Gamma}_{\omega}}+f_{-}|_{\widetilde{\Gamma}_{\omega}})\bigg{\\}}.\end{split}$

Note that
$\widetilde{S}_{\tau}^{m,c}=c\widetilde{S}_{\tau/c}^{mc,1}=cS_{\tau/c}$, where
$S_{\tau}$ is the operator considered in [30], when $m$ is replaced by $mc$.
Hence, by [30, Section 2] the operator $\widetilde{S}_{\tau}^{m,c}$ is closed
and symmetric, but it has deficiency indices
$\dim\ker((\widetilde{S}_{\tau}^{m,c})^{*}\mp i)=1$, i.e.
$\widetilde{S}_{\tau}^{m,c}$ is symmetric, but not self-adjoint.

Next, fix $L_{0}>0$ such that
$\Gamma\cap([L_{0},+\infty)\times\mathbb{R})=\widetilde{\Gamma}_{\omega}\cap([L_{0},+\infty)\times\mathbb{R})$,
i.e. $\Gamma$ and $\widetilde{\Gamma}_{\omega}$ coincide for $x\geq L_{0}$.
Consider for $L>L_{0}$ and $c_{1},\dots,c_{N}\in\mathbb{C}$ the function

$f(x,y):=\sum_{n=1}^{N}c_{n}u_{n}(x)v(y)w(x,y)$

with

$u_{n}(x):=\sin\left(\frac{2n\pi}{L}x\right)\chi_{[L,2L]}(x),$

where $\chi_{[L,2L]}$ denotes the indicator function for $[L,2L]$,

$v(y):=\begin{cases}1,&|y|\leq 2r,\\\ e^{-\gamma(|y|-2r)},&|y|>2r,\end{cases}$

where $r:=L\tan(\omega)$ and $\gamma:=-\frac{4mc^{2}\tau}{4c^{2}+\tau^{2}}>0$,
and

$w(x,y):=\begin{cases}\begin{pmatrix}1\\\
0\end{pmatrix},&(x,y)\in\Omega_{+},\\\
\begin{pmatrix}\tfrac{4c^{2}+\tau^{2}}{4c^{2}-\tau^{2}}\\\
\tfrac{4c\tau}{4c^{2}-\tau^{2}}e^{i\omega}\end{pmatrix},&(x,y)\in\Omega_{-}~{}\wedge~{}y>0,\\\
\begin{pmatrix}\tfrac{4c^{2}+\tau^{2}}{4c^{2}-\tau^{2}}\\\
-\tfrac{4c\tau}{4c^{2}-\tau^{2}}e^{-i\omega}\end{pmatrix},&(x,y)\in\Omega_{-}~{}\wedge~{}y<0.\end{cases}$

As in the proof of [30, Proposition 6.3] one sees that, when varying the
coefficients $c_{1},\dots,c_{N}$, the functions $f$ span an $N$ dimensional
space and since each $f$ is supported in $[L,2L]\times\mathbb{R}$, we find
that
$f\in\mathrm{dom}\,\widetilde{S}_{\tau}^{m,c}\cap\mathrm{dom}\,A_{0,\tau,0}^{m,c}$
and $\widetilde{S}_{\tau}^{m,c}f=A_{0,\tau,0}^{m,c}f$. Recall that by the
scaling property $\widetilde{S}_{\tau}^{m,c}=cS_{\tau/c}$ one has to replace
$m$ and $\tau$ in the formulas in [30] by $mc$ and $\tau/c$, respectively.
Hence, one gets with [30, equation (6.3)]

$\begin{split}\|A_{0,\tau,0}^{m,c}f&\|_{L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})}^{2}-\left(mc^{2}\frac{\tau^{2}-4c^{2}}{\tau^{2}+4c^{2}}\right)^{2}\|f\|_{L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})}^{2}\\\
&=\|\widetilde{S}_{\tau}^{m,c}f\|_{L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})}^{2}-\left(mc^{2}\frac{\tau^{2}-4c^{2}}{\tau^{2}+4c^{2}}\right)^{2}\|f\|_{L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})}^{2}\\\
&=c^{2}\left(\|S_{\tau/c}f\|_{L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})}^{2}-\left(mc\frac{\tau^{2}-4c^{2}}{\tau^{2}+4c^{2}}\right)^{2}\|f\|_{L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})}^{2}\right)\\\
&<c^{2}\sum_{n=1}^{N}|c_{n}|^{2}\bigg{\\{}\tan(\omega)\left[3+\frac{(4c^{2}+\tau^{2})^{2}+16c^{2}\tau^{2}}{(4c^{2}-\tau^{2})^{2}}\right](2N^{2}\pi^{2}+m^{2}c^{2}L^{2})\\\
&\qquad\qquad\qquad\qquad+\frac{4mc^{2}L\tau}{4c^{2}+\tau^{2}}-\frac{N^{2}\pi^{2}[(4c^{2}+\tau^{2})^{2}+16\tau^{2}c^{2}](4c^{2}+\tau^{2})}{2mc^{2}L\tau(4c^{2}-\tau^{2})^{2}}\bigg{\\}}.\end{split}$

Since we consider $m>0$ and $\tau<0$, the sum of the second and the third term
in the curly brackets is negative, when $L$ is sufficiently large. For these
values of $L$, one can choose $\omega_{\star}=\omega_{\star}(L)$ such that the
right hand side of the last displayed formula is negative for any
$\omega<\omega_{\star}$. As the space of all $f$’s has dimension $N$, we find
with the min-max principle that $(A_{0,\tau,0}^{m,c})^{2}$ has at least $N$
discrete eigenvalues below the bottom of its essential spectrum
$\big{(}mc^{2}\frac{\tau^{2}-4c^{2}}{\tau^{2}+4c^{2}}\big{)}^{2}$; cf.
Corollary 2.5. Therefore, by the spectral theorem, $A_{0,\tau,0}^{m,c}$ has at
least $N$ discrete eigenvalues in the gap of its essential spectrum. This
finishes the proof of this proposition. ∎

### Acknowledgements

J. Behrndt and M. Holzmann gratefully acknowledge financial support by the
Austrian Science Fund (FWF): P 33568-N. P. Exner and M. Tušek were supported
by the grant No. 21-07129S of the Czech Science Foundation (GAČR). J. Behrndt,
M. Holzmann, and M. Tušek thank the Ministry of Education, Youth and Sports of
the Czech Republic and the Austrian Agency for International Cooperation in
Education and Research (OeAD) for support within the project 8J23AT025 and CZ
16/2023, respectively. 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.

