# Spectral analysis of the Dirac operator with a singular interaction on a
broken line

Dale Frymark  Technische Universität Graz, Institut für Angewandte Mathematik,
Steyrergasse 30, 8010 Graz, Austria<EMAIL_ADDRESS>,  Markus Holzmann
Technische Universität Graz, Institut für Angewandte Mathematik, Steyrergasse
30, 8010 Graz, Austria<EMAIL_ADDRESS>and  Vladimir Lotoreichik
Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of
Sciences, 25068 Řež, Czech Republic<EMAIL_ADDRESS>
###### Abstract.

We consider the one-parametric family of self-adjoint realizations of the two-
dimensional massive Dirac operator with a Lorentz scalar $\delta$-shell
interaction of strength $\tau\in\mathbb{R}\setminus\\{-2,0,2\\}$ supported on
a broken line of opening angle $2\omega$ with $\omega\in(0,\frac{\pi}{2})$.
The essential spectrum of any such self-adjoint realization is symmetric with
respect to the origin with a gap around zero whose size depends on the mass
and, for $\tau<0$, also on the strength of the interaction, but does not
depend on $\omega$. As the main result, we prove that for any $N\in\mathbb{N}$
and strength $\tau\in(-\infty,0)\setminus\\{-2\\}$ the discrete spectrum of
any such self-adjoint realization has at least $N$ discrete eigenvalues, with
multiplicities taken into account, in the gap of the essential spectrum
provided that $\omega$ is sufficiently small. Moreover, we obtain an explicit
estimate on $\omega$ sufficient for this property to hold. For
$\tau\in(0,\infty)\setminus\\{2\\}$, the discrete spectrum consists of at most
one simple eigenvalue.

###### Key words and phrases:

Dirac operator; Lorentz scalar $\delta$-shell interaction; broken line;
essential spectrum; geometrically induced bound states; min-max principle

###### 2020 Mathematics Subject Classification:

47F05, 35P15, 35Q40

###### Contents

  1. 1 Introduction
  2. 2 Singular interaction supported on a broken line
  3. 3 Quadratic Form
  4. 4 Symmetry of the spectrum
  5. 5 Essential Spectrum
  6. 6 Discrete spectrum
  7. A On a Sobolev space with a piecewise constant weight
  8. B Bound for a one-dimensional auxiliary problem

##  1\. Introduction

Dirac operators with singular interactions supported on hypersurfaces in the
Euclidean space serve as an idealized model for more physically realistic
Dirac operators with regular potentials localized in the vicinity of a
hypersurface. The Dirac operator with a singular interaction supported on a
sphere in $\mathbb{R}^{3}$ was first analysed in [16], where separation of
variables was extensively used. The renewal of interest to this topic is
connected with the papers [3, 4, 5], where singular interactions supported on
general closed surfaces in $\mathbb{R}^{3}$ were analysed, see also [6, 7] for
further considerations in dimension three, [9, 15] for the study of the two-
dimensional case, and [10] for a recent unified treatment for space dimensions
two and three.

In this paper we are interested in spectral properties of Dirac operators with
singular potentials supported on a broken line, for which there are several
motivations. On the one hand, the analysis of two-dimensional Dirac operators
with singular interactions supported on curves with corner points revealed the
importance of the model Dirac operators with singular interactions supported
on unbounded broken lines [28, 35]. In particular, self-adjointness properties
of such Dirac operators with Lorentz scalar $\delta$-shell interactions
supported on star graphs were studied in [25].

On the other hand, the broken line is a typical example where the existence of
geometrically induced eigenvalues is observed. The relation of the spectral
properties of a differential operator and the geometry of the interaction
support or the domain in which the operator acts is one of the most classical
questions in spectral theory. In particular, one is often interested in
situations when a modification of the geometry generates bound states. There
is a huge number of results available for Laplacians with different boundary
conditions, see the monograph [21] and the references therein or [17, 18] for
the case of Dirichlet boundary conditions, [23, 32] for related results for
Laplacians with Robin boundary conditions, and [14, 22] for considerations on
the magnetic Laplacian with Neumann boundary conditions. One interesting
example of the above type is the non-relativistic counterpart of the Dirac
operator with a singular potential. Consider a Schrödinger operator with a
$\delta$-potential of strength $\alpha\in\mathbb{R}$ supported on a broken
line $\Gamma$ of opening angle $2\omega$ with $\omega\in(0,\frac{\pi}{2}]$,
i.e. the operator that is formally given by

$T_{\alpha}=-\Delta+\alpha\delta_{\Gamma}.$

With the help of the Birman-Schwinger principle it was established in [20]
that $\sigma(T_{\alpha})=\sigma_{\textup{ess}}(T_{\alpha})=[0,\infty)$ for
$\alpha\geq 0$, while for attractive interactions $\alpha<0$ one has
$\sigma_{\textup{ess}}(T_{\alpha})=[-\frac{\alpha^{2}}{4},\infty)$ and, if
$\Gamma$ is not a straight line, then the discrete spectrum of $T_{\alpha}$ is
non-empty. In [24] it was shown with the help of the min-max principle and
suitable test functions that for any $N\in\mathbb{N}$ the operator
$T_{\alpha}$ has at least $N$ discrete eigenvalues (taking multiplicities into
account), if the opening angle of $\Gamma$ is sufficiently small. Finally, it
was proved in [33] with a variational argument that $T_{\alpha}$ always has,
for $\alpha<0$, a non-empty discrete spectrum, when the interaction is
supported on a star graph which is not a straight line, see also [34] for
related considerations for Schrödinger operators with
$\delta^{\prime}$-interactions.

The main goal of this paper is to study the spectral properties and to
establish the existence of geometrically induced bound states for the two-
dimensional Dirac operator with a Lorentz scalar $\delta$-shell interaction
supported on the broken line $\Gamma$ that is formally given by

(1.1)
$\mathcal{A}_{\tau}=-i\sigma_{1}\partial_{1}-i\sigma_{2}\partial_{2}+m\sigma_{3}+\tau\sigma_{3}\delta_{\Gamma}.$

Here, $\tau\in\mathbb{R}\setminus\\{-2,0,2\\}$, $m>0$, and
$\sigma_{1},\sigma_{2}$, and $\sigma_{3}$ are the Pauli spin matrices defined
in (1.5) below. From a physical point of view, $\mathcal{A}_{\tau}$ describes
the propagation of a relativistic quantum particle with mass $m$ under the
influence of the Lorentz scalar potential $\tau\sigma_{3}\delta_{\Gamma}$. We
remark that $\tau=\pm 2$ is excluded, as in this case the operator
$\mathcal{A}_{\tau}$ decouples into two Dirac operators acting in the domains
with boundary $\Gamma$ and infinite mass boundary conditions on $\Gamma$, cf.
[9, 10]. In particular, in this confinement case the spectrum is computed in
[28, Proposition 1.12], by which the essential spectrum of any self-adjoint
realization of $\mathcal{A}_{2}$ is $(-\infty,-m]\cup[m,\infty)$ and there can
only be one eigenvalue having multiplicity one, while the spectrum of any
self-adjoint realization of $\mathcal{A}_{-2}$ is always the full real line;
note that $\tau=-2$ in the present manuscript corresponds to $m<0$ in [28],
which can be achieved via a unitary transformation.

Let us describe our results. The broken line decomposes $\mathbb{R}^{2}$ into
two regions. Let

(1.2) $\displaystyle\Omega_{+}\mathrel{\mathop{\mathchar
58\relax}}=\left\\{(r\cos{\theta},r\sin{\theta})\in\mathbb{R}^{2}\colon
r>0,~{}{\theta}\in(-\omega,\omega)\right\\}\subset\mathbb{R}^{2},$

be the wedge of opening angle $2\omega$, and
$\Omega_{-}\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{R}^{2}\setminus\overline{\Omega_{+}}$ be the open
complement of $\Omega_{+}$. Distinguish $\Gamma\mathrel{\mathop{\mathchar
58\relax}}=\partial\Omega_{+}$ as the joint boundary of $\Omega_{\pm}$, and
let $\nu=(\nu_{1},\nu_{2})$ be the unit normal vector field at $\Gamma$ which
is pointing outwards of $\Omega_{+}$. Note that setups given with angles
$\omega\in(\frac{\pi}{2},\pi)$ can have $\Omega_{+}$ and $\Omega_{-}$
interchanged and then rotated to yield a unitarily equivalent setup with
$\omega\in(0,\frac{\pi}{2}]$, as above, with the same spectrum. Hence, we will
always assume $\omega\in(0,\frac{\pi}{2}]$ in this paper.

In the following, functions $u\colon\mathbb{R}^{2}\rightarrow\mathbb{C}^{2}$
will often be restricted to the two regions of $\mathbb{R}^{2}$ and this is
denoted by $u_{\pm}\mathrel{\mathop{\mathchar 58\relax}}=u|_{\Omega_{\pm}}$.
In order to define the operator in (1.1) rigorously, we introduce, for
$\tau\in\mathbb{R}\setminus\\{-2,0,2\\}$ and $m>0$, the operator

(1.3)
$\begin{split}S_{\tau}u&=(-i\sigma_{1}\partial_{1}-i\sigma_{2}\partial_{2}+m\sigma_{3})u_{+}\oplus(-i\sigma_{1}\partial_{1}-i\sigma_{2}\partial_{2}+m\sigma_{3})u_{-},\\\
\operatorname{dom}S_{\tau}&=\left\\{u=u_{+}\oplus u_{-}\in
H^{1}(\Omega_{+};\mathbb{C}^{2})\oplus
H^{1}(\Omega_{-};\mathbb{C}^{2})\mathchar 58\relax
u_{-}|_{\Gamma}=Mu_{+}|_{\Gamma}\right\\},\end{split}$

where the matrix-valued function $M$ is given by

(1.4)
$M=\frac{4+\tau^{2}}{4-\tau^{2}}\sigma_{0}+\frac{4\tau}{4-\tau^{2}}i\sigma_{3}(\sigma_{1}\nu_{1}+\sigma_{2}\nu_{2})$

and $\sigma_{0}$ is the $2\times 2$ identity matrix.

If $\Omega_{+}$ is a smooth bounded domain, then the self-adjointness and some
spectral properties of $S_{\tau}$–including finiteness of the discrete
spectrum–were studied in [9], see also [4, 7, 10] for related results in
dimension three. In [26], the three-dimensional counterpart of $S_{\tau}$ with
$\Omega_{+}$ being a bounded $C^{4}$-domain was studied with the help of a
quadratic form associated with $S_{\tau}^{2}$ and for $\tau<0$ the asymptotic
expansion of the discrete eigenvalues was shown for $m\rightarrow\infty$, see
also [2, 29] for related considerations on Dirac operators on domains with
boundary conditions. Some of the ideas from [26] are used in the analysis in
the present paper as well. The case that $\Gamma\subset\mathbb{R}^{2}$ is the
straight line was considered in [11]. Eventually, we mention that in the
papers [1, 12, 29] results on the relation of the geometry of a domain
$\Omega\subset\mathbb{R}^{2}$ and the spectrum of the Dirac operator acting in
$\Omega$ with infinite mass boundary conditions are provided. In the case of
Dirac operators with zigzag boundary conditions there is a clear picture of
the geometrically induced spectrum [19, 36], as the spectrum of this operator
can be expressed via the spectrum of the Dirichlet Laplacian acting in the
same domain. However, to the best of our knowledge, there is no result in the
literature where the existence of geometrically induced bound states of Dirac
operators with singular potentials is shown, which is the main motivation in
the present paper.

Coming back to the case that $\Gamma$ is the broken line, it is known that the
operator $S_{\tau}$ is closed and symmetric, cf. [25]. Note that for
$\omega=\frac{\pi}{2}$ the curve $\Gamma$ is a straight line. In this case it
is known that $S_{\tau}$ is self-adjoint and that the spectrum of $S_{\tau}$
is purely absolutely continuous, cf. [11]. If $\omega\in(0,\frac{\pi}{2})$,
then the deficiency indices of $S_{\tau}$ are $\dim\ker(S_{\tau}^{*}\mp i)=1$.
Hence, $S_{\tau}$ admits a one-parametric family, labeled via $z$, of self-
adjoint extensions $A_{\tau,z}$, cf. [25] and Section 2 for details. The
spectral properties of these extensions are contained in the following,
central result of the article.

######  Theorem 1.1.

Let $\tau\in\mathbb{R}\setminus\\{-2,0,2\\}$, $\omega\in(0,\frac{\pi}{2})$,
$m>0$, and $A_{\tau,z}$ be any self-adjoint extension of $S_{\tau}$ in (1.3).
Then, the following is true:

  * (i)

If $\tau>0$, then
$\sigma_{\textup{ess}}(A_{\tau,z})=(-\infty,-m]\cup[m,\infty)$ and
$A_{\tau,z}$ has at most one discrete eigenvalue of multiplicity one.

  * (ii)

If $\tau<0$, then
$\sigma_{\textup{ess}}(A_{\tau,z})=\big{(}-\infty,-m\big{|}\frac{\tau^{2}-4}{\tau^{2}+4}\big{|}\big{]}\cup\big{[}m\big{|}\frac{\tau^{2}-4}{\tau^{2}+4}\big{|},\infty\big{)}$
and 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 $A_{\tau,z}$ has at least $N$
discrete eigenvalues with multiplicities taken into account.

The result relies on an analysis of the quadratic form of
$S_{\tau}^{*}S_{\tau}$, which is a restriction of the quadratic form of
$(A_{\tau,z})^{2}$, see Section 3. Repercussions for the essential spectrum of
$A_{\tau,z}$ are then obtained using perturbation theory and an auxiliary
result on the symmetry of the essential spectrum with respect to the origin in
Section 4. The statements on the essential spectrum in the theorem are then
obtained in Section 5 using Neumann bracketing and the method of singular
sequences.

The claim on the discrete spectrum for $\tau>0$ in Theorem 1.1 (i) relies on
the abstract perturbation theory of self-adjoint operators. The result on the
discrete spectrum for $\tau<0$ in Theorem 1.1 (ii) is shown in Section 6 with
the min-max principle applied to the quadratic form of $S_{\tau}^{*}S_{\tau}$.
An appropriate family of test functions is constructed via a modification of
the approach used in [24] to deal with the Schrödinger operator with a
$\delta$-interaction supported on a broken line. Moreover, we provide a
formula for a suitable value of the critical angle $\omega_{\star}$. It
remains an open problem whether the discrete spectrum of $A_{\tau,z}$ is
always non-empty provided that $\omega\neq\frac{\pi}{2}$ and
$\tau\in(-\infty,0)\setminus\\{-2\\}$.

### Notations

For a Hilbert space $\mathcal{H}$ we denote the associated inner product
(which is linear in the first entry) by
$\langle\cdot,\cdot\rangle_{\mathcal{H}}$ and the induced norm by
$\|\cdot\|_{\mathcal{H}}$. If $A$ is a closed linear operator in a Hilbert
space $\mathcal{H}$, then we denote its domain, range, kernel, spectrum, and
resolvent set by $\operatorname{dom}A$, $\operatorname{ran}A$, $\ker A$,
$\sigma(A)$, and $\rho(A)$, respectively. If $A$ is self-adjoint, then its
discrete and essential spectrum are $\sigma_{\textup{disc}}(A)$ and
$\sigma_{\textup{ess}}(A)$, respectively.

Let

(1.5) $\sigma_{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\sigma_{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\quad\sigma_{3}=\begin{pmatrix}1&0\\\ 0&-1\\\ \end{pmatrix},$

be the Pauli spin matrices and denote the $2\times 2$ identity matrix by
$\sigma_{0}$. They satisfy the anti-commutation relations

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

The notation

$\sigma\cdot\gradient\mathrel{\mathop{\mathchar
58\relax}}=\sigma_{1}\partial_{1}+\sigma_{2}\partial_{2}\quad\text{and}\quad\sigma\cdot
x=\sigma_{1}x_{1}+\sigma_{2}x_{2},$

will also be used for $x=(x_{1},x_{2})\in\mathbb{C}^{2}$.

If $\Omega\subset\mathbb{R}^{2}$ is a sufficiently smooth domain, then
$L^{2}(\Omega;\mathbb{C}^{n})$ and $L^{2}(\partial\Omega;\mathbb{C}^{n})$,
$n\in\mathbb{N}$, denote the sets of all square-integrable
$\mathbb{C}^{n}$-valued functions on $\Omega$ and $\partial\Omega$. To
simplify notation, we write $\|\cdot\|_{\Omega}\mathrel{\mathop{\mathchar
58\relax}}=\|\cdot\|_{L^{2}(\Omega;\mathbb{C}^{n})}$ and
$\|\cdot\|_{\partial\Omega}\mathrel{\mathop{\mathchar
58\relax}}=\|\cdot\|_{L^{2}(\partial\Omega;\mathbb{C}^{n})}$. Moreover, we use
for $s>0$ the symbols $H^{s}(\Omega;\mathbb{C}^{n})$ and
$H^{s}(\partial\Omega;\mathbb{C}^{n})$ for the $L^{2}$-based Sobolev spaces on
$\Omega$ and $\partial\Omega$, respectively. For $s\in[0,1]$, we will need the
space $H^{-s}(\partial\Omega;\mathbb{C}^{n})$, which is defined as the dual
space of $H^{s}(\partial\Omega;\mathbb{C}^{n})$, and we denote the associated
sesquilinear duality product by
$\langle\cdot,\cdot\rangle_{H^{s}(\partial\Omega;\mathbb{C}^{n})\times
H^{-s}(\partial\Omega;\mathbb{C}^{n})}$. For a function $u\in
H^{1}(\Omega;\mathbb{C}^{n})$ we denote its Dirichlet trace by
$u|_{\partial\Omega}\in H^{1/2}(\partial\Omega;\mathbb{C}^{n})$.

Recall that the wedge $\Omega_{+}$ is defined in (1.2), that
$\Omega_{-}=\mathbb{R}^{2}\setminus\overline{\Omega_{+}}$, and $\nu$ is the
outer unit normal vector field at $\Gamma\mathrel{\mathop{\mathchar
58\relax}}=\partial\Omega_{+}$ that is pointing outwards from $\Omega_{+}$,
cf. Figure 1.

Figure 1. Geometric setting. The domain $\Omega_{+}$ is depicted in gray.