##  Appendix A Boundary triples

In this section we briefly recall some notions related to the theory of
(ordinary) boundary triples. Boundary triples are an abstract tool in the
extension and spectral theory of symmetric and self-adjoint operators and they
are particularly useful to handle boundary value and transmission problems.
Here, we only state the definitions and results needed in the main part of the
paper, for proofs and further results we refer to [9, 21, 25, 26].

Throughout this section $\mathcal{H}$ is a complex Hilbert space and $S$ is a
densely defined closed symmetric operator in $\mathcal{H}$.

######  Definition A.1.

Let $\mathcal{G}$ be a complex Hilbert space and
$\Gamma_{0},\Gamma_{1}:\mathrm{dom}\,S^{*}\rightarrow\mathcal{G}$ be linear
mappings. Then $\\{\mathcal{G},\Gamma_{0},\Gamma_{1}\\}$ is called a boundary
triple for $S^{*}$, if the following conditions are fulfilled:

  * (i)

For all $f,g\in\mathrm{dom}\,S^{*}$ the abstract Green’s identity

$(S^{*}f,g)_{\mathcal{H}}-(f,S^{*}g)_{\mathcal{H}}=(\Gamma_{1}f,\Gamma_{0}g)_{\mathcal{G}}-(\Gamma_{0}f,\Gamma_{1}g)_{\mathcal{G}}$

holds.

  * (ii)

The map
$(\Gamma_{0},\Gamma_{1}):\mathrm{dom}\,S^{*}\rightarrow\mathcal{G}\times\mathcal{G}$
is surjective.

We note that a boundary triple for $S^{*}$ exists if and only if $S$ admits
self-adjoint extensions. In the following we always assume that this is the
case and that $\\{\mathcal{G},\Gamma_{0},\Gamma_{1}\\}$ is a boundary triple
associated with $S^{*}$. Then, the extension
$B_{0}:=S^{*}\upharpoonright\ker\Gamma_{0}$ of $S$ is self-adjoint and one has
the direct sum decomposition

$\mathrm{dom}\,S^{*}=\mathrm{dom}\,B_{0}\dot{+}\ker(S^{*}-z)=\ker\Gamma_{0}\dot{+}\ker(S^{*}-z),\quad
z\in\rho(B_{0}).$

Hence, the following mappings are well defined: The $\gamma$-field

$\rho(B_{0})\ni
z\mapsto\gamma(z):=\big{(}\Gamma_{0}\upharpoonright\ker(S^{*}-z)\big{)}^{-1},$

whose values $\gamma(z)$, $z\in\rho(B_{0})$, are bounded and everywhere
defined operators from $\mathcal{G}$ to $\mathcal{H}$, and the Weyl function

$\rho(B_{0})\ni z\mapsto
M(z):=\Gamma_{1}\big{(}\Gamma_{0}\upharpoonright\ker(S^{*}-z)\big{)}^{-1},$

whose values $M(z)$, $z\in\rho(B_{0})$, are bounded and everywhere defined
operators in $\mathcal{G}$. We remark that

$\gamma(z)^{*}=\Gamma_{1}(B_{0}-\overline{z})^{-1},\quad z\in\rho(B_{0}).$
(A.1)

In the following we discuss how the self-adjointness of certain extensions of
$S$ can be shown and how their spectral properties can be analyzed with the
help of a boundary triple. Let $\Theta$ be a linear operator in $\mathcal{G}$.
Then we define in $\mathcal{H}$ the linear operator $B_{\Theta}$ as the
restriction of $S^{*}$ onto

$\mathrm{dom}\,B_{\Theta}:=\big{\\{}f\in\mathrm{dom}\,S^{*}:\Gamma_{1}f=\Theta\Gamma_{0}f\big{\\}}.$
(A.2)

Note that in the above definition the equation $\Gamma_{1}f=\Theta\Gamma_{0}f$
contains the condition $\Gamma_{0}f\in\mathrm{dom}\,\Theta$. We remark that
for the description of all closed extensions of $S$ via (A.2) it is necessary
to extend the class of parameters $\Theta$ to the class of closed linear
relations in $\mathcal{G}$; cf. [9, Chapter 2.2]. The following theorem shows
that several properties of $B_{\Theta}$ are encoded in the parameter $\Theta$.

######  Theorem A.2.

Let $\\{\mathcal{G},\Gamma_{0},\Gamma_{1}\\}$ be a boundary triple for $S^{*}$
with associated $\gamma$-field $\gamma$ and Weyl function $M$. Let $\Theta$ be
a linear operator in $\mathcal{G}$ and let $B_{\Theta}$ be defined by (A.2).
Then $B_{\Theta}$ is self-adjoint in $\mathcal{H}$ if and only if $\Theta$ is
self-adjoint in $\mathcal{G}$. If $\Theta$ is self-adjoint, then the following
holds for all $z\in\rho(B_{0})$:

  * (i)

$z\in\sigma(B_{\Theta})$ if and only if $0\in\sigma(\Theta-M(z))$.

  * (ii)

$z\in\sigma_{\textup{p}}(B_{\Theta})$ if and only if
$0\in\sigma_{\textup{p}}(\Theta-M(z))$.

  * (iii)

$z\in\sigma_{\textup{ess}}(B_{\Theta})$ if and only if
$0\in\sigma_{\textup{ess}}(\Theta-M(z))$.