We are also going to make use of the notations

(1.7) $\begin{split}\Gamma_{l}\mathrel{\mathop{\mathchar
58\relax}}=\big{\\{}(r\cos\omega,r\sin\omega)\mathchar 58\relax
r>0\big{\\}}\quad\text{and}\quad\Gamma_{r}\mathrel{\mathop{\mathchar
58\relax}}=\big{\\{}(r\cos\omega,-r\sin\omega)\mathchar 58\relax
r>0\big{\\}},\end{split}$

so that $\Gamma=\overline{\Gamma_{l}\cup\Gamma_{r}}$. A function (spinor)
$u\in L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})$ can be decomposed into two parts
corresponding to the domains $\Omega_{\pm}$,

$\displaystyle u=u_{+}\oplus u_{-}\in L^{2}(\Omega_{+};\mathbb{C}^{2})\oplus
L^{2}(\Omega_{-};\mathbb{C}^{2}),~{}~{}\text{ where
}~{}~{}u_{\pm}\mathrel{\mathop{\mathchar 58\relax}}=u|_{\Omega_{\pm}}.$

##  2\. Singular interaction supported on a broken line

Self-adjoint extensions of the operator $S_{\tau}$ in equation (1.3) were
studied in [25, 35] along with the natural analog of $S_{\tau}$ with the
Lorentz scalar $\delta$-shell interaction supported on the star graph [25] and
the boundary of a curvilinear polygon [35]. These self-adjoint extensions are
the starting point for the spectral analysis in Sections 4-6.

Recall that $u\in H^{1}(\mathbb{R}^{2}\setminus\Gamma;\mathbb{C}^{2})$ belongs
to $\operatorname{dom}S_{\tau}$ if and only if
$u_{-}|_{\Gamma}=Mu_{+}|_{\Gamma}$, where the matrix $M$ is defined by (1.4).
We remark that for $\tau\neq\pm 2$ the matrix $M$ is invertible and the
transmission condition is equivalent to

(2.1) $u_{+}|_{\Gamma}=M^{-1}u_{-}|_{\Gamma}\quad\text{with}\quad
M^{-1}=\frac{4+\tau^{2}}{4-\tau^{2}}\sigma_{0}-\frac{4\tau}{4-\tau^{2}}i\sigma_{3}(\sigma\cdot\nu)$

and also to

(2.2)
$\displaystyle-i(\sigma\cdot\nu)\left(u_{+}|_{\Gamma}-u_{-}|_{\Gamma}\right)=\frac{\tau\sigma_{3}}{2}\left(u_{+}|_{\Gamma}+u_{-}|_{\Gamma}\right).$

The deficiency indices of $S_{\tau}$ were identified in [25, Corollary 2.16
(i)] by switching to polar coordinates and analyzing the orthogonal
decomposition of $S_{\tau}$. This orthogonal decomposition is only possible
for $m=0$, which we will assume in the following. However, since $m\sigma_{3}$
is a bounded perturbation, this does not affect the results about self-
adjointness. Moreover, since the straight line was already fully studied in
[11], we will assume that $\omega\in(0,\frac{\pi}{2})$ in the following.

The angular component of the aforementioned orthogonal decomposition of
$S_{\tau}$ is commonly referred to as the spin-orbit operator and is defined
by the linear map

(2.3) $\displaystyle J_{\tau}{\varphi}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\left(-i\sigma_{3}{\varphi}_{+}^{\prime}+\frac{{\varphi}_{+}}{2}\right)\oplus\left(-i\sigma_{3}{\varphi}_{-}^{\prime}+\frac{{\varphi}_{-}}{2}\right),$
$\displaystyle\operatorname{dom}J_{\tau}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\Big{\\{}{\varphi}={\varphi}_{+}\oplus{\varphi}_{-}\in
H^{1}((-\omega,\omega);\mathbb{C}^{2})\oplus
H^{1}((\omega,2\pi-\omega);\mathbb{C}^{2})\colon$ $\displaystyle\hskip
120.00018pt{\varphi}_{-}(\omega)=M_{l}{\varphi}_{+}(\omega),~{}{\varphi}_{-}(2\pi-\omega)=M_{r}{\varphi}_{+}(-\omega)\Big{\\}},$

where the matrices $M_{l}$ and $M_{r}$ are the restrictions of $M$ in (1.4)
onto $\Gamma_{l}$ and $\Gamma_{r}$, respectively, and given by

(2.4)
$M_{l}=\frac{1}{1-\frac{1}{4}\tau^{2}}\begin{pmatrix}1+\frac{1}{4}\tau^{2}&e^{-i\omega}\tau\\\
e^{i\omega}\tau&1+\frac{1}{4}\tau^{2}\end{pmatrix}\quad\text{and}\quad
M_{r}=\frac{1}{1-\frac{1}{4}\tau^{2}}\begin{pmatrix}1+\frac{1}{4}\tau^{2}&-e^{i\omega}\tau\\\
-e^{-i\omega}\tau&1+\frac{1}{4}\tau^{2}\end{pmatrix}.$

It was established in [25, Proposition 2.3] that the operator $J_{\tau}$ is
self-adjoint in the Hilbert space
$L^{2}((-\omega,\omega);\mathbb{C}^{2})\oplus
L^{2}((\omega,2\pi-\omega);\mathbb{C}^{2})$ with purely discrete spectrum.
Spectral properties of $J_{\tau}$ were determined in [25, Propositions 2.8,
2.11, 2.12] and will play a role later. In the following, we write
$n_{\lambda}\mathrel{\mathop{\mathchar 58\relax}}=\dim\ker(J_{\tau}-\lambda)$,
$\lambda\in\mathbb{R}$.

######  Lemma 2.1.

Let $\omega\in(0,\frac{\pi}{2})$ and $\tau\in\mathbb{R}\setminus\\{-2,0,2\\}$.
Then, $\sigma(J_{\tau})$ is symmetric about the origin,
$n_{\lambda}=n_{-{\lambda}}\leq 2$ for any ${\lambda}\in\sigma(J_{\tau})$,
$0,\frac{1}{2}\notin\sigma(J_{\tau})$, and $J_{\tau}$ has one simple
eigenvalue within the interval $(0,\frac{1}{2})$.

One can then show that $S_{\tau}$ (for $m=0$) is given by the orthogonal sum
of half-line Dirac operators with off-diagonal Coulomb potentials. These half-
line Dirac operators are closed, symmetric and densely defined operators in
the Hilbert space $L^{2}(\mathbb{R}_{+};\mathbb{C}^{2})$. They are denoted,
for ${\lambda}>0$, ${\lambda}\neq\frac{1}{2}$, by

(2.5) $\boldsymbol{d}_{\lambda}\psi\mathrel{\mathop{\mathchar
58\relax}}=\begin{pmatrix}0&-\frac{d}{dr}-\frac{{\lambda}}{r}\\\
\frac{d}{dr}-\frac{{\lambda}}{r}&0\end{pmatrix}\psi,\qquad\operatorname{dom}\boldsymbol{d}_{\lambda}\mathrel{\mathop{\mathchar
58\relax}}=H_{0}^{1}(\mathbb{R}_{+};\mathbb{C}^{2}).\\\ $

For $m=0$ the operator $S_{\tau}$ can then be decomposed by [25, Theorem 2.14]
as

$S_{\tau}\simeq\bigoplus_{{\lambda}\in\sigma(J_{\tau})\cap(0,\infty)}\bigoplus_{j=1}^{n_{\lambda}}\,\boldsymbol{d}_{\lambda}.$

In particular, each $\boldsymbol{d}_{\lambda}$ is self-adjoint when
${\lambda}>1/2$. When ${\lambda}\in(0,1/2)$, the deficiency indices of
$\boldsymbol{d}_{\lambda}$ are $(1,1)$. In particular, this shows that
$S_{\tau}$ has also deficiency indices $(1,1)$. The corresponding deficiency
elements are expressed in [25, Lemma 4.1] in terms of modified Bessel
functions of the second kind and order $\nu\in\mathbb{R}$, denoted
$K_{\nu}(\cdot)$. These modified Bessel functions play a key role in
determining self-adjoint extensions of $S_{\tau}$ in [25, Theorem 4.2], a
simplified variant of which is formulated below. For its formulation, let
$\varphi_{\lambda}$ be the eigenfunction of $J_{\tau}$ corresponding to the
unique simple eigenvalue ${\lambda}$ lying in the interval $(0,1/2)$; here we
extend $\varphi_{\lambda}$ to the interval $[0,2\pi)$ by periodicity, i.e.
$\varphi_{\lambda}({\theta}+2\pi)=\varphi_{\lambda}({\theta})$. We then define
the functions

(2.6)
$v_{\pm}(r\cos{\theta},r\sin{\theta})=K_{{\lambda}-1/2}(r){\varphi}_{\lambda}({\theta})\pm
K_{{\lambda}+1/2}(r)(\sigma_{1}\cos{\theta}+\sigma_{2}\sin{\theta}){\varphi}_{\lambda}({\theta}),$

for $r>0$, ${\theta}\in[0,2\pi)$.

######  Theorem 2.2.

Let $\omega\in(0,\frac{\pi}{2})$, $\tau\in\mathbb{R}\setminus\\{-2,0,2\\}$,
and $v_{\pm}$ be given by (2.6). All self-adjoint extensions of $S_{\tau}$ are
in one-to-one correspondence with complex numbers $z$ of modulus one.
Moreover, the self-adjoint extension $A_{\tau,z}$ of $S_{\tau}$ corresponding
to $z$ is characterized by

$\displaystyle\operatorname{dom}A_{\tau,z}$
$\displaystyle=\bigg{\\{}u=u_{0}+cv_{+}+czv_{-}\in
L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})\colon
u_{0}\in\operatorname{dom}S_{\tau},~{}c\in\mathbb{C}\bigg{\\}},$
$\displaystyle A_{\tau,z}u$
$\displaystyle=A_{\tau,z}(u_{0}+cv_{+}+czv_{-})=S_{\tau}u_{0}+(i\sigma_{0}+m\sigma_{3})cv_{+}+(-i\sigma_{0}+m\sigma_{3})czv_{-}.$

######  Remark 2.3.

The action of any self-adjoint extension $A_{\tau,z}$ of $S_{\tau}$ can be
easily determined via the definition of distributional derivatives as

$A_{\tau,z}u=(-i\sigma\cdot\gradient+m\sigma_{3})u_{+}\oplus(-i\sigma\cdot\gradient+m\sigma_{3})u_{-},\quad
u=u_{+}\oplus u_{-}\in\operatorname{dom}A_{\tau,z},$

where $\sigma\cdot\gradient u_{\pm}$ has to be understood in the
distributional sense.

The decomposition of general functions in the domain of $A_{\tau,z}$ given in
Theorem 2.2 provides a tool to classify the regularity of functions in the
domain. It is even possible to single out a unique distinguished self-adjoint
extension whose operator domain is the most regular in the scale of Sobolev
spaces. A similar statement is made for Dirac operators with infinite mass
boundary conditions in sectors in [28, Proposition 1.8]. For a proof of the
following result, we refer to [25, Theorem 4.4]. We remark that this
distinguished self-adjoint extension will also play a role in our spectral
analysis in Section 4.

######  Theorem 2.4.

Let $\omega\in(0,\frac{\pi}{2})$ and $\tau\in\mathbb{R}\setminus\\{-2,0,2\\}$.
The so called ‘distinguished’ self-adjoint extension of $S_{\tau}$, the only
extension such that

$\displaystyle\operatorname{dom}A_{\tau,z}\subset
H^{1/2}(\Omega_{+};\mathbb{C}^{2})\oplus H^{1/2}(\Omega_{-};\mathbb{C}^{2}),$

is the extension corresponding to $z=1$. In fact, the slightly stronger
statement

$\displaystyle\operatorname{dom}A_{\tau,1}\subset
H^{s}(\Omega_{+};\mathbb{C}^{2})\oplus H^{s}(\Omega_{-};\mathbb{C}^{2}),$

holds for all $s<1/2+\min\\{\sigma(J_{\tau})\cap(0,\infty)\\}$.

##  3\. Quadratic Form

Recall that for $\tau\in\mathbb{R}\setminus\\{-2,0,2\\}$ the operator
$S_{\tau}$ is defined by equation (1.3). The main goal in this section is to
find a convenient representation of $\|S_{\tau}u\|_{\mathbb{R}^{2}}^{2}$ for
$u\in\operatorname{dom}S_{\tau}$, see Theorem 3.2, which is the quadratic form
associated with the self-adjoint operator $S^{*}_{\tau}S_{\tau}$. This
operator will play a central role in the spectral analysis of the self-adjoint
extensions $A_{\tau,z}$ of $S_{\tau}$, as, e.g., the difference of the
resolvents of $S^{*}_{\tau}S_{\tau}$ and $(A_{\tau,z})^{2}$ will be compared
in Section 5 to determine the essential spectrum of $A_{\tau,z}$. Moreover,
this quadratic form will play a crucial role in the proof of the statements
about the discrete spectrum of $A_{\tau,z}$ in Theorem 1.1.

In the following we are going to make use of the tangential derivative
$\partial_{t}$ on the boundary of a Lipschitz hypograph
$\Omega\subset\mathbb{R}^{2}$. For its definition, let
$\gamma\colon\mathbb{R}\rightarrow\mathbb{R}^{2}$ be an arc-length
parametrization of $\partial\Omega$ such that the unit normal vector field
$\nu$, which is pointing outwards of $\Omega$, is given by
$\nu=(-\dot{\gamma}_{2},\dot{\gamma}_{1})$ almost everywhere. Then, for
$\varphi\in H^{1}(\partial\Omega)$ we define

(3.1) $\partial_{t}\varphi(x)\mathrel{\mathop{\mathchar
58\relax}}=(\varphi\circ\gamma)^{\prime}(x),\qquad x\in\mathbb{R}.$

Clearly, for any $s\in[0,1]$ the tangential derivative $\partial_{t}$ gives
rise to a bounded operator

$\partial_{t}\mathchar 58\relax H^{s}(\partial\Omega)\rightarrow
H^{s-1}(\partial\Omega).$

We will use that the tangential derivative of the trace of a function $u\in
H^{2}(\Omega)$ is given by

$\partial_{t}(u|_{\partial\Omega})=\dot{\gamma}\cdot(\gradient
u)|_{\partial\Omega}=\nu_{2}(\partial_{1}u)|_{\partial\Gamma}-\nu_{1}(\partial_{2}u)|_{\partial\Gamma}.$

Now, we are prepared to state some of the finer details for the proof of
Theorem 3.2.

######  Lemma 3.1.

Let $\Omega\subset\mathbb{R}^{2}$ be a Lipschitz hypograph and let
$\nu=(\nu_{1},\nu_{2})$ be the unit normal vector field at $\partial\Omega$
which points outwards of $\Omega$. Then for all $f,g\in
H^{1}(\Omega;\mathbb{C}^{2})$ the following holds:

(3.2)
$\displaystyle\big{\langle}(\sigma\cdot\gradient)f,(\sigma\cdot\gradient)g\big{\rangle}_{\Omega}=\big{\langle}\gradient
f,\gradient
g\big{\rangle}_{\Omega}-\big{\langle}f|_{\partial\Omega},i\sigma_{3}\partial_{t}(g|_{\partial\Omega})\big{\rangle}_{H^{1/2}(\partial\Omega;\mathbb{C}^{2})\times
H^{-1/2}(\partial\Omega;\mathbb{C}^{2})}.$

###### Proof.

Let $f\in H^{1}(\Omega;\mathbb{C}^{2})$ and $g\in
H^{2}(\Omega;\mathbb{C}^{2})$. Then, the identity
$(\sigma\cdot\gradient)^{2}=\Delta$ and an application of Gauss’ theorem yield

$\displaystyle\big{\langle}(\sigma\cdot\gradient)f,(\sigma\cdot\gradient)g\big{\rangle}_{\Omega}$
$\displaystyle=\big{\langle}(\sigma\cdot\nu)f|_{\partial\Omega},((\sigma\cdot\gradient)g)|_{\partial\Omega}\big{\rangle}_{\partial\Omega}-\big{\langle}f,(\sigma\cdot\gradient)^{2}g\big{\rangle}_{\Omega}$
$\displaystyle=\big{\langle}f|_{\partial\Omega},(\sigma\cdot\nu)((\sigma\cdot\gradient)g)|_{\partial\Omega}-\nu\cdot\gradient
g|_{\partial\Omega}\big{\rangle}_{\partial\Omega}+\big{\langle}\gradient
f,\gradient g\big{\rangle}_{\Omega}$
$\displaystyle=\big{\langle}f|_{\partial\Omega},\sigma_{1}\nu_{1}\sigma_{2}(\partial_{2}g)|_{\partial\Omega}+\sigma_{2}\nu_{2}\sigma_{1}(\partial_{1}g)|_{\partial\Omega}\big{\rangle}_{\partial\Omega}+\big{\langle}\gradient
f,\gradient g\big{\rangle}_{\Omega}$
$\displaystyle=\big{\langle}f|_{\partial\Omega},\sigma_{1}\sigma_{2}\left(\nu_{1}(\partial_{2}g)|_{\partial\Omega}-\nu_{2}(\partial_{1}g)|_{\partial\Omega}\right)\big{\rangle}_{\partial\Omega}+\big{\langle}\gradient
f,\gradient g\big{\rangle}_{\Omega}$
$\displaystyle=-\big{\langle}f|_{\partial\Omega},\sigma_{1}\sigma_{2}\partial_{t}(g|_{\partial\Omega})\big{\rangle}_{\partial\Omega}+\big{\langle}\gradient
f,\gradient g\big{\rangle}_{\Omega}.$

This expression agrees with equation (3.2), as
$\sigma_{1}\sigma_{2}=i\sigma_{3}$. Since $H^{2}(\Omega;\mathbb{C}^{2})$ is
dense in $H^{1}(\Omega;\mathbb{C}^{2})$, a continuity argument yields the
claim for $g\in H^{1}(\Omega;\mathbb{C}^{2})$. ∎

It is now possible to show a useful representation for the quadratic form
associated with the self-adjoint operator $S_{\tau}^{*}S_{\tau}$, i.e. of the
expression

$\displaystyle\|S_{\tau}u\|^{2}_{\mathbb{R}^{2}}$
$\displaystyle=\|(-i\sigma\cdot\gradient)u+m\sigma_{3}u\|^{2}_{\mathbb{R}^{2}\setminus\Gamma},\qquad
u\in\operatorname{dom}S_{\tau}.$

The following result can be seen as a two-dimensional variant of [26,
Proposition 3.1] with the interaction support being the broken line; cf. also
[2, Theorem 1.5] and [29, Proposition 14] for similar formulas.