  * (iv)

For $z\in\rho(B_{\Theta})$ one has

$(B_{\Theta}-z)^{-1}=(B_{0}-z)^{-1}+\gamma(z)\big{(}\Theta-M(z)\big{)}^{-1}\gamma(\overline{z})^{*}.$

Finally, we shortly recall an efficient way how a boundary triple can be
constructed, see [40, 41] or also [21, Section 1.4.2]. Let $B$ be a self-
adjoint operator in a Hilbert space $\mathcal{H}$, $\mathcal{G}$ be another
Hilbert space, and assume that
$\mathcal{T}:\mathrm{dom}\,B\rightarrow\mathcal{G}$ is surjective, continuous
(when $\mathrm{dom}\,B$ is endowed with the graph norm of $B$), and that
$\ker\mathcal{T}$ is dense in $\mathcal{H}$. Then,
$S:=B\upharpoonright\ker\mathcal{T}$ is a densely defined closed symmetric
operator in $\mathcal{H}$.

To proceed, define for $z\in\rho(B)$ the operator

$\mathcal{G}_{z}:=\big{(}\mathcal{T}(B-\overline{z})^{-1}\big{)}^{*}.$ (A.3)

It can be shown that $\mathcal{G}_{z}$ is bounded and injective. Moreover, one
has $\operatorname{ran}\mathcal{G}_{z}=\ker(S^{*}-z)$ and that for any
$z\in\rho(B)$ the direct sum decomposition

$\mathrm{dom}\,S^{*}=\mathrm{dom}\,B\dot{+}\ker(S^{*}-z)=\mathrm{dom}\,B\dot{+}\operatorname{ran}\mathcal{G}_{z}$

holds. Hence, for any $f\in\mathrm{dom}\,S^{*}$ there exist
$f_{z}\in\mathrm{dom}\,B$ and $\xi\in\mathcal{G}$ such that
$f=f_{z}+\mathcal{G}_{z}\xi$; it can be shown that $\xi$ is independent of the
choice of $z\in\rho(B)$. Then, the following result holds:

######  Proposition A.3.

Let $\zeta\in\rho(B)$ be fixed and define the mappings
$\Gamma_{0},\Gamma_{1}:\mathrm{dom}\,S^{*}\rightarrow\mathcal{G}$ acting on
$f=f_{\zeta}+\mathcal{G}_{\zeta}\xi=f_{\overline{\zeta}}+\mathcal{G}_{\overline{\zeta}}\xi$
with $f_{\zeta},f_{\overline{\zeta}}\in\mathrm{dom}\,B$, $\xi\in\mathcal{G}$,
as

$\Gamma_{0}f=\xi\quad\text{and}\quad\Gamma_{1}f=\frac{1}{2}\mathcal{T}(f_{\zeta}+f_{\overline{\zeta}}).$

Then, $\\{\mathcal{G},\Gamma_{0},\Gamma_{1}\\}$ is a boundary triple for
$S^{*}$. Moreover, the values of the $\gamma$-field and Weyl function are
given by $\gamma(z)=\mathcal{G}_{z}$ and
$M(z)=\mathcal{T}(\mathcal{G}_{z}-\frac{1}{2}(\mathcal{G}_{\zeta}+\mathcal{G}_{\overline{\zeta}}))$,
$z\in\rho(B)$.

##  Appendix B Proof of Proposition 3.2

First, we provide an abstract result about the compactness of integral
operators under suitable assumptions on the integral kernel that will be used
several times in the proof of Proposition 3.2.

######  Lemma B.1.

Let $k:\mathbb{R}^{2}\rightarrow\mathbb{C}$ be continuous in $\mathbb{R}^{2}$
and differentiable on the set $\\{(s,t)\in\mathbb{R}^{2}:s\neq t\\}$. Assume
that there exist constants $R,C,\kappa>0$ and $\alpha\in[0,1)$ such that for
$p\in\\{0,1\\}$ and all $(s,t)\in\mathbb{R}^{2}$ with $s\neq t$

$\max\left\\{\left|\frac{\partial^{p}}{\partial
s^{p}}k(s,t)\right|,\left|\frac{\partial^{p}}{\partial
s^{p}}\overline{k(t,s)}\right|\right\\}\leq
C\begin{cases}|s-t|^{-\alpha},&|s|,|t|\leq 3R,\\\
e^{-\kappa|s-t|},&|s-t|>2R,\\\ 0,&s,t>R\text{ or }s,t<-R,\end{cases}$ (B.1)

holds. Then, for any $r\in[-1,0]$ the mapping

$\mathcal{K}u(s)=\int_{\mathbb{R}}k(s,t)u(t)dt,\quad u\in
C_{0}^{\infty}(\mathbb{R};\mathbb{C}),~{}s\in\mathbb{R},$

can be extended to a compact operator
$\mathcal{K}:H^{r}(\mathbb{R};\mathbb{C})\rightarrow
H^{r+1}(\mathbb{R};\mathbb{C})$.

###### Proof.

Define for $p\in\\{0,1\\}$ the function
$k_{p}(s,t):=\frac{\partial^{p}}{\partial s^{p}}k(s,t)$, $s\neq t$, and the
integral operator $\mathcal{K}_{p}$

$\mathcal{K}_{p}:L^{2}(\mathbb{R};\mathbb{C})\rightarrow
L^{2}(\mathbb{R};\mathbb{C}),\qquad\mathcal{K}_{p}u(s):=\int_{\mathbb{R}}k_{p}(s,t)u(t)dt.$

The proof is split into several steps: In Step 1 we show that $k_{p}\in
L^{2}(\mathbb{R}^{2};\mathbb{C})$, which implies that $\mathcal{K}_{p}$ is a
Hilbert-Schmidt operator in $L^{2}(\mathbb{R};\mathbb{C})$ and hence, it is
well-defined and compact. In Step 2 we verify that for any $u\in
L^{2}(\mathbb{R};\mathbb{C})$ the function $\mathcal{K}u=\mathcal{K}_{0}u$ is
weakly differentiable and that
$\frac{d}{ds}\mathcal{K}_{0}u=\mathcal{K}_{1}u$. Using this, it is proved in
Step 3 that $\mathcal{K}$ gives rise to a compact operator from
$L^{2}(\mathbb{R};\mathbb{C})$ to $H^{1}(\mathbb{R};\mathbb{C})$. Then, in
Step 4 it is argued that $\mathcal{K}$ admits a compact extension from
$H^{-1}(\mathbb{R};\mathbb{C})$ to $L^{2}(\mathbb{R};\mathbb{C})$. Finally, in
Step 5 an interpolation argument is used to deduce that also for any
$r\in(-1,0)$ the map $\mathcal{K}$ has a compact extension from
$H^{r}(\mathbb{R};\mathbb{C})$ to $H^{r+1}(\mathbb{R};\mathbb{C})$.

Step 1. We claim that $k_{p}\in L^{2}(\mathbb{R}^{2};\mathbb{C})$. For two
intervals $I_{1}$ and $I_{2}$ set

$\mathcal{I}(I_{1},I_{2}):=\int_{I_{1}\times I_{2}}|k_{p}(s,t)|^{2}dsdt.$

Then

$\|k_{p}\|_{L^{2}(\mathbb{R}^{2};\mathbb{C})}^{2}=\mathcal{I}((-\infty,-R],\mathbb{R})+\mathcal{I}((-R,R),\mathbb{R})+\mathcal{I}([R,+\infty),\mathbb{R}).$
(B.2)

We further decompose

$\begin{split}\mathcal{I}((-\infty,-R],\mathbb{R})&=\mathcal{I}((-\infty,-R],(-\infty,-R])+\mathcal{I}((-\infty,-3R],(-R,R))\\\
&\qquad+\mathcal{I}((-3R,-R],(-R,R))+\mathcal{I}((-\infty,-R],[R,+\infty)).\end{split}$
(B.3)

By (B.1) we have

$\mathcal{I}((-\infty,-R],(-\infty,-R])=0$ (B.4)

and also

$\mathcal{I}((-\infty,-R],[R,+\infty))\leq
C^{2}\int_{-\infty}^{-R}\int_{R}^{+\infty}e^{2\kappa(t-s)}dsdt<+\infty.$ (B.5)

In the same way one finds that $\mathcal{I}((-\infty,-3R],(-R,R))<+\infty$.
Eventually, taking the properties of $k$ from (B.1) for $|s|,|t|\leq 3R$ into
account, one gets for bounded intervals $I_{1},I_{2}\subset[-3R,3R]$ that

$\mathcal{I}(I_{1},I_{2})<+\infty.$ (B.6)

Using (B.4)–(B.6) in (B.3) one concludes that
$\mathcal{I}((-\infty,-R],\mathbb{R})<+\infty$. In exactly the same way one
gets that $\mathcal{I}([R,+\infty),\mathbb{R})<+\infty$. Finally, employing
again similar estimates as in (B.5)–(B.6) one finds that

$\begin{split}\mathcal{I}((-R,R),\mathbb{R})=\mathcal{I}((-R,R),(-\infty,-3R])&+\mathcal{I}((-R,R),(-3R,3R))\\\
&+\mathcal{I}((-R,R),[3R,+\infty))<+\infty.\end{split}$

Hence, we get from (B.2) that $k_{p}\in L^{2}(\mathbb{R}^{2};\mathbb{C})$
showing that $\mathcal{K}_{p}$ is a Hilbert-Schmidt operator and thus, compact
in $L^{2}(\mathbb{R};\mathbb{C})$.

Step 2. Let $u\in C_{0}^{\infty}(\mathbb{R};\mathbb{C})$ be fixed. We claim
that $\mathcal{K}_{0}u\in H^{1}(\mathbb{R};\mathbb{C})$ and
$\frac{d}{ds}\mathcal{K}_{0}u=\mathcal{K}_{1}u$. Taking the result from Step 1
into account, it suffices to prove the latter identity for the weak
derivative. Let $\varphi\in C_{0}^{\infty}(\mathbb{R};\mathbb{C})$. By (B.1)
one has for any fixed $t\in\mathbb{R}$ that $\varphi^{\prime}k(\cdot,t)\in
L^{1}(\mathbb{R};\mathbb{C})$. Hence, we can apply Fubini’s theorem and the
dominated convergence theorem and get

$\begin{split}\int_{\mathbb{R}}\varphi^{\prime}(s)\mathcal{K}_{0}u(s)ds&=\int_{\mathbb{R}}\int_{\mathbb{R}}\varphi^{\prime}(s)k(s,t)ds\,u(t)dt\\\
&=\int_{\mathbb{R}}\lim_{\varepsilon\searrow
0}\int_{\mathbb{R}\setminus(t-\varepsilon,t+\varepsilon)}\varphi^{\prime}(s)k(s,t)ds\,u(t)dt.\end{split}$

Next, we use integration by parts, which is allowed, as for a fixed $t$ the
map $s\mapsto k(s,t)$ is differentiable in
$\mathbb{R}\setminus(t-\varepsilon,t+\varepsilon)$. Using the continuity of
$k$ and $\varphi$, the dominated convergence theorem, which is applicable,
since for any fixed $t\in\mathbb{R}$ one has $\varphi\frac{\partial}{\partial
s}k(\cdot,t)\in L^{1}(\mathbb{R};\mathbb{C})$ due to (B.1), and Fubini’s
theorem we find that

$\begin{split}\int_{\mathbb{R}}\varphi^{\prime}(s)\mathcal{K}_{0}u(s)ds&=\int_{\mathbb{R}}\lim_{\varepsilon\searrow
0}\left[-\varphi(s)k(s,t)\Big{|}_{t-\varepsilon}^{t+\varepsilon}-\int_{\mathbb{R}\setminus(t-\varepsilon,t+\varepsilon)}\varphi(s)\frac{\partial}{\partial
s}k(s,t)ds\right]u(t)dt\\\
&=-\int_{\mathbb{R}}\int_{\mathbb{R}}\varphi(s)\frac{\partial}{\partial
s}k(s,t)ds\,u(t)dt=-\int_{\mathbb{R}}\varphi(s)\mathcal{K}_{1}u(s)ds.\end{split}$

This shows the claim about the weak derivative of $\mathcal{K}_{0}u$. Note
that by density and continuity $\frac{d}{ds}\mathcal{K}_{0}u=\mathcal{K}_{1}u$
also holds for $u\in L^{2}(\mathbb{R};\mathbb{C})$.