######  Theorem 3.2.

Let $\tau\in\mathbb{R}\setminus\\{-2,0,2\\}$. Then, for any
$u\in\operatorname{dom}S_{\tau}$, the following holds:

(3.3) $\displaystyle\|S_{\tau}u\|^{2}_{\mathbb{R}^{2}}=\|\gradient
u\|^{2}_{\mathbb{R}^{2}\setminus\Gamma}+m^{2}\|u\|^{2}_{\mathbb{R}^{2}}+\frac{2m}{\tau}\|u_{+}|_{\Gamma}-u_{-}|_{\Gamma}\|^{2}_{\Gamma}.$

###### Proof.

Let $u\in\operatorname{dom}S_{\tau}$ be fixed. First, as $\nu$ is pointing
outside of $\Omega_{+}$ and inside of $\Omega_{-}$, Lemma 3.1 yields

(3.4) $\displaystyle\|S_{\tau}u\|^{2}_{\mathbb{R}^{2}}$
$\displaystyle=\|(-i\sigma\cdot\gradient)u+m\sigma_{3}u\|^{2}_{\mathbb{R}^{2}\setminus\Gamma}$
$\displaystyle=\|(\sigma\cdot\gradient)u\|^{2}_{\mathbb{R}^{2}\setminus\Gamma}+m^{2}\|u\|^{2}_{\mathbb{R}^{2}}+2\real\left(\langle-i(\sigma\cdot\gradient)u,m\sigma_{3}u\rangle_{\mathbb{R}^{2}\setminus\Gamma}\right)$
$\displaystyle=\|\gradient
u\|^{2}_{\mathbb{R}^{2}\setminus\Gamma}+m^{2}\|u\|^{2}_{\mathbb{R}^{2}}+2\real\left(\langle-i(\sigma\cdot\gradient)u,m\sigma_{3}u\rangle_{\mathbb{R}^{2}\setminus\Gamma}\right)$
$\displaystyle\qquad+\langle
u_{-}|_{\Gamma},i\sigma_{3}\partial_{t}(u_{-}|_{\Gamma})\rangle_{H^{1/2}(\Gamma;\mathbb{C}^{2})\times
H^{-1/2}(\Gamma;\mathbb{C}^{2})}$ $\displaystyle\qquad-\langle
u_{+}|_{\Gamma},i\sigma_{3}\partial_{t}(u_{+}|_{\Gamma})\rangle_{H^{1/2}(\Gamma;\mathbb{C}^{2})\times
H^{-1/2}(\Gamma;\mathbb{C}^{2})}.$

Recall that $M$ is defined by (1.4) and note that
$\sigma_{3}(\sigma\cdot\nu)=-(\sigma\cdot\nu)\sigma_{3}$, which holds by
(1.6), implies

$\sigma_{3}M=\left(\frac{4+\tau^{2}}{4-\tau^{2}}\sigma_{0}-\frac{4\tau}{4-\tau^{2}}i\sigma_{3}(\sigma\cdot\nu)\right)\sigma_{3}=M^{-1}\sigma_{3}.$

Since $\nu$ is piecewise constant, this yields

$\displaystyle\big{\langle}Mu_{+}|_{\Gamma},$ $\displaystyle
i\sigma_{3}(\partial_{t}(Mu_{+}|_{\Gamma})\big{\rangle}_{H^{1/2}(\Gamma;\mathbb{C}^{2})\times
H^{-1/2}(\Gamma;\mathbb{C}^{2})}$
$\displaystyle=\big{\langle}M^{-1}Mu_{+}|_{\Gamma},i\sigma_{3}\partial_{t}(u_{+}|_{\Gamma})\big{\rangle}_{H^{1/2}(\Gamma;\mathbb{C}^{2})\times
H^{-1/2}(\Gamma;\mathbb{C}^{2})},$

cf. (A.1) for details. This yields

(3.5) $\displaystyle\big{\langle}u_{-}|_{\Gamma},$ $\displaystyle
i\sigma_{3}\partial_{t}(u_{-}|_{\Gamma})\big{\rangle}_{H^{1/2}(\Gamma;\mathbb{C}^{2})\times
H^{-1/2}(\Gamma;\mathbb{C}^{2})}-\big{\langle}u_{+}|_{\Gamma},i\sigma_{3}\partial_{t}(u_{+}|_{\Gamma})\big{\rangle}_{H^{1/2}(\Gamma;\mathbb{C}^{2})\times
H^{-1/2}(\Gamma;\mathbb{C}^{2})}$
$\displaystyle=\big{\langle}Mu_{+}|_{\Gamma},i\sigma_{3}\partial_{t}(Mu_{+}|_{\Gamma})\big{\rangle}_{H^{1/2}(\Gamma;\mathbb{C}^{2})\times
H^{-1/2}(\Gamma;\mathbb{C}^{2})}$ $\displaystyle\hskip
60.00009pt-\big{\langle}u_{+}|_{\Gamma},i\sigma_{3}\partial_{t}(u_{+}|_{\Gamma})\big{\rangle}_{H^{1/2}(\Gamma;\mathbb{C}^{2})\times
H^{-1/2}(\Gamma;\mathbb{C}^{2})}$
$\displaystyle=\big{\langle}M^{-1}Mu_{+}|_{\Gamma},i\sigma_{3}\partial_{t}(u_{+}|_{\Gamma})\big{\rangle}_{H^{1/2}(\Gamma;\mathbb{C}^{2})\times
H^{-1/2}(\Gamma;\mathbb{C}^{2})}$ $\displaystyle\hskip
60.00009pt-\big{\langle}u_{+}|_{\Gamma},i\sigma_{3}\partial_{t}(u_{+}|_{\Gamma})\big{\rangle}_{H^{1/2}(\Gamma;\mathbb{C}^{2})\times
H^{-1/2}(\Gamma;\mathbb{C}^{2})}=0.$

This simplifies equation (3.4) and it remains only to recover the last term in
equation (3.3).

The identity

$\displaystyle\big{\langle}-i(\sigma\cdot\gradient)u_{\pm},m\sigma_{3}u_{\pm}\big{\rangle}_{{}_{\scriptstyle\Omega_{\pm}}}+\big{\langle}m\sigma_{3}u_{\pm},-i(\sigma\cdot\gradient)u_{\pm}\big{\rangle}_{{}_{\scriptstyle\Omega_{\pm}}}=\pm\big{\langle}-i(\sigma\cdot\nu)u_{\pm}|_{\Gamma},m\sigma_{3}u_{\pm}|_{\Gamma}\big{\rangle}_{\Gamma},$

and the fact that $M$ commutes with $-i\sigma_{3}(\sigma\cdot\nu)$ allow for
the following simplification:

$\displaystyle 2\real$
$\displaystyle\left(\langle-i(\sigma\cdot\gradient)u,m\sigma_{3}u\rangle_{\mathbb{R}^{2}\setminus\Gamma}\right)$
$\displaystyle=\big{\langle}-i(\sigma\cdot\gradient)u,m\sigma_{3}u\big{\rangle}_{\mathbb{R}^{2}\setminus\Gamma}+\big{\langle}m\sigma_{3}u,-i(\sigma\cdot\gradient)u\big{\rangle}_{\mathbb{R}^{2}\setminus\Gamma}$
$\displaystyle=m\big{[}\big{\langle}-i\sigma_{3}(\sigma\cdot\nu)u_{+}|_{\Gamma},u_{+}|_{\Gamma}\big{\rangle}_{\Gamma}-\big{\langle}-i\sigma_{3}(\sigma\cdot\nu)u_{-}|_{\Gamma},u_{-}|_{\Gamma}\big{\rangle}_{\Gamma}$
$\displaystyle\hskip
80.00012pt-\big{\langle}u_{+}|_{\Gamma},-i\sigma_{3}(\sigma\cdot\nu)Mu_{+}|_{\Gamma}\big{\rangle}_{\Gamma}+\big{\langle}u_{+}|_{\Gamma},-i\sigma_{3}(\sigma\cdot\nu)Mu_{+}|_{\Gamma}\big{\rangle}_{\Gamma}\Big{]}$
$\displaystyle=m\big{[}\big{\langle}-i\sigma_{3}(\sigma\cdot\nu)u_{+}|_{\Gamma},u_{+}|_{\Gamma}\big{\rangle}_{\Gamma}-\big{\langle}-i\sigma_{3}(\sigma\cdot\nu)u_{-}|_{\Gamma},u_{-}|_{\Gamma}\big{\rangle}_{\Gamma}$
$\displaystyle\hskip
80.00012pt-\big{\langle}u_{-}|_{\Gamma},-i\sigma_{3}(\sigma\cdot\nu)u_{+}|_{\Gamma}\big{\rangle}_{\Gamma}+\big{\langle}u_{+}|_{\Gamma},-i\sigma_{3}(\sigma\cdot\nu)u_{-}|_{\Gamma}\big{\rangle}_{\Gamma}\Big{]}$
$\displaystyle=m\big{[}\big{\langle}-i\sigma_{3}(\sigma\cdot\nu)(u_{+}|_{\Gamma}-u_{-}|_{\Gamma}),(u_{+}|_{\Gamma}+u_{-}|_{\Gamma})\big{\rangle}_{\Gamma}\big{]}$
$\displaystyle=m\left\langle-i\sigma_{3}(\sigma\cdot\nu)(u_{+}|_{\Gamma}-u_{-}|_{\Gamma}),-\frac{2}{\tau}i\sigma_{3}(\sigma\cdot\nu)(u_{+}|_{\Gamma}-u_{-}|_{\Gamma})\right\rangle_{\Gamma}=\frac{2m}{\tau}\big{\|}u_{+}|_{\Gamma}-u_{-}|_{\Gamma}\big{\|}^{2}_{\Gamma},$

where the second to last equality is due to the transmission condition in
equation (2.2). The claim follows. ∎

##  4\. Symmetry of the spectrum

Recall that the distinguished self-adjoint extension of $S_{\tau}$ is denoted
by $A_{\tau,1}$, cf. Theorems 2.2 and 2.4. In this section we show that the
spectrum of $A_{\tau,1}$ is symmetric with respect to the origin. Corollary
4.2 then shows that the essential spectrum of any self-adjoint extension of
$S_{\tau}$ is symmetric, which will be helpful in determining further spectral
properties in Sections 5 and 6.

######  Theorem 4.1.

Let $\omega\in(0,\frac{\pi}{2})$ and $\tau\in\mathbb{R}\setminus\\{-2,0,2\\}$.
Then the spectrum of $A_{\tau,1}$ is symmetric about the origin, i.e.
${\lambda}\in\sigma(A_{\tau,1})$ if and only if
$-{\lambda}\in\sigma(A_{\tau,1})$. In particular, this symmetry also holds for
the discrete and the essential spectrum of $A_{\tau,1}$.

###### Proof.

First, recall that the spin-orbit operator $J_{\tau}$ is given in equation
(2.3). Let ${\lambda}$ be the unique simple eigenvalue of $J_{\tau}$ in the
interval $(0,\frac{1}{2})$ from Lemma 2.1 and ${\varphi}_{\lambda}$ the
corresponding eigenvector. Define

$\displaystyle v(x(r,{\theta}),y(r,{\theta}))\mathrel{\mathop{\mathchar
58\relax}}=K_{{\lambda}-1/2}(r)\cdot{\varphi}_{\lambda}({\theta}),$

where $x(r,{\theta})=r\cos{\theta}$ and $y(r,{\theta})=r\sin{\theta}$. Then,
by Theorem 2.4 the distinguished self-adjoint extension of $S_{\tau}$ can be
written as

$\displaystyle A_{\tau,1}u$
$\displaystyle=(-i(\sigma\cdot\gradient)+m\sigma_{3})u_{+}\oplus(-i(\sigma\cdot\gradient)+m\sigma_{3})u_{-},$
$\displaystyle\operatorname{dom}A_{\tau,1}$
$\displaystyle=\operatorname{dom}S_{\tau}+\operatorname{span}\\{v\\}.$

Define the charge conjugation operator by $Cf\mathrel{\mathop{\mathchar
58\relax}}=\sigma_{1}\overline{f}$. We show that $C$ maps
$\operatorname{dom}A_{\tau,1}$ into itself. Note that
$\overline{\sigma_{1}}=\sigma_{1}$, $\overline{\sigma_{2}}=-\sigma_{2}$, and
$\overline{\sigma_{3}}=\sigma_{3}$. First, for
$u\in\operatorname{dom}S_{\tau}$, the transmission condition in equation (1.4)
and the anti-commutation relations in (1.6) yield

$\displaystyle 0$
$\displaystyle=C\left[\left(\frac{4+\tau^{2}}{4-\tau^{2}}\sigma_{0}+\frac{4\tau}{4-\tau^{2}}i\sigma_{3}(\sigma\cdot\nu)\right)u_{+}|_{\Gamma}-u_{-}|_{\Gamma}\right]$
$\displaystyle=\sigma_{1}\left[\left(\frac{4+\tau^{2}}{4-\tau^{2}}\sigma_{0}-\frac{4\tau}{4-\tau^{2}}i\sigma_{3}(\sigma_{1}\nu_{1}-\sigma_{2}\nu_{2})\right)\overline{u_{+}|_{\Gamma}}-\overline{u_{-}|_{\Gamma}}\right]$
$\displaystyle=\left[\left(\frac{4+\tau^{2}}{4-\tau^{2}}\sigma_{0}+\frac{4\tau}{4-\tau^{2}}i\sigma_{3}(\sigma_{1}\nu_{1}+\sigma_{2}\nu_{2})\right)\sigma_{1}\overline{u_{+}|_{\Gamma}}-\sigma_{1}\overline{u_{-}|_{\Gamma}}\right]$
$\displaystyle=\left[\left(\frac{4+\tau^{2}}{4-\tau^{2}}\sigma_{0}+\frac{4\tau}{4-\tau^{2}}i\sigma_{3}(\sigma\cdot\nu)\right)Cu_{+}|_{\Gamma}-Cu_{-}|_{\Gamma}\right],$

and so $Cu\in\operatorname{dom}S_{\tau}$. Next, as $K_{{\lambda}-1/2}$ is
real-valued and scalar, we have

$\displaystyle
Cv(x(r,{\theta}),y(r,{\theta}))=K_{{\lambda}-1/2}(r)C{\varphi}_{\lambda}({\theta}).$

To show that $C{\varphi}_{\lambda}\in\ker(J_{\tau}-{\lambda})$, we first
verify that $C{\varphi}_{{\lambda}}\in\operatorname{dom}J_{\tau}$. As $M_{l}$
is a restriction of $M$, a similar consideration as above shows first
$\sigma_{1}\overline{M_{l}}=M_{l}\sigma_{1}$ and further

$\displaystyle C{\varphi}_{{\lambda},-}(\omega)$
$\displaystyle=\sigma_{1}\overline{{\varphi}_{{\lambda},-}}(\omega)=\sigma_{1}\overline{M_{l}{\varphi}_{{\lambda},+}(\omega)}=M_{l}\sigma_{1}\overline{{\varphi}_{{\lambda},+}(\omega)}=M_{l}C{\varphi}_{{\lambda},+}(\omega).$

An analogous argument shows that
$C{\varphi}_{{\lambda},-}(2\pi-\omega)=M_{r}C{\varphi}_{{\lambda},+}(-\omega)$,
and hence $C{\varphi}_{\lambda}\in\operatorname{dom}J_{\tau}$. Moreover,

$\begin{split}J_{\tau}C{\varphi}_{\lambda}&=\left(-i\sigma_{3}\sigma_{1}\overline{{\varphi}_{{\lambda},+}}^{\prime}+\frac{\sigma_{1}\overline{{\varphi}_{{\lambda},+}}}{2}\right)\oplus\left(-i\sigma_{3}\sigma_{1}\overline{{\varphi}_{{\lambda},-}}^{\prime}+\frac{\sigma_{1}\overline{{\varphi}_{{\lambda},-}}}{2}\right)\\\
&=\sigma_{1}\left[\left(i\sigma_{3}\overline{{\varphi}_{{\lambda},+}^{\prime}}+\frac{\overline{{\varphi}_{{\lambda},+}}}{2}\right)\oplus\left(i\sigma_{3}\overline{{\varphi}_{{\lambda},-}^{\prime}}+\frac{\overline{{\varphi}_{{\lambda},-}}}{2}\right)\right]\\\
&=\sigma_{1}\overline{\left[\left(-i\sigma_{3}{\varphi}_{{\lambda},+}^{\prime}+\frac{{\varphi}_{{\lambda},+}}{2}\right)\oplus\left(-i\sigma_{3}{\varphi}_{{\lambda},-}^{\prime}+\frac{{\varphi}_{{\lambda},-}}{2}\right)\right]}\\\
&=\sigma_{1}\overline{{\lambda}{\varphi}_{\lambda}}={\lambda}C{\varphi}_{\lambda}.\end{split}$

Since ${\lambda}$ is a simple eigenvalue of $J_{\tau}$ and
${\varphi}_{\lambda},C{\varphi}_{\lambda}\in\ker(J_{\tau}-{\lambda})$, we find
that $C{\varphi}_{\lambda}=a{\varphi}_{\lambda}$ for some $a\in\mathbb{C}$.
Hence, $Cv\in\operatorname{span}\\{v\\}$ and for any
$u\in\operatorname{dom}A_{\tau,1}$, $Cu\in\operatorname{dom}A_{\tau,1}$.
Finally, using $\sigma_{2}=-\overline{\sigma_{2}}$, we calculate for
$u\in\operatorname{dom}A_{\tau,1}$

$\displaystyle(A_{\tau,1}Cu)_{\pm}$
$\displaystyle=[(-i(\sigma\cdot\gradient)+m\sigma_{3}]Cu_{\pm}=[-i(\sigma\cdot\gradient)+m\sigma_{3}]\sigma_{1}\overline{u}_{\pm}$
$\displaystyle=\sigma_{1}(-i\sigma_{1}\partial_{1}+i\sigma_{2}\partial_{2}-m\sigma_{3})\overline{u}_{\pm}=-\sigma_{1}\overline{(-i\sigma_{1}\partial_{1}-i\sigma_{2}\partial_{2}+m\sigma_{3})u_{\pm}}$
$\displaystyle=-C(-i(\sigma\cdot\gradient)+m\sigma_{3})u_{\pm}=-C(A_{\tau,1}u)_{\pm}.$

If we let ${\lambda}\in\sigma(A_{\tau,1})$ and
$(u_{n})_{n=1}^{\infty}\subset\operatorname{dom}A_{\tau,1}$ such that
$\|u_{n}\|_{\mathbb{R}^{2}}=1$ and $(A_{\tau,1}-{\lambda})u_{n}\to 0$, then
$Cu_{n}\in\operatorname{dom}A_{\tau,1}$, $\|Cu_{n}\|_{\mathbb{R}^{2}}=1$ and

$\displaystyle(A_{\tau,1}+{\lambda})Cu_{n}=C(-A_{\tau,1}+{\lambda})u_{n}\to
0,$

so that $-{\lambda}\in\sigma(A_{\tau,1})$. The result follows. ∎

######  Corollary 4.2.

Let $\omega\in(0,\frac{\pi}{2})$ and $\tau\in\mathbb{R}\setminus\\{-2,0,2\\}$.
Let $z\in\mathbb{C}$ with $|z|=1$ and $A_{\tau,z}$ be the corresponding self-
adjoint extension of $S_{\tau}$ via Theorem 2.2. Then the essential spectrum
of $A_{\tau,z}$ is symmetric with respect to the origin, i.e.
$\sigma_{\scriptstyle\text{\rm
ess}}(A_{\tau,z})=-\sigma_{\scriptstyle\text{\rm ess}}(A_{\tau,z})$.