Step 3. We show the claim of the lemma for $r=0$, i.e. that the mapping

$\mathcal{K}:L^{2}(\mathbb{R};\mathbb{C})\rightarrow
H^{1}(\mathbb{R};\mathbb{C}),\qquad\mathcal{K}u(s):=\int_{\mathbb{R}}k(s,t)u(t)dt,$

is compact. First, note that $\mathcal{K}$ is well-defined by the results in
Step 1 & 2. Next, consider the map

$\mathcal{D}:H^{1}(\mathbb{R};\mathbb{C})\rightarrow
L^{2}(\mathbb{R};\mathbb{C}),\qquad\mathcal{D}f=f+f^{\prime}.$

With the help of the Fourier transform it is not difficult to see that
$\mathcal{D}$ is bijective. Using the results from Step 1 & 2 we find that
$\mathcal{D}\mathcal{K}=\mathcal{K}_{0}+\mathcal{K}_{1}$ and the right hand
side defines a compact operator in $L^{2}(\mathbb{R};\mathbb{C})$. Therefore,
$\mathcal{K}=\mathcal{D}^{-1}(\mathcal{K}_{0}+\mathcal{K}_{1})$ is compact as
an operator from $L^{2}(\mathbb{R};\mathbb{C})$ to
$H^{1}(\mathbb{R};\mathbb{C})$.

Step 4. We claim that $\mathcal{K}$ admits a compact extension
$\widetilde{\mathcal{K}}:H^{-1}(\mathbb{R};\mathbb{C})\rightarrow
L^{2}(\mathbb{R};\mathbb{C})$. Due to the assumptions on the function
$\overline{k(t,s)}$ in (B.1) it follows as in Step 1–3 that the map

$\mathcal{L}:L^{2}(\mathbb{R};\mathbb{C})\rightarrow
H^{1}(\mathbb{R};\mathbb{C}),\qquad\mathcal{L}u(s):=\int_{\mathbb{R}}\overline{k(t,s)}u(t)dt,$

is compact. Hence, also the anti-dual operator

$\widetilde{\mathcal{K}}:=\mathcal{L}^{\prime}:H^{-1}(\mathbb{R};\mathbb{C})\rightarrow
L^{2}(\mathbb{R};\mathbb{C})$

is compact. We claim that $\widetilde{\mathcal{K}}$ is an extension of
$\mathcal{K}$. To see this, we note that $\mathcal{L}u=\mathcal{K}_{0}^{*}u$
holds for any $u\in L^{2}(\mathbb{R};\mathbb{C})$, and compute for $u,v\in
L^{2}(\mathbb{R};\mathbb{C})$

$\begin{split}\big{(}\widetilde{\mathcal{K}}u,v\big{)}_{L^{2}(\mathbb{R};\mathbb{C})}=\big{(}u,\mathcal{L}v\big{)}_{H^{-1}(\mathbb{R};\mathbb{C})\times
H^{1}(\mathbb{R};\mathbb{C})}&=\big{(}u,\mathcal{L}v\big{)}_{L^{2}(\mathbb{R};\mathbb{C})}\\\
&=\big{(}\mathcal{K}_{0}u,v\big{)}_{L^{2}(\mathbb{R};\mathbb{C})}=\big{(}\mathcal{K}u,v\big{)}_{L^{2}(\mathbb{R};\mathbb{C})};\end{split}$

in the latter computation we used the symbol
$(\cdot,\cdot)_{H^{-1}(\mathbb{R};\mathbb{C})\times
H^{1}(\mathbb{R};\mathbb{C})}$ for the sesquilinear duality product in
$H^{-1}(\mathbb{R};\mathbb{C})\times H^{1}(\mathbb{R};\mathbb{C})$. Since this
is true for all $v\in L^{2}(\mathbb{R};\mathbb{C})$, we conclude that
$\widetilde{\mathcal{K}}u=\mathcal{K}u$, which shows that
$\widetilde{\mathcal{K}}$ is a compact extension of $\mathcal{K}$.

Step 5. By the results in Step 3 & 4 the claim of the lemma is true for $r=-1$
and $r=0$. By interpolation, one can apply, e.g., [24, Theorem 10] with
$X_{0}=H^{-1}(\mathbb{R};\mathbb{C})$, $X_{1}=L^{2}(\mathbb{R};\mathbb{C})$,
and $W=H^{-2}(\mathbb{R};\mathbb{C})$, it follows that also for $r\in(-1,0)$
the map $\mathcal{K}$ gives rise to a compact operator from
$H^{r}(\mathbb{R};\mathbb{C})$ to $H^{r+1}(\mathbb{R};\mathbb{C})$. ∎

###### Proof of Proposition 3.2.

Define the operator

$\mathcal{K}:=U_{\gamma}V(\sigma\cdot\nu)\mathcal{C}_{z}^{m,c}(\sigma\cdot\nu)V^{*}U_{\gamma}^{-1}-U_{x}\sigma_{2}\mathcal{C}_{z}^{m,c,\Sigma}\sigma_{2}U_{x}^{-1}.$

The map $\mathcal{K}$ is an integral operator and we will do a fine analysis
of the integral kernel of $\mathcal{K}$, so that we can apply Lemma B.1 for
$r=-\frac{1}{2}$ to each of the entries of the $2\times 2$ block operator
matrix $\mathcal{K}$ to conclude the claimed result.

Recall that $\gamma$ is given by (2.1). We use for
$y=(y_{1},y_{2})\in\mathbb{R}^{2}$ the notation $\boldsymbol{y}=y_{1}+iy_{2}$,
implying that $|y|_{\mathbb{R}^{2}}=|\boldsymbol{y}|_{\mathbb{C}}$.
Furthermore, for $s\in\mathbb{R}$ one has
$\boldsymbol{t}(\gamma(s))=\dot{\boldsymbol{\gamma}}(s)$ and
$\boldsymbol{\nu}(\gamma(s))=\dot{\gamma}_{2}(s)-i\dot{\gamma_{1}}(s)=-i\dot{\boldsymbol{\gamma}}(s)$,
and thus

$V(\gamma(s))=\begin{pmatrix}1&0\\\
0&\overline{\dot{\boldsymbol{\gamma}}(s)}\end{pmatrix}\quad\text{and}\quad\sigma\cdot\nu(\gamma(s))=\begin{pmatrix}0&i\overline{\dot{\boldsymbol{\gamma}}(s)}\\\
-i\dot{\boldsymbol{\gamma}}(s)&0\end{pmatrix}.$

Therefore, by using the definition of $U_{\gamma},U_{x}$, and
$\mathcal{C}_{z}^{m,c}$ we see for $u\in L^{2}(\mathbb{R};\mathbb{C}^{2})$
that

$\begin{split}\mathcal{K}u(s)&=\lim_{\varepsilon\searrow
0}\int_{\mathbb{R}\setminus B(s,\varepsilon)}k(s,t)u(t)dt,\end{split}$ (B.7)

where

$\begin{split}k(s,t):=\begin{pmatrix}0&i\overline{\dot{\boldsymbol{\gamma}}(s)}\\\
-i&0\end{pmatrix}G_{z}^{m,c}\big{(}\gamma(s)-\gamma(t)\big{)}\begin{pmatrix}0&i\\\
-i\dot{\boldsymbol{\gamma}}(t)&0\end{pmatrix}-\sigma_{2}G_{z}^{m,c}(s-t,0)\sigma_{2}.\end{split}$

Define the number $\zeta(z):=-i\sqrt{\tfrac{z^{2}}{c^{2}}-(mc)^{2}}$ and note
that

$\begin{split}\frac{\sigma\cdot(\gamma(s)-\gamma(t))}{|\gamma(s)-\gamma(t)|}&=\frac{1}{|\gamma(s)-\gamma(t)|}\begin{pmatrix}0&\overline{\boldsymbol{\gamma}(s)}-\overline{\boldsymbol{\gamma}(t)}\\\
\boldsymbol{\gamma}(s)-\boldsymbol{\gamma}(t)&0\end{pmatrix}\\\
&=|\gamma(s)-\gamma(t)|\begin{pmatrix}0&\frac{1}{\boldsymbol{\gamma}(s)-\boldsymbol{\gamma}(t)}\\\
\frac{1}{\overline{\boldsymbol{\gamma}(s)}-\overline{\boldsymbol{\gamma}(t)}}&0\end{pmatrix}.\end{split}$

Then, by the definition of $G_{z}^{m,c}$ in (3.2), the function $k$ can be
further decomposed as $k(s,t)=k_{1}(s,t)+k_{2}(s,t)+k_{3}(s,t)$ with

$\begin{split}k_{1}(s,t)&:=\frac{1}{2\pi
c}\overline{\dot{\boldsymbol{\gamma}}(s)}\big{(}\dot{\boldsymbol{\gamma}}(t)-\dot{\boldsymbol{\gamma}}(s))\begin{pmatrix}\tfrac{z}{c}-mc&0\\\
0&0\end{pmatrix}K_{0}\left(\zeta(z)|\gamma(s)-\gamma(t)|\right),\\\
k_{2}(s,t)&:=\frac{1}{2\pi
c}\left(\frac{z}{c}\sigma_{0}-mc\sigma_{3}\right)\big{(}K_{0}\left(\zeta(z)|\gamma(s)-\gamma(t)|\right)-K_{0}\left(\zeta(z)|s-t|\right)\big{)},\\\
k_{3}(s,t)&:=\frac{i\zeta(z)}{2\pi
c}\Bigg{(}K_{1}\left(\zeta(z)|s-t|\right)|s-t|\begin{pmatrix}0&\frac{1}{s-t}\\\
\frac{1}{s-t}&0\end{pmatrix}\\\ &\qquad-
K_{1}\left(\zeta(z)|\gamma(s)-\gamma(t)|\right)|\gamma(s)-\gamma(t)|\Bigg{(}\begin{matrix}0&\tfrac{\overline{\dot{\boldsymbol{\gamma}}(s)}}{\overline{\boldsymbol{\gamma}(s)}-\overline{\boldsymbol{\gamma}(t)}}\\\
\tfrac{\dot{\boldsymbol{\gamma}}(t)}{\boldsymbol{\gamma}(s)-\boldsymbol{\gamma}(t)}&0\end{matrix}\Bigg{)}\Bigg{)}.\end{split}$

Denote the corresponding integral operators by $\mathcal{K}_{j}$,
$j\in\\{1,2,3\\}$, i.e.

$\mathcal{K}_{j}\varphi(s)=\lim_{\varepsilon\searrow
0}\int_{\mathbb{R}\setminus B(s,\varepsilon)}k_{j}(s,t)\varphi(t)dt,\quad
s\in\mathbb{R},~{}\varphi\in C_{0}^{\infty}(\mathbb{R};\mathbb{C}^{2}).$

We show that the kernels $k_{j}$ satisfy the assumptions in Lemma B.1 implying
that each of these operators is not strongly singular (i.e. the limit for
$\varepsilon\searrow 0$ can be removed) and has the claimed mapping
properties.

Let $M>0$ be the number specified in the definition of $\gamma$ in (2.1).
First, we note that, due to the asymptotic properties of the modified Bessel
functions and their derivatives for large arguments [1, Sections 9.6 and 9.7]
and (2.2), there exist constants $M_{1},\kappa_{1}>0$ such that for
$j\in\\{1,2,3\\}$ and $p\in\\{0,1\\}$

$\max\left\\{\left|\frac{\partial^{p}k_{j}(s,t)}{\partial
s^{p}}\right|,\left|\frac{\partial^{p}\overline{k_{j}(t,s)}}{\partial
s^{p}}\right|\right\\}\leq M_{1}e^{\kappa_{1}|s-t|},\quad|s-t|\geq M.$ (B.8)

Next, taking the power series representation of $K_{0}$ from [1, equation
9.6.13] into account, there exist a constant $M_{2}$ and holomorphic functions
$g_{1},g_{2}$ such that

$K_{0}(\xi)=-\log\xi+M_{2}+g_{1}(\xi^{2})+\xi^{2}g_{2}(\xi^{2})\log\xi=-\log\xi+g_{3}(\xi),\quad\xi\in\mathbb{C}\setminus\\{0\\},$
(B.9)

with the $C^{1}$-smooth function
$g_{3}(\xi):=M_{2}+g_{1}(\xi^{2})+\xi^{2}g_{2}(\xi^{2})\log\xi$; cf. the proof
of [11, Lemma 3.2] for a similar consideration. We will often use that the
function

$\log\big{(}\zeta(z)|\gamma(s)-\gamma(t)|\big{)}-\log\big{(}\zeta(z)|s-t|\big{)}=\log\left|\frac{\boldsymbol{\gamma}(s)-\boldsymbol{\gamma}(t)}{s-t}\right|$
(B.10)

is $C^{\infty}$-smooth. For $s\neq t$, this follows from the injectivity and
the smoothness of $\gamma$, and for $s=t$ from a Taylor series expansion of
$\boldsymbol{\gamma}$ and (2.2).