###### Proof.

The claim holds for the distinguished self-adjoint extension $A_{\tau,1}$ by
Theorem 4.1 and since

$\displaystyle(A_{\tau,z}-i)^{-1}-(A_{\tau,1}-i)^{-1}$

has finite rank, it follows that $\sigma_{\scriptstyle\text{\rm
ess}}(A_{\tau,z})=\sigma_{\scriptstyle\text{\rm ess}}(A_{\tau,1})$. ∎

##  5\. Essential Spectrum

In this section we show the statements about the essential spectrum of any
self-adjoint extension of $S_{\tau}$ in Theorem 1.1. We begin with a general
result for symmetric operators. Then, in Proposition 5.2 we show that
$(-\infty,-m]\cup[m,\infty)\subset\sigma_{\scriptstyle\text{\rm
ess}}(A_{\tau,z})$ holds for any self-adjoint extension $A_{\tau,z}$ of
$S_{\tau}$. Together with Theorem 3.2 and Proposition 5.1 this implies already
that this inclusion is an equality if $\tau\in(0,\infty)\setminus\\{2\\}$.
Then, in Propositions 5.4 and 5.5 we show with a Neumann bracketing argument
and suitable singular sequences that for $\tau\in(-\infty,0)\setminus\\{-2\\}$
one has

$\sigma_{\scriptstyle\text{\rm
ess}}(S_{\tau}^{*}S_{\tau})=\Bigg{[}m^{2}\left(\frac{\tau^{2}-4}{\tau^{2}+4}\right)^{2},\infty\Bigg{)}.$

This allows us finally with Proposition 5.1 and Corollary 4.2 to determine
$\sigma_{\textup{ess}}(A_{\tau,z})$ also for
$\tau\in(-\infty,0)\setminus\\{-2\\}$.

######  Proposition 5.1.

Let $S$ be a closed, densely defined, symmetric operator in the Hilbert space
$\mathcal{H}$ with deficiency indices $(n,n)$, $n\in\mathbb{N}$, and $D$ be a
self-adjoint extension of $S$. Then the resolvent difference

$W\mathrel{\mathop{\mathchar 58\relax}}=(S^{*}S+1)^{-1}-(D^{2}+1)^{-1}$

is a finite-rank operator with rank at most $n$.

###### Proof.

The self-adjoint extension $D$ can be described by the von Neumann extension
theory of symmetric operators. According to [37, Theorem 13.10] there exists a
unitary operator $U\mathchar 58\relax\ker(S^{*}-i)\to\ker(S^{*}+i)$ such that
the self-adjoint extension $D$ is defined as

$Du=Su_{0}+i\widetilde{u}-iU\widetilde{u},\qquad\operatorname{dom}D=\\{u_{0}+\widetilde{u}+U\widetilde{u}\mathchar
58\relax u_{0}\in\operatorname{dom}S,\,\widetilde{u}\in\ker(S^{*}-i)\\}.$

Likewise, [37, Proposition 3.7] says that $u\in\operatorname{dom}S^{*}$ admits
a unique decomposition $u=u_{0}+\widetilde{u}+\widehat{u}$ for
$u_{0}\in\operatorname{dom}S$, $\widetilde{u}\in\ker(S^{*}-i)$ and
$\widehat{u}\in\ker(S^{*}+i)$. Define the linear operator $\Pi\mathchar
58\relax\operatorname{dom}S^{*}\to\ker(S^{*}-i)$ such that $\Pi
u\mathrel{\mathop{\mathchar 58\relax}}=\widetilde{u}$ and let
$f,g\in\mathcal{H}$ be arbitrary. We set

$u\mathrel{\mathop{\mathchar 58\relax}}=(S^{*}S+1)^{-1}f,\qquad
v\mathrel{\mathop{\mathchar 58\relax}}=(D^{2}+1)^{-1}g,$

which, in particular, means that $u\in\operatorname{dom}(S^{*}S)$ and
$v\in\operatorname{dom}(S^{*})$. We calculate that

$\displaystyle(Wf,g)_{\mathcal{H}}$
$\displaystyle=((S^{*}S+1)^{-1}f,g)_{\mathcal{H}}-(f,(D^{2}+1)^{-1}g)_{\mathcal{H}}$
$\displaystyle=(u,D^{2}v)_{\mathcal{H}}-(S^{*}Su,v)_{\mathcal{H}}$
$\displaystyle=(u,S^{*}Sv_{0}-\widetilde{v}-U\widetilde{v})_{\mathcal{H}}-(S^{*}Su,v_{0}+\widetilde{v}+U\widetilde{v})_{\mathcal{H}}$
$\displaystyle=(u,-\widetilde{v}-U\widetilde{v})_{\mathcal{H}}-(S^{*}Su,\widetilde{v}+U\widetilde{v})_{\mathcal{H}}$
$\displaystyle=-((S^{*}S+1)u,\widetilde{v}+U\widetilde{v})_{\mathcal{H}}$
$\displaystyle=-(f,(1+U)\Pi(D^{2}+1)^{-1}g)_{\mathcal{H}},$

where the symmetry of $S^{*}S$ was used to cancel two terms between the third
and fourth equalities. Hence,

$W=-(1+U)\Pi(D^{2}+1)^{-1}$

and because $\operatorname{ran}\Pi=\ker(S^{*}-i)$, the rank of $W$ can be at
most $n$. ∎

In the next proposition we show that the inclusion
$(-\infty,-m]\cup[m,\infty)\subset\sigma_{\scriptstyle\text{\rm
ess}}(A_{\tau,z})$ holds for any self-adjoint extension $A_{\tau,z}$ of
$S_{\tau}$.

######  Proposition 5.2.

Let $A_{\tau,z}$ be any self-adjoint extension of $S_{\tau}$ defined in (1.3).
Then the inclusion
$(-\infty,-m]\cup[m,\infty)\subset\sigma_{\scriptstyle\text{\rm
ess}}(A_{\tau,z})$ holds.

###### Proof.

Let ${\lambda}\in(-\infty,-m)\cup(m,\infty)$ be fixed and fix a vector
$\zeta\in\mathbb{C}^{2}\setminus\\{0\\}$ such that
$(\sqrt{{\lambda}^{2}-m^{2}}\sigma_{1}+m\sigma_{3}+{\lambda}\sigma_{0})\zeta\neq
0$. Also let $\chi\in C_{0}^{\infty}(\mathbb{R})$ be chosen so that
$\chi(r)=1$ for $|r|<1/2$, $\chi(r)=0$ for $|r|>1$, and
$x_{n}=(-1-n^{2},0)^{T}$. Now, consider the sequence

(5.1)
$\displaystyle\psi_{n}(x)=\frac{1}{n}\chi\left(\frac{1}{n}|x-x_{n}|\right)e^{i\sqrt{{\lambda}^{2}-m^{2}}x\cdot
e_{1}}\left(\sqrt{{\lambda}^{2}-m^{2}}\sigma_{1}+m\sigma_{3}+{\lambda}\sigma_{0}\right)\zeta,\qquad
n\in{\mathbb{N}},$

where $e_{1}=(1,0)^{T}$. As in [8, Theorem 5.7(i)], it is possible to show
that

  * (i)

$\psi_{n}\in\operatorname{dom}S_{\tau}$,

  * (ii)

$\|\psi_{n}\|^{2}_{\mathbb{R}^{2}}=c>0$ for all $n$ with a constant $c$
independent of $n$,

  * (iii)

$\psi_{n}$ converges weakly to zero and,

  * (iv)

$(S_{\tau}-{\lambda})\psi_{n}\to 0$ as $n\to\infty$.

The sequence $\\{\psi_{n}\\}_{n=1}^{\infty}$ is thus a Weyl sequence for
${\lambda}$ and any self-adjoint extension $A_{\tau,z}$ of $S_{\tau}$, which
implies $(-\infty,-m)\cup(m,\infty)\subset\sigma_{\scriptstyle\text{\rm
ess}}(A_{\tau,z})$. Since the essential spectrum of any self-adjoint operator
is always closed, the result follows. ∎

Proposition 5.2 can now be combined with Theorem 3.2 to characterize the
essential spectrum for any positive interaction strength $\tau$ such that
$\tau\neq 2$.

######  Corollary 5.3.

Let $\tau\in(0,\infty)\setminus\\{2\\}$ and $A_{\tau,z}$ be any self-adjoint
extension of $S_{\tau}$, defined in (1.3). Then $\sigma_{\scriptstyle\text{\rm
ess}}(A_{\tau,z})=(-\infty,-m]\cup[m,\infty)$.

###### Proof.

Propositions 5.1 and 5.2 immediately imply that
$[m^{2},\infty)\subset\sigma_{\textup{ess}}(S_{\tau}^{*}S_{\tau})$. Moreover,
Theorem 3.2 says that for any $u\in\operatorname{dom}(S_{\tau}^{*}S_{\tau})$

$\begin{split}(S_{\tau}^{*}S_{\tau}u,u)_{\mathbb{R}^{2}}&=\|S_{\tau}u\|^{2}_{\mathbb{R}^{2}}=\|\gradient
u\|^{2}_{\mathbb{R}^{2}\setminus\Gamma}+m^{2}\|u\|^{2}_{\mathbb{R}^{2}}+\frac{2m}{\tau}\big{\|}u_{+}|_{\Gamma}-u_{-}|_{\Gamma}\big{\|}^{2}_{\Gamma}\geq
m^{2}\|u\|^{2}_{\mathbb{R}^{2}},\end{split}$

which yields the reverse inclusion
$\sigma_{\textup{ess}}(S_{\tau}^{*}S_{\tau})\subset[m^{2},\infty)$. Hence,
$\sigma_{\textup{ess}}(S_{\tau}^{*}S_{\tau})=[m^{2},\infty)$. Proposition 5.1
says that $S^{*}_{\tau}S_{\tau}$ and $(A_{\tau,z})^{2}$ differ by a finite-
rank perturbation and so their essential spectra are equal. Corollary 4.2
further shows that the essential spectrum of $A_{\tau,z}$ is symmetric about
$0$ and the result follows. ∎

We now turn our attention to the case where $\tau$ is negative such that
$\tau\neq-2$. The corresponding claim about the essential spectrum of any
self-adjoint extension of $S_{\tau}$ from Theorem 1.1 will be shown in three
steps. Begin by setting $\varepsilon_{\tau}\mathrel{\mathop{\mathchar
58\relax}}=m\frac{\tau^{2}-4}{\tau^{2}+4}$. First, a Neumann bracketing
argument is used to prove that
$[\varepsilon_{\tau}^{2},\infty)\subset\sigma_{\textup{ess}}(S_{\tau}^{*}S_{\tau})$.
The reverse inclusion is contained in Proposition 5.5 and uses a particular
Weyl sequence construction. The final statements then follow by a similar
argument as the proof of Corollary 5.3.

######  Proposition 5.4.

Let $\tau\in(-\infty,0)\setminus\\{-2\\}$. Then

$\displaystyle\sigma_{\scriptstyle\text{\rm
ess}}(S_{\tau}^{*}S_{\tau})\subset\Bigg{[}m^{2}\left(\frac{\tau^{2}-4}{\tau^{2}+4}\right)^{2},\infty\Bigg{)}.$

###### Proof.

As indicated in Figure 2, we define for a fixed $\gamma>0$ the following
domains $\Omega_{j}\subset\mathbb{R}^{2}$, $j\in\\{1,\dots,5\\}$:

$\begin{split}\Omega_{1}&\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\left(\frac{\gamma}{\sin\omega}+r\cos\varphi,r\sin\varphi\right)\mathchar
58\relax r>0,\varphi\in(-\omega,\omega)\right\\},\\\
\Omega_{2}&\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\left(-\frac{\gamma}{\sin\omega}+r\cos\varphi,r\sin\varphi\right)\mathchar
58\relax r>0,\varphi\in(\omega,2\pi-\omega)\right\\},\\\
\Omega_{3}&\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\left(r\cos\omega+s\sin\omega,r\sin\omega-s\cos\omega\right)\mathchar
58\relax r>\frac{\gamma}{\tan\omega},s\in(-\gamma,\gamma)\right\\},\\\
\Omega_{4}&\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\left(r\cos\omega+s\sin\omega,-r\sin\omega+s\cos\omega\right)\mathchar
58\relax r>\frac{\gamma}{\tan\omega},s\in(-\gamma,\gamma)\right\\},\\\
\Omega_{5}&\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{R}^{2}\setminus\overline{\big{(}\Omega_{1}\cup\Omega_{2}\cup\Omega_{3}\cup\Omega_{4}\big{)}}.\end{split}$

Figure 2. Decomposition of $\mathbb{R}^{2}$ for Neumann bracketing. The
domains $\Omega_{3}$ and $\Omega_{4}$ are depicted in light gray, the set
$\Omega_{5}$ in dark gray.

In each of the regions $\Omega_{j}$, we define a quadratic form in
$L^{2}(\Omega_{j})$: for $j=1,2$ by

$\displaystyle\mathfrak{b}_{j}[u]$ $\displaystyle=\|\gradient
u\|^{2}_{{}_{\scriptstyle\Omega_{j}}}+m^{2}\|u\|^{2}_{{}_{\scriptstyle\Omega_{j}}},\quad\operatorname{dom}\mathfrak{b}_{j}=H^{1}(\Omega_{j};\mathbb{C}^{2}),$

and for $j=3,4,5$ with $\Gamma_{j}\mathrel{\mathop{\mathchar
58\relax}}=\Gamma\cap\Omega_{j}$ by

$\displaystyle\mathfrak{b}_{j}[u]$ $\displaystyle=\|\gradient
u\|^{2}_{\Omega_{j}\setminus\Gamma_{j}}+m^{2}\|u\|^{2}_{\Omega_{j}}+\frac{2m}{\tau}\|u_{+}|_{\Gamma_{j}}-u_{-}|_{\Gamma_{j}}\|_{\Gamma_{j}}^{2},$
$\displaystyle\operatorname{dom}\mathfrak{b}_{j}$ $\displaystyle=\big{\\{}u\in
H^{1}(\Omega_{j}\setminus\Gamma_{j};\mathbb{C}^{2})~{}\mathchar
58\relax~{}u_{-}|_{\Gamma_{j}}=Mu_{+}|_{\Gamma_{j}}\big{\\}},$

where functions $u\in L^{2}(\Omega_{j};\mathbb{C}^{2})$ are written with the
notation $u_{\pm}=u\upharpoonright(\Omega_{j}\cap\Omega_{\pm})$. The quadratic
forms $\mathfrak{b}_{j}$, $j\in\\{1,\dots,5\\}$, are closed and semibounded
from below. The associated self-adjoint operators are denoted by $B_{j}$. If
we define $T=\bigoplus_{j=1}^{5}B_{j}$, then we have, in the sense of
semibounded self-adjoint operators, $S_{\tau}^{*}S_{\tau}\geq T$; i.e. for any
$u\in\operatorname{dom}S_{\tau}$ there holds
$u\in\operatorname{dom}\mathfrak{t}$ and
$\|S_{\tau}u\|_{\mathbb{R}^{2}}^{2}\geq\mathfrak{t}[u]$, where $\mathfrak{t}$
is the quadratic form in $L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})$ associated
with the operator $T$. The min-max principle implies

(5.2)
$\inf\sigma_{\textup{ess}}(S_{\tau}^{*}S_{\tau})\geq\inf\sigma_{\textup{ess}}(T).$

Begin by observing that $B_{1}$ and $B_{2}$ are shifted Neumann Laplacians in
$\Omega_{1}$ and $\Omega_{2}$, respectively, and hence

(5.3) $B_{1},B_{2}\geq m^{2}.$

Moreover, as $\Omega_{5}$ is a bounded Lipschitz domain,
$\operatorname{dom}\mathfrak{b}_{5}$ is compactly embedded in
$L^{2}(\Omega_{5};\mathbb{C}^{2})$ and thus,

(5.4) $\sigma_{\textup{ess}}(B_{5})=\emptyset.$

It therefore remains to estimate $\inf\sigma_{\textup{ess}}(B_{j})$,
$j\in\\{3,4\\}$. Let $\Omega_{\gamma}\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{R}_{+}\times(-\gamma,\gamma)$ and

$\displaystyle
M_{j}=\frac{4+\tau^{2}}{4-\tau^{2}}\sigma_{0}+\frac{4\tau}{4-\tau^{2}}i\sigma_{3}(\sigma\cdot\nu_{j}),\qquad\nu_{j}=\nu\upharpoonright(\Gamma\cap\Omega_{j}).$

Define $\widetilde{\Gamma}\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{R}_{+}\times\\{0\\}$. Then, a rotation by $\pm\omega$,
translation by $(-\frac{\gamma}{\tan\omega},0)$, and finally for $\Omega_{3}$
a change of variables $y\rightarrow-y$ shows that $B_{j}$, $j\in\\{3,4\\}$, is
unitarily equivalent to a self-adjoint operator $\widetilde{B}_{j}$ associated
with the closed and semibounded quadratic form