In the following, the properties of $\mathcal{K}_{1}$ and its kernel $k_{1}$
are further analyzed. With (B.9) and (B.10) it is not difficult to see that
$k_{1}$ can be written as

$\begin{split}k_{1}(s,t)&=\frac{1}{2\pi
c}\overline{\dot{\boldsymbol{\gamma}}(s)}\big{(}\dot{\boldsymbol{\gamma}}(t)-\dot{\boldsymbol{\gamma}}(s)\big{)}\begin{pmatrix}\tfrac{z}{c}-mc&0\\\
0&0\end{pmatrix}\\\
&\qquad\cdot\left(-\log\big{(}\zeta(z)|s-t|\big{)}-\log\left|\frac{\boldsymbol{\gamma}(s)-\boldsymbol{\gamma}(t)}{s-t}\right|+g_{3}\big{(}\zeta(z)|\gamma(s)-\gamma(t)|\big{)}\right).\end{split}$

Since $\boldsymbol{\gamma}$ is smooth, the function
$(\dot{\boldsymbol{\gamma}}(t)-\dot{\boldsymbol{\gamma}}(s))\log(\zeta(z)|s-t|)$
is continuous in $\mathbb{R}^{2}$, differentiable for $s\neq t$, and its first
derivatives are bounded by
$\widetilde{M}_{1}\big{|}\log|s-t|\big{|}+\widetilde{M}_{2}$ for some
constants $\widetilde{M}_{1},\widetilde{M}_{2}>0$. Hence, $k_{1}$ is
continuous in $\mathbb{R}^{2}$ and differentiable for $s\neq t$, for
$p\in\\{0,1\\}$ one has

$\max\left\\{\left|\frac{\partial^{p}k_{1}(s,t)}{\partial
s^{p}}\right|,\left|\frac{\partial^{p}\overline{k_{1}(t,s)}}{\partial
s^{p}}\right|\right\\}\leq M_{3}\big{|}\log|s-t|\big{|}+M_{4},\quad|s|,|t|\leq
2M,$ (B.11)

where $M_{3},M_{4}$ are positive constants, and $k_{1}(s,t)=0$, if $s,t<-M$ or
$s,t>M$, as in that case
$\dot{\boldsymbol{\gamma}}(s)=\dot{\boldsymbol{\gamma}}(t)$. Since $k_{1}$
fulfils (B.8), this function satisfies the assumptions in Lemma B.1 and thus,
$\mathcal{K}_{1}:H^{-1/2}(\mathbb{R};\mathbb{C}^{2})\rightarrow
H^{1/2}(\mathbb{R};\mathbb{C}^{2})$ is compact.

Next, $\mathcal{K}_{2}$ and its kernel $k_{2}$ are considered. With (B.9) and
(B.10) one finds that $k_{2}\in C^{1}(\mathbb{R}^{2};\mathbb{C}^{2})$. In
particular, this shows for a constant $M_{5}>0$ that

$\max\left\\{\left|\frac{\partial^{p}k_{2}(s,t)}{\partial
s^{p}}\right|,\left|\frac{\partial^{p}\overline{k_{2}(t,s)}}{\partial
s^{p}}\right|\right\\}\leq M_{5},\quad|s|,|t|\leq 2M.$

Finally, since for $s,t\geq M$

$|\gamma(s)-\gamma(t)|=\left|\begin{pmatrix}a_{+}\\\
c_{+}\end{pmatrix}(s-t)\right|=|s-t|$ (B.12)

holds, we get that $k_{2}(s,t)=0$, if $s,t\geq M$. A similar consideration can
be done also for $s,t\leq-M$. With (B.8) one sees that $k_{2}$ satisfies (B.1)
and hence, by Lemma B.1 the map
$\mathcal{K}_{2}:H^{-1/2}(\mathbb{R};\mathbb{C}^{2})\rightarrow
H^{1/2}(\mathbb{R};\mathbb{C}^{2})$ is compact.

It remains to analyze $\mathcal{K}_{3}$ and its kernel $k_{3}$. By the power
series representation of $K_{1}$ in [1, equation 9.6.11] there exist
holomorphic functions $g_{4},g_{5}$ such that

$K_{1}(\xi)=\frac{1}{\xi}+\xi g_{4}(\xi^{2})\log\xi+\xi
g_{5}(\xi^{2}),\quad\xi\in\mathbb{C}\setminus\\{0\\};$ (B.13)

cf. the proof of [11, Lemma 3.2] for a similar consideration. Hence, we can
further decompose $k_{3}(s,t)$ as
$k_{3}(s,t)=k_{4}(s,t)+k_{5}(s,t)+k_{6}(s,t)$ with