$\displaystyle\widetilde{\mathfrak{b}}_{j}[u]$ $\displaystyle=\|\gradient
u\|^{2}_{\Omega_{\gamma}\setminus\widetilde{\Gamma}}+m^{2}\|u\|^{2}_{\Omega_{\gamma}}+\frac{2m}{\tau}\big{\|}u_{+}|_{\widetilde{\Gamma}}-u_{-}|_{\widetilde{\Gamma}}\big{\|}_{\widetilde{\Gamma}}^{2},$
$\displaystyle\operatorname{dom}\widetilde{\mathfrak{b}}_{j}$
$\displaystyle=\big{\\{}u\in
H^{1}(\Omega_{\gamma}\setminus\widetilde{\Gamma};\mathbb{C}^{2})\colon
u_{-}|_{\widetilde{\Gamma}}=M_{j}u_{+}|_{\widetilde{\Gamma}}\big{\\}},$

where $u_{+}=u|_{{\mathbb{R}_{+}\times(0,\gamma)}}$ and
$u_{-}=u|_{\mathbb{R}_{+}\times(-\gamma,0)}$. Set
$\Theta_{j}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{\sqrt{2}}(\sigma_{0}+i(\sigma\cdot\nu_{j}))$. A direct
calculation shows that $\Theta_{j}$ is a unitary matrix and that

(5.5)
$\Theta_{j}^{*}M_{j}\Theta_{j}=\frac{4+\tau^{2}}{4-\tau^{2}}\sigma_{0}-\frac{4\tau}{4-\tau^{2}}\sigma_{3}=\mathchar
58\relax\widetilde{M}.$

We define for $j\in\\{3,4\\}$

$\mathfrak{c}[u]\mathrel{\mathop{\mathchar
58\relax}}=\widetilde{\mathfrak{b}}_{j}[\Theta_{j}u],\qquad\operatorname{dom}\mathfrak{c}\mathrel{\mathop{\mathchar
58\relax}}=\Theta_{j}^{*}\big{(}\operatorname{dom}\widetilde{\mathfrak{b}}_{j}\big{)},$

and note that this form is independent of $j$ and explicitly given by

$\displaystyle\mathfrak{c}[u]$ $\displaystyle=\|\nabla
u\|^{2}_{\Omega_{\gamma}\setminus\widetilde{\Gamma}}+m^{2}\|u\|^{2}_{\Omega_{\gamma}}+\frac{2m}{\tau}\big{\|}u_{+}|_{\widetilde{\Gamma}}-u_{-}|_{\widetilde{\Gamma}}\big{\|}_{\widetilde{\Gamma}}^{2},$
$\displaystyle\operatorname{dom}\mathfrak{c}$ $\displaystyle=\left\\{u\in
H^{1}(\Omega_{\gamma}\setminus\widetilde{\Gamma};\mathbb{C}^{2})\colon
u_{-}|_{\widetilde{\Gamma}}=\widetilde{M}u_{+}|_{\widetilde{\Gamma}}\right\\}.$

Since $\Theta_{j}$ is unitary, we have for $j\in\\{3,4\\}$

(5.6)
$\inf\sigma_{\textup{ess}}(B_{j})\geq\inf\left\\{\frac{\mathfrak{c}[u]}{\|u\|_{\Omega_{\gamma}}^{2}}\colon
u\in\operatorname{dom}\mathfrak{c}\setminus\\{0\\}\right\\}.$

Let $u\in\operatorname{dom}\mathfrak{c}$. Then, it follows in a similar way as
in the proof of [26, Lemma 4.13], cf. also Appendix B for details, for almost
every $x>0$ that

(5.7)
$\begin{split}\int_{(-\gamma,\gamma)\setminus\\{0\\}}&\big{(}|\partial_{x}u(x,y)|^{2}+|\partial_{y}u(x,y)|^{2}\big{)}dy+m^{2}\int_{(-\gamma,\gamma)}|u(x,y)|^{2}dy+\frac{2m}{\tau}|u(x,0^{+})-u(x,0^{-})|^{2}\\\
&\geq\int_{(-\gamma,\gamma)\setminus\\{0\\}}|\partial_{y}u(x,y)|^{2}dy+m^{2}\int_{(-\gamma,\gamma)}|u(x,y)|^{2}dy+\frac{2m}{\tau}|u(x,0^{+})-u(x,0^{-})|^{2}\\\
&\geq E(\gamma)\int_{(-\gamma,\gamma)}|u(x,y)|^{2}dy,\end{split}$

where $E(\gamma)\in[0,m^{2})$ such that

$\displaystyle\lim_{\gamma\rightarrow\infty}E(\gamma)=m^{2}\left(\frac{\tau^{2}-4}{\tau^{2}+4}\right)^{2}.$

This implies that

$\begin{split}\mathfrak{c}[u]&=\int_{\mathbb{R}_{+}}\bigg{(}\int_{(-\gamma,\gamma)\setminus\\{0\\}}\big{(}|\partial_{x}u(x,y)|^{2}+|\partial_{y}u(x,y)|^{2}\big{)}dy\\\
&\qquad\qquad\qquad+m^{2}\int_{(-\gamma,\gamma)}|u(x,y)|^{2}dy+\frac{2m}{\tau}|u(x,0^{+})-u(x,0^{-})|^{2}\bigg{)}dx\\\
&\geq\int_{\mathbb{R}_{+}}E(\gamma)\int_{(-\gamma,\gamma)}|u(x,y)|^{2}dydx.\end{split}$

Therefore, equations (5.2)–(5.4) and (5.6) imply that with
$\gamma\rightarrow\infty$ we have $\inf\sigma_{\scriptstyle\text{\rm
ess}}(S_{\tau}^{*}S_{\tau})\geq
m^{2}\big{(}\frac{\tau^{2}-4}{\tau^{2}+4}\big{)}^{2}$. ∎

We now prove the opposite inclusion.

######  Proposition 5.5.

Let $\tau\in(-\infty,0)\setminus\\{-2\\}$. Then

$\displaystyle\Bigg{[}m^{2}\left(\frac{\tau^{2}-4}{\tau^{2}+4}\right)^{2},\infty\Bigg{)}\subset\sigma_{\scriptstyle\text{\rm
ess}}(S_{\tau}^{*}S_{\tau}).$

###### Proof.

Set $z=-m\frac{4\tau}{\tau^{2}+4}$, $x_{n}=n^{2}+K$ for $n\in\mathbb{N}$ and
some $K>0$, $\lambda>0$, and choose $\chi\in C_{0}^{\infty}(\mathbb{R})$ such
that $\chi(r)=1$ for $|r|\leq 1/2$ and $\chi(r)=0$ for $|r|>1$. Define the
function

$\displaystyle\psi_{n}(x,y)$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{\sqrt{n}}\chi\left(\frac{\xi-
x_{n}}{n}\right)\chi\left(\frac{c\zeta}{n}\right)e^{i{\lambda}\xi}v(\zeta),$
$\displaystyle\text{ with }\quad
v(\zeta)=\begin{cases}ae^{-z\zeta},&\zeta>0,\\\
M_{l}ae^{z\zeta},&\zeta<0,\end{cases}\qquad\begin{pmatrix}\xi\\\
\zeta\end{pmatrix}=\begin{pmatrix}\cos\omega&\sin\omega\\\
\sin\omega&-\cos\omega\end{pmatrix}\begin{pmatrix}x\\\ y\end{pmatrix},$

where $0\neq a\in\mathbb{C}^{2}$, $M_{l}$ is the matrix in (2.4) and $c>0$ is
chosen so that $\operatorname{supp}(\psi_{n})\cap\Gamma_{r}=\emptyset$. We
will show that

  1. (i)

$\psi_{n}\in\operatorname{dom}S_{\tau}$ for all $n$,

  2. (ii)

$c_{1}\leq\|\psi_{n}\|^{2}_{\mathbb{R}^{2}}\leq c_{2}$ for constants
$c_{2}\geq c_{1}>0$,

  3. (iii)

$\psi_{n}\perp\psi_{m}$ for $m\neq n$, so that $\psi_{n}$ converges weakly to
zero and,

  4. (iv)

there exists a constant $c_{3}>0$ such that for any
$u\in\operatorname{dom}S_{\tau}$,

$\displaystyle\left|\left\langle\left(S_{\tau}+\sqrt{m^{2}-z^{2}+{\lambda}^{2}}\right)u,\left(S_{\tau}-\sqrt{m^{2}-z^{2}+{\lambda}^{2}}\right)\psi_{n}\right\rangle_{\mathbb{R}^{2}}\right|\leq\frac{c_{3}}{n}\left(\|S_{\tau}u\|^{2}_{\mathbb{R}^{2}}+\|u\|^{2}_{\mathbb{R}^{2}}\right)^{1/2}.$

Combined, these facts show that $\psi_{n}$ satisfies Weyl’s criterion for
quadratic forms in the sense of [27, Appendix] and
$m^{2}-z^{2}+{\lambda}^{2}=m^{2}\big{(}\frac{\tau^{2}-4}{\tau^{2}+4}\big{)}^{2}+\lambda^{2}\in\sigma_{\scriptstyle\text{\rm
ess}}(S_{\tau}^{*}S_{\tau})$. The choice of ${\lambda}>0$ was arbitrary and
the essential spectrum is always closed, so the result follows.

It remains to show claims (i)–(iv) above. Item (i) is clear by construction.
To see claim (ii), we compute

$\begin{split}\|\psi_{n}\|_{\mathbb{R}^{2}}^{2}&=\int_{\mathbb{R}^{2}}\frac{1}{n}\left|\chi\left(\frac{\xi-
x_{n}}{n}\right)\chi\left(\frac{c\zeta}{n}\right)\right|^{2}|v(\zeta)|^{2}dxdy\\\
&=\int_{-1}^{1}|\chi(x^{\prime})|^{2}dx^{\prime}\int_{-n}^{n}\left|\chi\left(\frac{c\zeta}{n}\right)\right|^{2}|v(\zeta)|^{2}d\zeta,\end{split}$

where the substitution $x^{\prime}=\frac{\xi-x_{n}}{n}$ was done. The
properties of the function $v$ then imply that claim (ii) is true.

In light of claim (ii), it is easy to see that claim (iii) holds true, as the
functions $\psi_{n}$ and $\psi_{m}$ for $n\neq m$ have disjoint supports.

Claim (iv) makes use of Theorem 3.2 and the transformation from the variables
$(x,y)$ to $(\xi,\zeta)$. Let $u\in\operatorname{dom}S_{\tau}$ and set
$\tilde{u}(\xi,\zeta)\mathrel{\mathop{\mathchar
58\relax}}=u(x(\xi,\zeta),y(\xi,\zeta))$,
$\tilde{\psi}_{n}(\xi,\zeta)\mathrel{\mathop{\mathchar
58\relax}}=\psi_{n}(x(\xi,\zeta),y(\xi,\zeta))$, and
$\widetilde{\Gamma}\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{R}_{+}\times\\{0\\}$. These facts allow for the definition

$\begin{split}I\mathrel{\mathop{\mathchar
58\relax}}=\bigg{\langle}&\left(S_{\tau}+\sqrt{m^{2}-z^{2}+{\lambda}^{2}}\right)u,\left(S_{\tau}-\sqrt{m^{2}-z^{2}+{\lambda}^{2}}\right)\psi_{n}\bigg{\rangle}_{\mathbb{R}^{2}}=\left(z^{2}-\lambda^{2}\right)(\tilde{u},\tilde{\psi}_{n})_{\mathbb{R}^{2}}\\\
&+(\gradient\tilde{u},\gradient\tilde{\psi}_{n})_{\mathbb{R}^{2}\setminus\widetilde{\Gamma}}+\frac{2m}{\tau}\int_{x_{n}-n}^{x_{n}+n}(\tilde{u}_{+}|_{\widetilde{\Gamma}}(\xi)-\tilde{u}_{-}|_{\widetilde{\Gamma}}(\xi))\overline{(\tilde{\psi}_{n,+}|_{\widetilde{\Gamma}}(\xi)-\tilde{\psi}_{n,-}|_{\widetilde{\Gamma}}(\xi))}d\xi.\end{split}$

Note that $u,\psi_{n}\in\operatorname{dom}S_{\tau}$ implies
$\tilde{u}_{-}|_{\widetilde{\Gamma}}(\xi)=M_{l}\tilde{u}_{+}|_{\widetilde{\Gamma}}(\xi)$,
$\tilde{\psi}_{n,-}|_{\widetilde{\Gamma}}(\xi)=M_{l}\tilde{\psi}_{n,+}|_{\widetilde{\Gamma}}(\xi)$
and $M_{l}=M_{l}^{*}$. Integration by parts then yields

(5.8)
$\begin{split}I&=(\tilde{u},-\Delta\tilde{\psi}_{n})_{\mathbb{R}^{2}\setminus\widetilde{\Gamma}}+\left(z^{2}-\lambda^{2}\right)(\tilde{u},\tilde{\psi}_{n})_{\mathbb{R}^{2}}\\\
&+\int_{x_{n}-n}^{x_{n}+n}\tilde{u}_{+}|_{\widetilde{\Gamma}}(\xi)\overline{\left(\frac{2m}{\tau}(\sigma_{0}-M_{l})^{2}\tilde{\psi}_{n,+}|_{\widetilde{\Gamma}}(\xi)+\left(M_{l}\frac{\partial\tilde{\psi}_{n,-}}{\partial\zeta}\bigg{|}_{\widetilde{\Gamma}}(\xi)-\frac{\partial\tilde{\psi}_{n,+}}{\partial\zeta}\bigg{|}_{\widetilde{\Gamma}}(\xi)\right)\right)}d\xi.\end{split}$

We then calculate

$\displaystyle\frac{\partial\tilde{\psi}_{n,+}}{\partial\zeta}(\xi,\zeta)$
$\displaystyle=\frac{c}{n^{3/2}}\chi\left(\frac{\xi-
x_{n}}{n}\right)\chi^{\prime}\left(\frac{c\zeta}{n}\right)e^{i\lambda\xi}ae^{-z\zeta}$
$\displaystyle\hskip 120.00018pt-\frac{z}{n^{1/2}}\chi\left(\frac{\xi-
x_{n}}{n}\right)\chi\left(\frac{c\zeta}{n}\right)e^{i\lambda\xi}ae^{-z\zeta},$
$\displaystyle\frac{\partial\tilde{\psi}_{n,-}}{\partial\zeta}(\xi,\zeta)$
$\displaystyle=\frac{c}{n^{3/2}}\chi\left(\frac{\xi-
x_{n}}{n}\right)\chi^{\prime}\left(\frac{c\zeta}{n}\right)e^{i\lambda\xi}M_{l}ae^{z\zeta}$
$\displaystyle\hskip 120.00018pt+\frac{z}{n^{1/2}}\chi\left(\frac{\xi-
x_{n}}{n}\right)\chi\left(\frac{c\zeta}{n}\right)e^{i\lambda\xi}M_{l}ae^{z\zeta}.$

Using that $\chi(0)=1$ and $\chi^{\prime}(0)=0$ we infer

$\begin{split}M_{l}\frac{\partial\tilde{\psi}_{n,-}}{\partial\zeta}\big{|}_{\widetilde{\Gamma}}(\xi)-\frac{\partial\tilde{\psi}_{n,+}}{\partial\zeta}\big{|}_{\widetilde{\Gamma}}(\xi)=\frac{z}{n^{1/2}}\chi\left(\frac{\xi-
x_{n}}{n}\right)e^{i\lambda\xi}(M_{l}^{2}+\sigma_{0})a=z(M_{l}^{2}+\sigma_{0})\tilde{\psi}_{n,+}|_{\widetilde{\Gamma}}(\xi).\end{split}$

A direct calculation shows that
$z(M_{l}^{2}+\sigma_{0})=-\frac{8m\tau}{4-\tau^{2}}M_{l}$ and
$\frac{2m}{\tau}(\sigma_{0}-M_{l})^{2}=\frac{8m\tau}{4-\tau^{2}}M_{l}$. Hence,
we obtain that

(5.9)
$\int_{x_{n}-n}^{x_{n}+n}\tilde{u}_{+}|_{\widetilde{\Gamma}}(\xi)\overline{\left(\frac{2m}{\tau}(\sigma_{0}-M_{l})^{2}\tilde{\psi}_{n,+}|_{\widetilde{\Gamma}}(\xi)+\left(M_{l}\frac{\partial\tilde{\psi}_{n,-}}{\partial\zeta}\bigg{|}_{\widetilde{\Gamma}}(\xi)-\frac{\partial\tilde{\psi}_{n,+}}{\partial\zeta}\bigg{|}_{\widetilde{\Gamma}}(\xi)\right)\right)}d\xi=0.$

Returning to the analysis of other terms in $I$, we calculate

$\begin{split}\Delta\tilde{\psi}_{n}(\xi,\zeta)&=(z^{2}-\lambda^{2})\tilde{\psi}_{n}(\xi,\zeta)+\frac{1}{n^{5/2}}\chi^{\prime\prime}\left(\frac{\xi-
x_{n}}{n}\right)\chi\left(\frac{c\zeta}{n}\right)e^{i\lambda\xi}v(\zeta)\\\
&\quad+\frac{2i\lambda}{n^{3/2}}\chi^{\prime}\left(\frac{\xi-
x_{n}}{n}\right)\chi\left(\frac{c\zeta}{n}\right)e^{i\lambda\xi}v(\zeta)+\frac{c^{2}}{n^{5/2}}\chi\left(\frac{\xi-
x_{n}}{n}\right)\chi^{\prime\prime}\left(\frac{c\zeta}{n}\right)e^{i\lambda\xi}v(\zeta)\\\
&\quad+\frac{2c}{n^{3/2}}\chi\left(\frac{\xi-
x_{n}}{n}\right)\chi^{\prime}\left(\frac{c\zeta}{n}\right)e^{i\lambda\xi}v^{\prime}(\zeta).\end{split}$

The Cauchy-Schwarz inequality yields

(5.10)
$\begin{split}\bigg{|}&(\tilde{u},-\Delta\tilde{\psi}_{n})_{\mathbb{R}^{2}\setminus\widetilde{\Gamma}}+\left(z^{2}-\lambda^{2}\right)(\tilde{u},\tilde{\psi}_{n})_{\mathbb{R}^{2}}\bigg{|}\\\
&=\bigg{|}\int_{\mathbb{R}^{2}}\tilde{u}(\xi,\zeta)\overline{\bigg{(}\frac{1}{n^{5/2}}\chi^{\prime\prime}\left(\frac{\xi-
x_{n}}{n}\right)\chi\left(\frac{c\zeta}{n}\right)e^{i\lambda\xi}v(\zeta)+\frac{2i\lambda}{n^{3/2}}\chi^{\prime}\left(\frac{\xi-
x_{n}}{n}\right)\chi\left(\frac{c\zeta}{n}\right)e^{i\lambda\xi}v(\zeta)}\\\
&\quad\qquad\overline{+\frac{c^{2}}{n^{5/2}}\chi\left(\frac{\xi-
x_{n}}{n}\right)\chi^{\prime\prime}\left(\frac{c\zeta}{n}\right)e^{i\lambda\xi}v(\zeta)+\frac{2c}{n^{3/2}}\chi\left(\frac{\xi-
x_{n}}{n}\right)\chi^{\prime}\left(\frac{c\zeta}{n}\right)e^{i\lambda\xi}v^{\prime}(\zeta)\bigg{)}}d\xi
d\zeta\bigg{|}\\\ &\leq\frac{C}{n}\|u\|_{\mathbb{R}^{2}}.\end{split}$

The absolute value of the expression $I$ in equation (5.8) can thus be
estimated by equations (5.9) and (5.10) and we have

(5.11)
$|I|\leq\frac{C}{n}\|u\|_{\mathbb{R}^{2}}\leq\frac{C}{n}\left(\|S_{\tau}u\|^{2}_{\mathbb{R}^{2}}+\|u\|^{2}_{\mathbb{R}^{2}}\right)^{1/2},$

which is claim (iv). The proof of the proposition is therefore complete. ∎

In analogy to Corollary 5.3, it is possible to combine Corollary 4.2 and
Propositions 5.4 and 5.5 in order to show the following result:

######  Corollary 5.6.

Let $\tau\in(-\infty,0)\setminus\\{-2\\}$ and let $A_{\tau,z}$ be any self-
adjoint extension of $S_{\tau}$ in (1.3). Then

$\sigma_{\scriptstyle\text{\rm
ess}}(A_{\tau,z})=\bigg{(}-\infty,-m\left|\frac{\tau^{2}-4}{\tau^{2}+4}\right|\bigg{]}\cup\bigg{[}m\left|\frac{\tau^{2}-4}{\tau^{2}+4}\right|,\infty\bigg{)}.$

##  6\. Discrete spectrum

In this section we show the statements about the discrete spectrum of any
self-adjoint extension of $S_{\tau}$ defined in (1.3) in Theorem 1.1. We begin
with the simpler case of a positive interaction strength.

######  Proposition 6.1.

Let $\tau\in(0,\infty)\setminus\\{2\\}$, $\omega\in(0,\frac{\pi}{2})$, and let
$A_{\tau,z}$ be any self-adjoint extension of $S_{\tau}$ in (1.3). Then
$A_{\tau,z}$ has at most one discrete eigenvalue of multiplicity one.

###### Proof.

First, Proposition 5.1, Corollary 5.3, and Theorem 3.2 imply that
$\sigma(S_{\tau}^{*}S_{\tau})=[m^{2},\infty)$. By Corollary 5.3 one has
$\sigma_{\rm ess}(A_{\tau,z})=(-\infty,-m]\cup[m,\infty)$. Recall that by
Theorem 2.2 the deficiency indices of $S_{\tau}$ are $(1,1)$. Hence, by
Proposition 5.1 the resolvent difference
$(S_{\tau}^{*}S_{\tau}+1)^{-1}-\left[(A_{\tau,z})^{2}+1\right]^{-1}$ is a
rank-one operator. The perturbation result [13, §9.3, Theorem 3] implies that
the dimension of the spectral subspace of the self-adjoint operator
$(A_{\tau,z})^{2}$ corresponding to the interval $(0,m^{2})$ is at most one.
As the spectral subspace for $(A_{\tau,z})^{2}$ corresponding to the interval
$(0,m^{2})$ coincides with the spectral subspace for $A_{\tau,z}$
corresponding to the interval $(-m,m)$, the result follows. ∎

######  Remark 6.2.

As a consequence of the above proposition and of the symmetry of the spectrum
of the distinguished self-adjoint extension $A_{\tau,1}$ stated in Theorem
4.1, we infer that for any $\tau\in(0,\infty)\setminus\\{2\\}$ the discrete
spectrum of $A_{\tau,1}$ in the gap $(-m,m)$ can either be empty or consist of
one simple eigenvalue at $0$.

The case where $\tau<0$ is more complicated. The proof of the existence of
discrete eigenvalues is based on the min-max principle and on the construction
of a family of test functions. This is a modification of the approach of [24],
applied there to Schrödinger operators with $\delta$-interactions supported on
broken lines. We remark that the angle $\omega_{\star}$ that is implicitly
given in the proposition below is computed explicitly after the proof. Since
the explicit value is rather complicated, we prefer to state the implicit
value here.

######  Proposition 6.3.

Let $\tau\in(-\infty,0)\setminus\\{-2\\}$ and $N\in{\mathbb{N}}$. Define the
number $\omega_{\star}\in(0,\frac{\pi}{2})$ by

$\omega_{\star}=\omega_{\star}(\tau,N)\mathrel{\mathop{\mathchar
58\relax}}=\max_{L\in(0,\infty)}\arctan\left\\{\frac{N^{2}\pi^{2}(4+\tau^{2})^{2}(16\tau^{2}+(4+\tau^{2})^{2})-8m^{2}L^{2}\tau^{2}(4-\tau^{2})^{2}}{(2N^{2}\pi^{2}+m^{2}L^{2})[(4-\tau^{2})^{2}+(4+\tau^{2})^{2}]4m\tau
L(4+\tau^{2})}\right\\}.$

If $\omega\in(0,\omega_{\star}]$, then any self-adjoint extension $A_{\tau,z}$
of $S_{\tau}$ has at least $N$ discrete eigenvalues with multiplicities taken
into account.

###### Proof.

Our strategy is to apply the min-max principle to the non-negative operator
$(A_{\tau,z})^{2}$ using a family of test functions lying in the domain of the
symmetric operator $S_{\tau}$. The test functions will be supported in the
strip $\Pi_{L}\mathrel{\mathop{\mathchar 58\relax}}=[L,2L]\times\mathbb{R}$
for some $L>0$. We will also use the notation $d\mathrel{\mathop{\mathchar
58\relax}}=L\tan(\omega)$. The intersection $\Pi_{L}\cap\Omega_{-}$ is the
union of two disjoint domains: the subdomain of $\Pi_{L}\cap\Omega_{-}$ which
lies above $\Gamma_{l}$ will be denoted by $\Omega^{l}_{-}$ and the subdomain
of $\Pi_{L}\cap\Omega_{-}$ below $\Gamma_{r}$ will be denoted by
$\Omega^{r}_{-}$; see Figure 3.

Figure 3. Decomposition of $\mathbb{R}^{2}$ used for the test functions
$u_{n}(x,y)$. The subdomains $\Omega_{-}^{l}$ and $\Omega_{-}^{r}$ are
depicted in light gray.

Recall that by Corollary 5.6 the essential spectra of $(A_{\tau,z})^{2}$ and
of $S_{\tau}^{*}S_{\tau}$ are identified with

(6.1) $\displaystyle\sigma_{\rm ess}((A_{\tau,z})^{2})=\sigma_{\rm
ess}(S_{\tau}^{*}S_{\tau})=\big{[}\varepsilon_{\tau}^{2},\infty\big{)},\qquad\text{where}\,\,\,\varepsilon_{\tau}=m\frac{\tau^{2}-4}{\tau^{2}+4}.$

Fix $N\in\mathbb{N}$ and define the following abbreviations:

$\displaystyle a\mathrel{\mathop{\mathchar
58\relax}}=\frac{4+\tau^{2}}{4-\tau^{2}}\hskip 20.00003pt\text{ and }\hskip
20.00003ptb\mathrel{\mathop{\mathchar 58\relax}}=\frac{4\tau}{4-\tau^{2}}.$

We introduce a family of test functions
$u_{n}\colon\mathbb{R}^{2}\rightarrow\mathbb{C}^{2}$ of the form
$u_{n}(x,y)\mathrel{\mathop{\mathchar 58\relax}}=f_{n}(x)g(y)h(x,y)$,
$n\in\\{1,2,\dots,N\\}$. Here, the functions
$f_{n}\colon\mathbb{R}\rightarrow\mathbb{R}$, $n\in\\{1,2,\dots,N\\}$, are
defined by

$f_{n}(x)=\sin\left(\frac{2n\pi}{L}x\right)\cdot\chi_{{}_{\scriptstyle[L,2L]}}(x),$

where $\chi_{[L,2L]}$ is the characteristic function of the interval $[L,2L]$.
Furthermore, the function $g\mathchar 58\relax\mathbb{R}\rightarrow\mathbb{R}$
is introduced by

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

with $\gamma=-\frac{4m\tau}{4+\tau^{2}}$. Finally, the function $h\mathchar
58\relax\mathbb{R}^{2}\rightarrow\mathbb{C}^{2}$ is given by

$h(x,y)=\begin{cases}\begin{pmatrix}1\\\
0\end{pmatrix},&(x,y)\in\Omega_{+},\\\ \begin{pmatrix}a\\\
be^{i\omega}\end{pmatrix},&(x,y)\in\Omega^{l}_{-},\\\ \begin{pmatrix}a\\\
-be^{-i\omega}\end{pmatrix},&(x,y)\in\Omega^{r}_{-}.\end{cases}$

The vector-valued function $h$ ensures that the transmission condition for
$\operatorname{dom}S_{\tau}$ is satisfied, and so by construction
$\\{u_{n}\\}_{n=1}^{N}\subset\operatorname{dom}S_{\tau}$. Since the functions
$f_{n}$ are linearly independent, also the functions $\\{u_{n}\\}_{n=1}^{N}$
are linearly independent and the subspace
$\mathcal{L}_{N}\mathrel{\mathop{\mathchar 58\relax}}={\rm
span}\,\\{u_{1},u_{2},\dots,u_{N}\\}$ of $\operatorname{dom}S_{\tau}$ has
dimension $N$. A generic element $u\in\mathcal{L}_{N}$ is therefore given by
$u(x,y)=\sum_{n=1}^{N}c_{n}u_{n}(x,y)$ with some complex coefficients
$\\{c_{n}\\}_{n=1}^{N}$. We are going to show that, for
$\omega\in(0,\omega_{\star}]$ and the corresponding $L$ that maximizes the
expression in the definition of $\omega_{\star}$, one has
$\|S_{\tau}u\|_{\mathbb{R}^{2}}^{2}<\varepsilon_{\tau}^{2}\|u\|_{\mathbb{R}^{2}}^{2}$
for all $u\in\mathcal{L}_{N}$.

We will compute all the terms in $\|S_{\tau}u\|^{2}_{{\mathbb{R}}^{2}}$ from
Theorem 3.2 for $u\in\mathcal{L}_{N}$. Begin by noticing that

$\displaystyle\int_{\Gamma_{l}}\big{|}u_{+}|_{\Gamma_{l}}-u_{-}|_{\Gamma_{l}}\big{|}^{2}ds=\int_{\Gamma_{r}}\big{|}u_{+}|_{\Gamma_{r}}-u_{-}|_{\Gamma_{r}}\big{|}^{2}ds.$

Recall that the edge $\Gamma_{l}$ is parametrized by
$\\{(s\cos(\omega),s\sin(\omega))\mathchar 58\relax s\in\mathbb{R}_{+}\\}$. We
calculate that

$\displaystyle c(\tau)\mathrel{\mathop{\mathchar
58\relax}}=\Bigg{|}\begin{pmatrix}a-1\\\
be^{i\omega}\end{pmatrix}\Bigg{|}^{2}=(a-1)^{2}+b^{2}=\frac{4\tau^{2}(\tau^{2}+4)}{(\tau^{2}-4)^{2}}.$

Hence, we find that

$\displaystyle\int_{\Gamma_{l}}\big{|}u_{+}|_{\Gamma_{l}}-u_{-}|_{\Gamma_{l}}\big{|}^{2}ds$
$\displaystyle=\int_{{}_{\scriptstyle
L/\cos(\omega)}}^{2L/\cos(\omega)}c(\tau)\Bigg{|}\sum_{n=1}^{N}c_{n}\sin\left(\frac{2ns\pi}{L}\cos(\omega)\right)\Bigg{|}^{2}ds.$

The change of variables $t=(s\pi\cos(\omega))/L$ and the orthogonality of the
functions $\sin(2nt)$, $n\in\\{1,\dots,N\\}$, in $L^{2}(\pi,2\pi)$ then yields

$\displaystyle\int_{\Gamma_{l}}\big{|}u_{+}|_{\Gamma_{l}}-u_{-}|_{\Gamma_{l}}\big{|}^{2}ds$
$\displaystyle=c(\tau)\frac{L}{\pi\cos(\omega)}\int_{\pi}^{2\pi}\Big{|}\sum_{n=1}^{N}c_{n}\sin(2nt)\Big{|}^{2}dt$
$\displaystyle=c(\tau)\frac{L}{\pi\cos(\omega)}\sum_{n=1}^{N}\int_{\pi}^{2\pi}|c_{n}|^{2}\cdot|\sin(2nt)|^{2}dt$
$\displaystyle=c(\tau)\frac{L}{\pi\cos(\omega)}\sum_{n=1}^{N}\frac{\pi}{2}|c_{n}|^{2}=\frac{c(\tau)L}{2\cos(\omega)}\sum_{n=1}^{N}|c_{n}|^{2}.$

This immediately implies that

(6.2)
$\displaystyle\big{\|}u_{+}|_{\Gamma}-u_{-}|_{\Gamma}\big{\|}^{2}_{\Gamma}=2\int_{\Gamma_{l}}\big{|}u_{+}|_{\Gamma_{l}}-u_{-}|_{\Gamma_{l}}\big{|}^{2}ds=\frac{c(\tau)L}{\cos(\omega)}\sum_{n=1}^{N}|c_{n}|^{2}.$

For the calculation of $\|u\|^{2}_{\mathbb{R}^{2}}$, we define

$\displaystyle\kappa(\tau)=a^{2}+b^{2}=\frac{(4+\tau^{2})^{2}+16\tau^{2}}{(4-\tau^{2})^{2}},$

and notice that due to the symmetry of $|u|$ with respect to the $x$-axis one
has

$\|u\|^{2}_{{\mathbb{R}}^{2}}=2\int_{\Pi_{L}^{+}}|u|^{2}dxdy,$

where $\Pi_{L}^{+}\mathrel{\mathop{\mathchar
58\relax}}=[L,2L]\times[0,\infty)$. We define three disjoint subregions of
$\Pi_{L}^{+}$ by

$\displaystyle\Pi_{L,1}^{+}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\Pi_{L}^{+}\cap\Omega_{+},$ $\displaystyle\Pi_{L,2}^{+}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\Pi_{L}^{+}\cap\\{(x,y)\in\Omega^{l}_{-}\colon y\leq 2d\\},$
$\displaystyle\Pi_{L,3}^{+}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\Pi_{L}^{+}\cap\\{(x,y)\in\Omega^{l}_{-}\colon y>2d\\}.$

Note that $\overline{\Pi_{L}^{+}}=\overline{\cup_{j=1}^{3}\Pi_{L,j}^{+}}$.
Using the symmetry of $|f_{n}(x)|$ about $x=3L/2$ we obtain that

$\displaystyle\|u\|^{2}_{\mathbb{R}^{2}}$
$\displaystyle=2\int_{\Pi_{L,1}^{+}}|u|^{2}dxdy+2\int_{\Pi_{L,2}^{+}}|u|^{2}dxdy+2\int_{\Pi_{L,3}^{+}}|u|^{2}dxdy$
$\displaystyle=2\int_{L}^{2L}\Big{|}\sum_{n=1}^{N}c_{n}\sin(\frac{2n\pi}{L}x)\Big{|}^{2}dx\left\\{\frac{3L\tan(\omega)}{2}+\frac{L\tan(\omega)}{2}\kappa(\tau)+\int_{2d}^{\infty}\kappa(\tau)e^{-2\gamma(y-2d)}dy\right\\}$
$\displaystyle=2\sum_{n=1}^{N}|c_{n}|^{2}\int_{L}^{2L}\sin^{2}\left(\frac{2n\pi}{L}x\right)dx\left\\{\frac{L\tan(\omega)[3+\kappa(\tau)]}{2}+\frac{\kappa(\tau)}{2\gamma}\right\\}$
$\displaystyle=\left\\{\frac{L^{2}\tan(\omega)[3+\kappa(\tau)]}{2}+\frac{L\kappa(\tau)}{2\gamma}\right\\}\sum_{n=1}^{N}|c_{n}|^{2}.$

Finally, we calculate $\|\gradient u\|^{2}_{\mathbb{R}^{2}\setminus\Gamma}$
for a generic $u\in\mathcal{L}_{N}$. A direct computation yields

$\displaystyle\partial_{x}u(x,y)$
$\displaystyle=\sum_{n=1}^{N}c_{n}\frac{2n\pi}{L}\cos\left(\frac{2n\pi}{L}x\right)\chi_{{}_{\scriptstyle[L,2L]}}(x)g(y)h(x,y),$
$\displaystyle\partial_{y}u(x,y)$
$\displaystyle=-\sum_{n=1}^{N}c_{n}\sin\left(\frac{2n\pi}{L}x\right)\chi_{[L,2L]}(x)\chi_{{}_{\scriptstyle(-\infty,-2d]\cup[2d,\infty)}}(y)\text{sgn}(y)\gamma
e^{-\gamma(|y|-2d)}h(x,y),$

where it was used in between that $h(x,y)$ is piecewise constant. A
calculation similar to that of $\|u\|^{2}_{\mathbb{R}^{2}}$ above then yields

$\displaystyle\|\partial_{x}u\|^{2}_{\mathbb{R}^{2}\setminus\Gamma}$
$\displaystyle=2\sum_{n=1}^{N}|c_{n}|^{2}\left(\frac{2n\pi}{L}\right)^{2}\int_{L}^{2L}\cos^{2}\left(\frac{2n\pi}{L}x\right)dx\left\\{\frac{L\tan(\omega)[3+\kappa(\tau)]}{2}+\frac{\kappa(\tau)}{2\gamma}\right\\}$
$\displaystyle=\sum_{n=1}^{N}|c_{n}|^{2}\frac{(2n\pi)^{2}}{L}\left\\{\frac{L\tan(\omega)[3+\kappa(\tau)]}{2}+\frac{\kappa(\tau)}{2\gamma}\right\\}.$