$\begin{split}k_{4}(s,t)&:=\frac{i}{2\pi c}\begin{pmatrix}0&\frac{1}{s-t}\\\
\frac{1}{s-t}&0\end{pmatrix}-\frac{i}{2\pi
c}\begin{pmatrix}0&\frac{\overline{\dot{\boldsymbol{\gamma}}(s)}}{\overline{\boldsymbol{\gamma}(s)}-\overline{\boldsymbol{\gamma}(t)}}\\\
\frac{\dot{\boldsymbol{\gamma}}(t)}{\boldsymbol{\gamma}(s)-\boldsymbol{\gamma}(t)}&0\end{pmatrix},\\\
k_{5}(s,t)&:=\frac{i\zeta(z)^{2}}{2\pi
c}\left(g_{4}(\zeta(z)^{2}|s-t|^{2})\log(\zeta(z)|s-t|)\begin{pmatrix}0&s-t\\\
s-t&0\end{pmatrix}\right.\\\ &\qquad\qquad\qquad-
g_{4}(\zeta(z)^{2}|\gamma(s)-\gamma(t)|^{2})\log(\zeta(z)|\gamma(s)-\gamma(t)|)\\\
&\qquad\qquad\qquad\qquad\cdot\left.\begin{pmatrix}0&\overline{\dot{\boldsymbol{\gamma}}(s)}(\boldsymbol{\gamma}(s)-\boldsymbol{\gamma}(t))\\\
\dot{\boldsymbol{\gamma}}(t)(\overline{\boldsymbol{\gamma}(s)}-\overline{\boldsymbol{\gamma}(t)})&0\end{pmatrix}\right),\\\
k_{6}(s,t)&:=\frac{i\zeta(z)^{2}}{2\pi
c}\left(g_{5}(\zeta(z)^{2}|s-t|^{2})\begin{pmatrix}0&s-t\\\
s-t&0\end{pmatrix}\right.\\\
&\qquad-\left.g_{5}(\zeta(z)^{2}|\gamma(s)-\gamma(t)|^{2})\begin{pmatrix}0&\overline{\dot{\boldsymbol{\gamma}}(s)}(\boldsymbol{\gamma}(s)-\boldsymbol{\gamma}(t))\\\
\dot{\boldsymbol{\gamma}}(t)(\overline{\boldsymbol{\gamma}(s)}-\overline{\boldsymbol{\gamma}(t)})&0\end{pmatrix}\right).\end{split}$

Since $g_{5}$ is analytic, we observe that $k_{6}$ can be extended to a
$C^{\infty}$-smooth function in $\mathbb{R}^{2}$. The same is also true for
$k_{4}$. Indeed, employing the fundamental theorem of calculus twice, we get

$\begin{split}\frac{1}{s-t}-\frac{\dot{\boldsymbol{\gamma}}(t)}{\boldsymbol{\gamma}(s)-\boldsymbol{\gamma}(t)}&=\frac{\boldsymbol{\gamma}(s)-\boldsymbol{\gamma}(t)-\dot{\boldsymbol{\gamma}}(t)(s-t)}{(s-t)(\boldsymbol{\gamma}(s)-\boldsymbol{\gamma}(t))}\\\
&=\frac{\int_{0}^{1}\dot{\boldsymbol{\gamma}}(t+\xi(s-t))(s-t)d\xi-\dot{\boldsymbol{\gamma}}(t)(s-t)}{(s-t)(\boldsymbol{\gamma}(s)-\boldsymbol{\gamma}(t))}\\\
&=\int_{0}^{1}\int_{0}^{1}\ddot{\boldsymbol{\gamma}}(t+\zeta\xi(s-t))\xi
d\zeta
d\xi\cdot\frac{s-t}{\boldsymbol{\gamma}(s)-\boldsymbol{\gamma}(t)}.\end{split}$

The first factor in the last line is a smooth function everywhere in
$\mathbb{R}^{2}$, and due to (2.2) the same is true also for the second
factor. Therefore, $k_{4}$ can be extended to a $C^{\infty}$-function at
$s=t$.

Next, we analyze the properties of $k_{5}(s,t)$ for $s,t$ being close to each
other. In a very similar way as in the consideration of $k_{1}$ one gets that
$k_{5}$ is continuous everywhere in $\mathbb{R}^{2}$ and that there exist
constants $M_{6},M_{7}>0$ such that for $p\in\\{0,1\\}$

$\max\left\\{\left|\frac{\partial^{p}k_{5}(s,t)}{\partial
s^{p}}\right|,\left|\frac{\partial^{p}\overline{k_{5}(t,s)}}{\partial
s^{p}}\right|\right\\}\leq M_{6}\big{|}\log|s-t|\big{|}+M_{7},\quad|s|,|t|\leq
2M,$

holds; cf. (B.11). Since $k_{4}$ and $k_{6}$ are smooth functions and
$k_{3}=k_{4}+k_{5}+k_{6}$, we conclude that there exist constants
$M_{8},M_{9}>0$ such that for $p\in\\{0,1\\}$

$\max\left\\{\left|\frac{\partial^{p}k_{3}(s,t)}{\partial
s^{p}}\right|,\left|\frac{\partial^{p}\overline{k_{3}(t,s)}}{\partial
s^{p}}\right|\right\\}\leq M_{8}\big{|}\log|s-t|\big{|}+M_{9},\quad|s|,|t|\leq
2M,$ (B.14)

is satisfied.

Finally, we show that $k_{3}(s,t)=0$ for $s,t>M$ or $s,t<-M$. Note that for
$s,t\geq M$ we have (B.12) and

$\frac{\dot{\boldsymbol{\gamma}}(t)}{\boldsymbol{\gamma}(s)-\boldsymbol{\gamma}(t)}=\frac{a_{+}+ic_{+}}{a_{+}(s-t)+ic_{+}(s-t)}=\frac{1}{s-t}.$

Similar identities are also true for $s,t\leq-M$. Hence, we conclude from the
definition of $k_{3}$ that $k_{3}(s,t)=0$ for $s,t>M$ or $s,t<-M$. Combining
this with (B.8) and (B.14), it follows from Lemma B.1 that also
$\mathcal{K}_{3}:H^{-1/2}(\mathbb{R};\mathbb{C}^{2})\rightarrow
H^{1/2}(\mathbb{R};\mathbb{C}^{2})$ is compact. Together with the mapping
properties of $\mathcal{K}_{1}$ and $\mathcal{K}_{2}$ shown above this yields
that

$\mathcal{K}=\mathcal{K}_{1}+\mathcal{K}_{2}+\mathcal{K}_{3}:H^{-1/2}(\mathbb{R};\mathbb{C}^{2})\rightarrow
H^{1/2}(\mathbb{R};\mathbb{C}^{2})$

is a compact operator. The proof of Proposition 3.2 is complete. ∎

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