Furthermore, we obtain that

$\displaystyle\|\partial_{y}u\|^{2}_{\mathbb{R}^{2}\setminus\Gamma}$
$\displaystyle=\sum_{n=1}^{N}|c_{n}|^{2}\int_{L}^{2L}\sin^{2}\left(\frac{2n\pi}{L}x\right)dx\cdot
2\int_{2d}^{\infty}\gamma^{2}\kappa(\tau)e^{-2\gamma(y-2d)}dy$
$\displaystyle=\sum_{n=1}^{N}|c_{n}|^{2}\cdot\frac{L}{2}\cdot\gamma\kappa(\tau).$

After subtracting the bottom of the essential spectrum, we finally can
calculate that

$\displaystyle\|S_{\tau}u\|^{2}_{\mathbb{R}^{2}}-\varepsilon_{\tau}^{2}\|u\|^{2}_{\mathbb{R}^{2}}$
$\displaystyle=\|\gradient
u\|^{2}_{\mathbb{R}^{2}\setminus\Gamma}+m^{2}\left[1-\left(\frac{\tau^{2}-4}{\tau^{2}+4}\right)^{2}\right]\|u\|^{2}_{\mathbb{R}^{2}}+\frac{2m}{\tau}\big{\|}u_{+}|_{\Gamma}-u_{-}|_{\Gamma}\big{\|}^{2}_{\Gamma}$
$\displaystyle=\sum_{n=1}^{N}|c_{n}|^{2}\Bigg{\\{}\frac{(2n\pi)^{2}}{L}\left\\{\frac{L\tan(\omega)[3+\kappa(\tau)]}{2}+\frac{\kappa(\tau)}{2\gamma}\right\\}+\frac{L\gamma\kappa(\tau)}{2}$
$\displaystyle\qquad+\frac{16m^{2}\tau^{2}}{(\tau^{2}+4)^{2}}\left\\{\frac{L^{2}\tan(\omega)[3+\kappa(\tau)]}{2}+\frac{L\kappa(\tau)}{2\gamma}\right\\}+\frac{2mLc(\tau)}{\tau\cos(\omega)}\Bigg{\\}}.$

Next, by using that for $\omega<\frac{\pi}{2}$ one has $\cos\omega<1$,
$\frac{8\tau^{2}}{(\tau^{2}+4)^{2}}\leq 1$, and

$\displaystyle\frac{L\kappa(\tau)}{2\gamma}$
$\displaystyle\left[\frac{16m^{2}\tau^{2}}{(\tau^{2}+4)^{2}}+\gamma^{2}\right]+\frac{2mLc(\tau)}{\tau}=\frac{L\kappa(\tau)}{2\gamma}\cdot
2\gamma^{2}+\frac{2mLc(\tau)}{\tau}$
$\displaystyle=\frac{4mL\tau}{(4-\tau^{2})^{2}(4+\tau^{2})}\left[-16\tau^{2}+(4+\tau^{2})^{2}\right]=\frac{4mL\tau}{(4-\tau^{2})^{2}(4+\tau^{2})}(4-\tau^{2})^{2}=\frac{4mL\tau}{4+\tau^{2}},$

we can estimate for $u\in\mathcal{L}_{N}\setminus\\{0\\}$

(6.3)
$\displaystyle\|S_{\tau}u\|^{2}_{\mathbb{R}^{2}}-\varepsilon_{\tau}^{2}\|u\|^{2}_{\mathbb{R}^{2}}<\sum_{n=1}^{N}|c_{n}|^{2}\left\\{\tan(\omega)[3+\kappa(\tau)](2N^{2}\pi^{2}+m^{2}L^{2})+\frac{4mL\tau}{4+\tau^{2}}+\frac{2N^{2}\pi^{2}\kappa(\tau)}{L\gamma}\right\\}.$

Hence, if we choose for $L>0$

$\begin{split}\omega&=\omega(L)\mathrel{\mathop{\mathchar
58\relax}}=\arctan\left\\{\frac{-4mL^{2}\tau\gamma-2N^{2}\pi^{2}\kappa(\tau)(4+\tau^{2})}{(2N^{2}\pi^{2}+m^{2}L^{2})[3+\kappa(\tau)]L\gamma(4+\tau^{2})}\right\\}\\\
&=\arctan\left\\{\frac{N^{2}\pi^{2}(4+\tau^{2})^{2}(16\tau^{2}+(4+\tau^{2})^{2})-8m^{2}L^{2}\tau^{2}(4-\tau^{2})^{2}}{(2N^{2}\pi^{2}+m^{2}L^{2})[(4-\tau^{2})^{2}+(4+\tau^{2})^{2}]4m\tau
L(4+\tau^{2})}\right\\},\end{split}$

then the right hand side in (6.3) is zero, implying that
$\|S_{\tau}u\|^{2}_{\mathbb{R}^{2}}-\varepsilon_{\tau}^{2}\|u\|^{2}_{\mathbb{R}^{2}}<0$
for all $u\in\mathcal{L}_{N}\setminus\\{0\\}$. Note that the function
$\omega(L)$ has a unique positive maximum for $L>0$ and hence, this claim
remains true for the maximum $\omega_{\star}\mathrel{\mathop{\mathchar
58\relax}}=\max_{L>0}\omega(L)$.

Let $A_{\tau,z}$ be any self-adjoint extension of $S_{\tau}$. The inclusion
$\mathcal{L}_{N}\subset\operatorname{dom}A_{\tau,z}$ then holds, as
$\operatorname{dom}S_{\tau}\subset\operatorname{dom}A_{\tau,z}$. Let
$L_{\star}>0$ denote the argument for which the maximum for $\omega_{\star}$
is attained. Then, for any $u\in\mathcal{L}_{N}\setminus\\{0\\}$ and
$\omega\in(0,\omega_{\star}]$, we get

$\|A_{\tau,z}u\|^{2}_{\mathbb{R}^{2}}-\varepsilon_{\tau}^{2}\|u\|_{\mathbb{R}^{2}}^{2}=\|S_{\tau}u\|^{2}_{\mathbb{R}^{2}}-\varepsilon_{\tau}^{2}\|u\|_{\mathbb{R}^{2}}^{2}<0.$

The characterization of the essential spectrum in (6.1) and the min-max
principle then imply that the self-adjoint operator $(A_{\tau,z})^{2}$ has at
least $N$ discrete eigenvalues, with multiplicities taken into account. Thus,
we conclude by Corollary 5.6 that $A_{\tau,z}$ has the same lower bound on the
number of discrete eigenvalues with multiplicities taken into account. ∎

The maximum in Proposition 6.3 can be found explicitly. Let us introduce the
functions

$\displaystyle F(\tau)$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=N^{2}\pi^{2}(4+\tau^{2})^{2}(16\tau^{2}+(4+\tau^{2})^{2}),\qquad
G(\tau)\mathrel{\mathop{\mathchar
58\relax}}=[(4-\tau^{2})^{2}+(4+\tau^{2})^{2}]4|\tau|(4+\tau^{2}),$
$\displaystyle H(\tau)$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=8\tau^{2}(4-\tau^{2})^{2}.$

Clearly, the maximum is attained for $L>0$ satisfying the condition
$m^{2}L^{2}H(\tau)>F(\tau)$. Let us introduce a new variable $x>0$ through the
relation $x=m^{2}L^{2}H(\tau)-F(\tau)$. Then we have

$\omega_{\star}=\max_{x\in(0,\infty)}\arctan\left\\{\frac{xH(\tau)^{3/2}}{G(\tau)\big{(}2N^{2}\pi^{2}H(\tau)+F(\tau)+x\big{)}\sqrt{F(\tau)+x}}\right\\}.$

Thus, we need to find the point $x>0$ that maximizes the function

$J(x)\mathrel{\mathop{\mathchar
58\relax}}=\frac{x}{\big{(}2N^{2}\pi^{2}H(\tau)+F(\tau)+x\big{)}\sqrt{F(\tau)+x}}.$

Differentiating with respect to $x$ we find

$\displaystyle
J^{\prime}(x)=\frac{-x^{2}+x\big{[}2N^{2}\pi^{2}H(\tau)+F(\tau)\big{]}+2F(\tau)\big{[}2N^{2}\pi^{2}H(\tau)+F(\tau)\big{]}}{2\big{[}2N^{2}\pi^{2}H(\tau)+F(\tau)+x\big{]}^{2}(F(\tau)+x)^{3/2}}.$

The maximum of $J(x)$ is located at the unique positive solution of the
equation $J^{\prime}(x)=0$, which is given by

$x_{\star}(\tau)=N^{2}\pi^{2}H(\tau)+\frac{F(\tau)}{2}+\sqrt{\left(N^{2}\pi^{2}H(\tau)+\frac{F(\tau)}{2}\right)^{2}+4F(\tau)\left(N^{2}\pi^{2}H(\tau)+\frac{F(\tau)}{2}\right)}.$

Hence, we arrive at

$\omega_{\star}=\arctan\left\\{\frac{x_{\star}(\tau)H(\tau)^{3/2}}{G(\tau)\big{(}2N^{2}\pi^{2}H(\tau)+F(\tau)+x_{\star}(\tau)\big{)}\sqrt{F(\tau)+x_{\star}(\tau)}}\right\\}.$

From this formula, one gets that $\omega_{\star}\rightarrow 0$ in the limits
$\tau\rightarrow 0^{-}$, $\tau\rightarrow-2$, and $\tau\rightarrow-\infty$ for
any fixed $N\in\mathbb{N}$.

##  Appendix A On a Sobolev space with a piecewise constant weight

Recall that $M$ is defined by (1.4) and that $\partial_{t}$ is given by (3.1).
In this appendix we prove for any $u\in\operatorname{dom}S_{\tau}$ the
identity

(A.1) $\displaystyle\big{\langle}Mu_{+}|_{\Gamma},$ $\displaystyle
i\sigma_{3}(\partial_{t}(Mu_{+}|_{\Gamma})\big{\rangle}_{H^{1/2}(\Gamma;\mathbb{C}^{2})\times
H^{-1/2}(\Gamma;\mathbb{C}^{2})}$
$\displaystyle=\big{\langle}M^{-1}Mu_{+}|_{\Gamma},i\sigma_{3}\partial_{t}(u_{+}|_{\Gamma})\big{\rangle}_{H^{1/2}(\Gamma;\mathbb{C}^{2})\times
H^{-1/2}(\Gamma;\mathbb{C}^{2})},$

which appears in the proof of Theorem 3.2 in equation (3.5). Note that
$u_{+}|_{\Gamma}\in H^{1/2}(\Gamma;\mathbb{C}^{2})$ and
$Mu_{+}|_{\Gamma}=u_{-}|_{\Gamma}\in H^{1/2}(\Gamma;\mathbb{C}^{2})$. Let
$\gamma\colon\mathbb{R}\rightarrow\mathbb{R}^{2}$ be an arc length
parametrization of $\Gamma$ such that $\gamma(\mathbb{R}_{-})=\Gamma_{r}$ and
$\gamma(\mathbb{R}_{+})=\Gamma_{l}$, set
$\widetilde{\nu}\mathrel{\mathop{\mathchar 58\relax}}=\nu\circ\gamma$, and
define for $s\in[0,1]$ the weighted Sobolev space

(A.2)
$H^{s}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})\mathrel{\mathop{\mathchar
58\relax}}=\big{\\{}f\in H^{s}(\mathbb{R};\mathbb{C}^{2})\mathchar 58\relax
i\sigma_{3}(\sigma\cdot\widetilde{\nu})f\in
H^{s}(\mathbb{R};\mathbb{C}^{2})\big{\\}},$

which is a Hilbert space when endowed with the norm

$\|f\|_{H^{s}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})}^{2}\mathrel{\mathop{\mathchar
58\relax}}=\|f\|_{H^{s}(\mathbb{R};\mathbb{C}^{2})}^{2}+\|i\sigma_{3}(\sigma\cdot\widetilde{\nu})f\|_{H^{s}(\mathbb{R};\mathbb{C}^{2})}^{2},\qquad
f\in H^{s}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2}).$

The definition of $M$ in (1.4) then implies that
$(u_{+}|_{\Gamma})\circ\gamma\in
H^{1/2}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})$. Hence, we conclude that
(A.1) is true if we can show the following proposition:

######  Proposition A.1.

Let $f,g\in H^{1/2}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})$. Then,

$\begin{split}\big{\langle}(i\sigma_{3}(\sigma\cdot\widetilde{\nu})f)^{\prime},g&\big{\rangle}_{H^{-1/2}(\mathbb{R};\mathbb{C}^{2})\times
H^{1/2}(\mathbb{R};\mathbb{C}^{2})}\\\
&=\big{\langle}f^{\prime},i\sigma_{3}(\sigma\cdot\widetilde{\nu})g\big{\rangle}_{H^{-1/2}(\mathbb{R};\mathbb{C}^{2})\times
H^{1/2}(\mathbb{R};\mathbb{C}^{2})}\end{split}$

holds.

For the proof of Proposition A.1, some preliminary considerations on the
spaces $H^{s}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})$ are necessary.
First, we consider the cases $s=0$ and $s=1$. In the following, we denote by
$[X,Y]_{\theta}$, $\theta\in[0,1]$, the interpolation space between two
Hilbert spaces $X\subset Y$ with continuous and dense embedding; see [30,
Chapter I, §15] and also [31, Appendix B].

######  Lemma A.2.

Let the space $H^{s}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})$,
$s\in[0,1]$, be defined as in (A.2). Then, the following holds:

  * (i)

$H^{0}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})=L^{2}(\mathbb{R};\mathbb{C}^{2})$,
with equivalent norms, and multiplication by
$i\sigma_{3}(\sigma\cdot\widetilde{\nu})$ defines a bounded map in
$L^{2}(\mathbb{R};\mathbb{C}^{2})$.

  * (ii)

$H^{1}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})=H^{1}_{0}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})$,
with equivalent norms, one has

(A.3)
$\big{(}i\sigma_{3}(\sigma\cdot\widetilde{\nu})f\big{)}^{\prime}=i\sigma_{3}(\sigma\cdot\widetilde{\nu})f^{\prime}$

for all $f\in H^{1}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})$, and
multiplication by $i\sigma_{3}(\sigma\cdot\widetilde{\nu})$ defines a bounded
map in $H^{1}_{0}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})$.

In particular, for any $\theta\in[0,1]$ the map

(A.4)
$i\sigma_{3}(\sigma\cdot\widetilde{\nu})\colon\big{[}H^{1}_{0}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2}),L^{2}(\mathbb{R};\mathbb{C}^{2})\big{]}_{\theta}\rightarrow\big{[}H^{1}_{0}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2}),L^{2}(\mathbb{R};\mathbb{C}^{2})\big{]}_{\theta}$

is bounded.

###### Proof.

(i) Since $|\widetilde{\nu}|=1$, the anti-commutation relations in (1.6) imply
that $i\sigma_{3}(\sigma\cdot\widetilde{\nu})$ is pointwise unitary. Hence,
the claims in item (i) are clear.

(ii) Let $f\in H^{1}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})$. Then, by
definition $f\in H^{1}(\mathbb{R};\mathbb{C}^{2})$ and
$i\sigma_{3}(\sigma\cdot\widetilde{\nu})f\in
H^{1}(\mathbb{R};\mathbb{C}^{2})$. This implies for any $\varphi\in
C_{0}^{\infty}(\mathbb{R};\mathbb{C}^{2})$ that

$\begin{split}\int_{\mathbb{R}}&\big{(}i\sigma_{3}(\sigma\cdot\widetilde{\nu})f\big{)}^{\prime}\varphi
dx=-\int_{\mathbb{R}}i\sigma_{3}(\sigma\cdot\widetilde{\nu})f\varphi^{\prime}dx\\\
&=\int_{\mathbb{R_{-}}}i\sigma_{3}(\sigma\cdot\widetilde{\nu}_{r})f^{\prime}\varphi
dx+\int_{\mathbb{R_{+}}}i\sigma_{3}(\sigma\cdot\widetilde{\nu}_{l})f^{\prime}\varphi
dx-i\sigma_{3}(\sigma\cdot(\widetilde{\nu}_{r}-\widetilde{\nu}_{l}))f(0)\varphi(0),\end{split}$

where the notation $\widetilde{\nu}_{l/r}=\widetilde{\nu}|_{\mathbb{R}_{+/-}}$
and integration by parts in $\mathbb{R}_{\pm}$ were used in the last step.
Since $\widetilde{\nu}_{l}\neq\widetilde{\nu}_{r}$, the anti-commutation
relations in (1.6) imply that
$i\sigma_{3}(\sigma\cdot(\widetilde{\nu}_{r}-\widetilde{\nu}_{l}))$ is
invertible and hence $f(0)$ must be $0$, i.e. $f\in
H^{1}_{0}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})$. Moreover, the above
calculation shows (A.3) for $f\in
H^{1}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})$.

On the other hand, a similar consideration as above yields for $f\in
H^{1}_{0}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})$ that $f\in
H^{1}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})$ and that (A.3) holds in
this case as well. Eventually, since $i\sigma_{3}(\sigma\cdot\widetilde{\nu})$
is pointwise unitary, it follows from (A.3) that the norms in
$H^{1}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})$ and
$H^{1}_{0}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})$ are equivalent.

Finally, as the multiplication by $i\sigma_{3}(\sigma\cdot\widetilde{\nu})$
gives rise to a bounded map in both spaces
$H^{1}_{0}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})$ and
$L^{2}(\mathbb{R};\mathbb{C}^{2})$ by items (ii) and (i), respectively, it
follows by interpolation that the map in (A.4) is well defined and bounded. ∎

In the next lemma we characterize
$H^{1/2}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})$. For this purpose,
recall that the space
$H^{1/2}_{00}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})$ is defined by

$H^{1/2}_{00}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})=\left\\{f\in
H^{1/2}(\mathbb{R};\mathbb{C}^{2})\mathchar
58\relax\int_{\mathbb{R}}\frac{|f(x)|^{2}}{|x|}dx<\infty\right\\}$

and that this space, endowed with the norm

$\|f\|_{H^{1/2}_{00}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})}^{2}\mathrel{\mathop{\mathchar
58\relax}}=\|f\|_{H^{1/2}(\mathbb{R};\mathbb{C}^{2})}^{2}+\int_{\mathbb{R}}\frac{|f(x)|^{2}}{|x|}dx,\qquad
f\in H^{1/2}_{00}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2}),$

is a Hilbert space. We are going to use

(A.5)
$H^{1/2}_{00}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})=\big{[}H^{1}_{0}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2}),L^{2}(\mathbb{R};\mathbb{C}^{2})\big{]}_{1/2},$

cf. [30, Chapter I, Theorem 11.7 and Remark 11.5]. In particular, by
elementary properties of interpolation spaces this implies that
$H^{1}_{0}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})$ is dense in
$H^{1/2}_{00}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})$.

######  Lemma A.3.

Let $H^{1/2}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})$ be defined by
(A.2). Then,
$H^{1/2}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})=H^{1/2}_{00}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})$.

###### Proof.

First, it follows from (A.4) and (A.5) that
$H^{1/2}_{00}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})\subset
H^{1/2}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})$. To show the converse
inclusion, let $f\in H^{1/2}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})$.
Set $\widetilde{\nu}_{r}=\widetilde{\nu}(-1)$ and
$\widetilde{\nu}_{l}=\widetilde{\nu}(1)$. Then, as $\widetilde{\nu}_{r}$ is
constant, $f\in H^{1/2}(\mathbb{R};\mathbb{C}^{2})$ implies that
$i\sigma_{3}(\sigma\cdot\widetilde{\nu}_{r})f\in
H^{1/2}(\mathbb{R};\mathbb{C}^{2})$ and thus also

$i\sigma_{3}(\sigma\cdot(\widetilde{\nu}-\widetilde{\nu}_{r}))f=i\sigma_{3}(\sigma\cdot(\widetilde{\nu}_{l}-\widetilde{\nu}_{r}))\chi_{\mathbb{R}_{+}}f\in
H^{1/2}(\mathbb{R};\mathbb{C}^{2}),$

where $\chi_{\mathbb{R}_{+}}$ denotes the characteristic function for
$\mathbb{R}_{+}$. Since the anti-commutation relations in (1.6) imply that
$i\sigma_{3}(\sigma\cdot(\widetilde{\nu}_{l}-\widetilde{\nu}_{r}))$ is
invertible, we conclude that $\chi_{\mathbb{R}_{+}}f\in
H^{1/2}(\mathbb{R};\mathbb{C}^{2})$. Making use of the equivalent Sobolev-
Slobodeckii norm in $H^{1/2}(\mathbb{R};\mathbb{C}^{2})$, this implies that

$\begin{split}\infty&>\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{|(\chi_{\mathbb{R}_{+}}f)(x)-(\chi_{\mathbb{R}_{+}}f)(y)|^{2}}{(y-x)^{2}}dydx\geq\int_{0}^{\infty}\int_{-\infty}^{0}\frac{|f(x)|^{2}}{(y-x)^{2}}dydx=\int_{0}^{\infty}\frac{|f(x)|^{2}}{x}dx.\end{split}$

In a similar way as above, the relations
$i\sigma_{3}(\sigma\cdot\widetilde{\nu}_{l})f\in
H^{1/2}(\mathbb{R};\mathbb{C}^{2})$ and
$i\sigma_{3}(\sigma\cdot\widetilde{\nu})f\in
H^{1/2}(\mathbb{R};\mathbb{C}^{2})$ imply that $\chi_{\mathbb{R}_{-}}f\in
H^{1/2}(\mathbb{R};\mathbb{C}^{2})$ and

$\begin{split}-\int_{-\infty}^{0}\frac{|f(x)|^{2}}{x}dx<\infty.\end{split}$

Hence, $f\in H^{1/2}_{00}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})$, which
finishes the proof of this lemma. ∎

Now, we are prepared to prove Proposition A.1:

###### Proof of Proposition A.1.

Let $f,g\in H^{1/2}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})$. Since
$H^{1}_{0}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})$ is dense in the space
$H^{1/2}_{00}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})=H^{1/2}_{\widetilde{\nu}}(\mathbb{R};\mathbb{C}^{2})$,
there exists a sequence $\\{f_{n}\\}_{n\in\mathbb{N}}\subset
H^{1}_{0}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})$ such that

$\|f_{n}-f\|_{H^{1/2}_{00}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})}\rightarrow
0,\quad\text{as}\quad n\rightarrow\infty.$

Taking the definition of the norm in
$H^{1/2}_{00}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})$ into account this
implies, in particular, that $f_{n}\rightarrow f$ in
$H^{1/2}(\mathbb{R};\mathbb{C}^{2})$ and thus, also $f_{n}^{\prime}\rightarrow
f^{\prime}$ in $H^{-1/2}(\mathbb{R};\mathbb{C}^{2})$.

Next, as the multiplication by $i\sigma_{3}(\sigma\cdot\widetilde{\nu})$ gives
rise to a bounded operator in
$H^{1/2}_{00}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})$ by (A.4) and (A.5),
we get that

$\big{\|}i\sigma_{3}(\sigma\cdot\widetilde{\nu})(f_{n}-f)\big{\|}_{H^{1/2}(\mathbb{R};\mathbb{C}^{2})}\leq\big{\|}i\sigma_{3}(\sigma\cdot\widetilde{\nu})(f_{n}-f)\big{\|}_{H^{1/2}_{00}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})}\rightarrow
0,\quad\text{as}\quad n\rightarrow\infty.$

In particular, $i\sigma_{3}(\sigma\cdot\widetilde{\nu})f_{n}\rightarrow
i\sigma_{3}(\sigma\cdot\widetilde{\nu})f$ in
$H^{1/2}(\mathbb{R};\mathbb{C}^{2})$ and
$(i\sigma_{3}(\sigma\cdot\widetilde{\nu})f_{n})^{\prime}\rightarrow(i\sigma_{3}(\sigma\cdot\widetilde{\nu})f)^{\prime}$
in $H^{-1/2}(\mathbb{R};\mathbb{C}^{2})$. Hence, using (A.3), we conclude

$\begin{split}\big{\langle}(i\sigma_{3}(\sigma\cdot\widetilde{\nu})f)^{\prime},g&\big{\rangle}_{H^{-1/2}(\mathbb{R};\mathbb{C}^{2})\times
H^{1/2}(\mathbb{R};\mathbb{C}^{2})}\\\
&=\lim_{n\rightarrow\infty}\big{\langle}(i\sigma_{3}(\sigma\cdot\widetilde{\nu})f_{n})^{\prime},g\big{\rangle}_{H^{-1/2}(\mathbb{R};\mathbb{C}^{2})\times
H^{1/2}(\mathbb{R};\mathbb{C}^{2})}\\\
&=\lim_{n\rightarrow\infty}\int_{\mathbb{R}}i\sigma_{3}(\sigma\cdot\widetilde{\nu})f_{n}^{\prime}\overline{g}dx=\lim_{n\rightarrow\infty}\int_{\mathbb{R}}f_{n}^{\prime}\overline{i\sigma_{3}(\sigma\cdot\widetilde{\nu})g}dx\\\
&=\lim_{n\rightarrow\infty}\big{\langle}f_{n}^{\prime},i\sigma_{3}(\sigma\cdot\widetilde{\nu})g\big{\rangle}_{H^{-1/2}(\mathbb{R};\mathbb{C}^{2})\times
H^{1/2}(\mathbb{R};\mathbb{C}^{2})}\\\
&=\big{\langle}f^{\prime},i\sigma_{3}(\sigma\cdot\widetilde{\nu})g\big{\rangle}_{H^{-1/2}(\mathbb{R};\mathbb{C}^{2})\times
H^{1/2}(\mathbb{R};\mathbb{C}^{2})},\end{split}$

which is the claimed identity. ∎

######  Remark A.4.

The property $u_{\pm}|_{\Gamma}\in H^{1/2}(\mathbb{R};\mathbb{C}^{2})$ for any
$u\in\operatorname{dom}S_{\tau}$ immediately follows from the trace theorem.
However, the construction in this appendix obtains the stronger statement that
$(u_{\pm}|_{\Gamma}\circ\gamma)\in
H^{1/2}_{00}(\mathbb{R}\setminus\\{0\\};\mathbb{C}^{2})$ for any
$u\in\operatorname{dom}S_{\tau}$ as a by-product. As before we identify
$\Gamma$ with $\mathbb{R}$ here in such a way that the origin of the real axis
coincides with the corner of the broken line.

##  Appendix B Bound for a one-dimensional auxiliary problem

Define for $\gamma>0$ and $\tau\in(-\infty,0)\setminus\\{-2\\}$ in
$L^{2}((-\gamma,\gamma);\mathbb{C}^{2})$ the semibounded and closed quadratic
form $\mathfrak{d}_{\gamma}$ by

$\begin{split}\mathfrak{d}_{\gamma}[f]&=\int_{(-\gamma,\gamma)\setminus\\{0\\}}|f^{\prime}(x)|^{2}dx+m^{2}\int_{(-\gamma,\gamma)}|f(x)|^{2}dx+\frac{2m}{\tau}|f(0^{+})-f(0^{-})|^{2},\\\
\operatorname{dom}\mathfrak{d}_{\gamma}&=\big{\\{}u\in
H^{1}((-\gamma,\gamma)\setminus\\{0\\};\mathbb{C}^{2})\colon
f(0^{-})=\widetilde{M}f(0^{+})\big{\\}},\end{split}$

where $\widetilde{M}$ is given by (5.5). This appendix is devoted to the proof
that for all $f\in\operatorname{dom}\mathfrak{d}_{\gamma}$ one has the bound

(B.1) $\begin{split}\mathfrak{d}_{\gamma}[f]\geq
E(\gamma)\int_{(-\gamma,\gamma)}|f(x)|^{2}dx\end{split}$

with $E(\gamma)\in[0,m^{2})$ such that

(B.2)
$\lim_{\gamma\rightarrow\infty}E(\gamma)=m^{2}\left(\frac{\tau^{2}-4}{\tau^{2}+4}\right)^{2},$

which is used in (5.7) in the proof of Proposition 5.4. To verify (B.1) we
follow closely the proof of the very similar result [26, Lemma 4.13], but
since the situation in [26] is slightly different from ours and for
completeness we give a full proof here.

The strategy to show (B.1)–(B.2) is to determine the self-adjoint operator
$D_{\gamma}$ that is associated with $\mathfrak{d}_{\gamma}$ via Kato’s first
representation theorem and to estimate its spectrum. As
$\operatorname{dom}D_{\gamma}\subset\operatorname{dom}\mathfrak{d}_{\gamma}$
is compactly embedded in $L^{2}((-\gamma,\gamma);\mathbb{C}^{2})$, its
spectrum is purely discrete and hence, it suffices to estimate the smallest
eigenvalue $E(\gamma)$ of $D_{\gamma}$. Since one has
$C_{0}^{\infty}((-\gamma,\gamma)\setminus\\{0\\};\mathbb{C}^{2})\subset\operatorname{dom}\mathfrak{d}_{\gamma}$,
one finds that $\operatorname{dom}D_{\gamma}\subset
H^{2}((-\gamma,\gamma)\setminus\\{0\\};\mathbb{C}^{2})\cap\operatorname{dom}\mathfrak{d}_{\gamma}$
and that the action of $D_{\gamma}$ is $D_{\gamma}f=-f^{\prime\prime}+m^{2}f$
in $(-\gamma,\gamma)\setminus\\{0\\}$ for $f\in\operatorname{dom}D_{\gamma}$.
Moreover, using integration by parts, one gets for any
$f\in\operatorname{dom}D_{\gamma}$ and any
$g\in\operatorname{dom}\mathfrak{d}_{\gamma}$

$\begin{split}\mathfrak{d}_{\gamma}[f,g]&=\int_{(-\gamma,\gamma)\setminus\\{0\\}}(-f^{\prime\prime}+m^{2}f)\overline{g}dx+f^{\prime}(\gamma)\overline{g(\gamma)}-f^{\prime}(0^{+})\overline{g(0^{+})}\\\
&\qquad+f^{\prime}(0^{-})\overline{g(0^{-})}-f^{\prime}(-\gamma)\overline{g(-\gamma)}+\frac{2m}{\tau}(f(0^{+})-f(0^{-}))\overline{(g(0^{+})-g(0^{-}))}\\\
&=\int_{(-\gamma,\gamma)\setminus\\{0\\}}D_{\gamma}f\overline{g}dx+f^{\prime}(\gamma)\overline{g(\gamma)}-f^{\prime}(-\gamma)\overline{g(-\gamma)}\\\
&\qquad+\left[\widetilde{M}f^{\prime}(0^{-})-f^{\prime}(0^{+})+\frac{2m}{\tau}(\sigma_{0}-\widetilde{M})^{2}f(0^{+})\right]\overline{g(0^{+})},\end{split}$

where in the second step the transmission condition for
$f,g\in\operatorname{dom}\mathfrak{d}_{\gamma}$ was taken into account. Thus,
we conclude that

$\begin{split}\operatorname{dom}D_{\gamma}=\bigg{\\{}f\in
H^{2}((-\gamma,\gamma)\setminus\\{0\\};\mathbb{C}^{2})&\colon
f(0^{-})=\widetilde{M}f(0^{+}),f^{\prime}(\gamma)=f^{\prime}(-\gamma)=0,\\\
&\qquad\widetilde{M}f^{\prime}(0^{-})-f^{\prime}(0^{+})+\frac{2m}{\tau}(\sigma_{0}-\widetilde{M})^{2}f(0^{+})=0\bigg{\\}}.\end{split}$

To compute the smallest eigenvalue $E=E(\gamma)$ of $D_{\gamma}$, we set
$k=\sqrt{m^{2}-E}$ and note first that a corresponding eigenfunction $f_{E}$
must be of the form

$f_{E}(x)=\begin{cases}a_{+}e^{-kx}+b_{+}e^{kx},&x\in(0,\gamma),\\\
a_{-}e^{kx}+b_{-}e^{-kx},&x\in(-\gamma,0),\end{cases}$

for some $a_{\pm},b_{\pm}\in\mathbb{C}^{2}$ and it has to fulfil the
conditions for $f\in\operatorname{dom}D_{\gamma}$. From the condition
$f_{E}^{\prime}(\pm\gamma)=0$ we conclude that $a_{\pm}=b_{\pm}e^{2k\gamma}$.
Taking this into account, one finds that
$f_{E}(0^{-})=\widetilde{M}f_{E}(0^{+})$ yields $b_{-}=\widetilde{M}b_{+}$.
Hence, the last condition for $f_{E}\in\operatorname{dom}D_{\gamma}$
simplifies to

(B.3)
$\begin{split}0&=\widetilde{M}f^{\prime}_{E}(0^{-})-f^{\prime}_{E}(0^{+})+\frac{2m}{\tau}(\sigma_{0}-\widetilde{M})^{2}f_{E}(0^{+})\\\
&=k(e^{2k\gamma}-1)(\widetilde{M}^{2}+\sigma_{0})b_{+}+\frac{2m}{\tau}(e^{2k\gamma}+1)(\sigma_{0}-\widetilde{M})^{2}b_{+}.\end{split}$

By the definition of $\widetilde{M}$ one finds with a direct calculation that
$\widetilde{M}^{2}+\sigma_{0}=2\frac{4+\tau^{2}}{4-\tau^{2}}\widetilde{M}$ and
$(\sigma_{0}-\widetilde{M})^{2}=\frac{4\tau^{2}}{4-\tau^{2}}\widetilde{M}$.
Since $\widetilde{M}$ is invertible, we find that (B.3) is equivalent to

(B.4)
$-\frac{4m\tau}{4+\tau^{2}}=k\frac{e^{k\gamma}-e^{-k\gamma}}{e^{k\gamma}+e^{-k\gamma}}=k\tanh(k\gamma)=\mathchar
58\relax F_{\gamma}(k).$

Since

$F_{\gamma}^{\prime}(k)=\tanh(k\gamma)+\frac{k\gamma}{\cosh^{2}(k\gamma)}>0\quad\text{for}\quad
k>0,$

$\lim_{k\rightarrow 0^{+}}F(k\gamma)=0$, and
$\lim_{k\rightarrow\infty}F(k\gamma)=\infty$, we find that
$F_{\gamma}\mathchar 58\relax(0,\infty)\rightarrow(0,\infty)$ is bijective.
Hence, the equation (B.4) has exactly one solution $k_{\gamma}>0$, which is
associated with the smallest eigenvalue of $D_{\gamma}$; we remark that (B.4)
has infinitely many solutions $k$ of the form $k=ix$ with $x\in\mathbb{R}$,
which are, due to $k=\sqrt{m^{2}-E}$, associated with eigenvalues of
$D_{\gamma}$ that are larger than $m$.

Finally, to compute the limit of the solution $k_{\gamma}$ of (B.4) for
$\gamma\rightarrow\infty$, we note first that $\tanh(x)<1$ for all $x>0$ and
(B.4) yield that $k_{\gamma}>-\frac{4m\tau}{4+\tau^{2}}$ for all $\gamma>0$.
Thus, $\gamma k_{\gamma}\rightarrow\infty$, as $\gamma\rightarrow\infty$. By
using now once more that $k_{\gamma}$ satisfies (B.4), we obtain

$\lim_{\gamma\rightarrow\infty}k_{\gamma}=\lim_{\gamma\rightarrow\infty}\left(-\frac{4m\tau}{4+\tau^{2}}\frac{1}{\tanh(k_{\gamma}\gamma)}\right)=-\frac{4m\tau}{4+\tau^{2}}.$

Hence, we conclude for the smallest eigenvalue $E(\gamma)$ of $D_{\gamma}$ via
the relation $k_{\gamma}=\sqrt{m^{2}-E(\gamma)}$ that

$\lim_{\gamma\rightarrow\infty}E(\gamma)=\lim_{\gamma\rightarrow\infty}(m^{2}-k_{\gamma}^{2})=m^{2}\left(\frac{\tau^{2}-4}{\tau^{2}+4}\right)^{2},$

which is exactly the claim in (B.2).

### Acknowledgement

DF and MH gratefully acknowledge financial support by the Austrian Science
Fund (FWF): P33568-N. VL acknowledges the support by the grant No. 21-07129S
of the Czech Science Foundation (GAČR). 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.

## References

  * [1] P. Antunes, R. Benguria, V. Lotoreichik, T. Ourmières-Bonafos, A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalities, Comm. Math. Phys. 386(2) (2021), 781–818.
  * [2] N. Arrizabalaga, L. Le Treust, N. Raymond, On the MIT bag model in the non-relativistic limit, Comm. Math. Phys. 354(2) (2017), 641–669.
  * [3] N. Arrizabalaga, A. Mas, L. Vega, Shell interactions for Dirac operators, J. Math. Pures Appl. 102 (2014), 617–639.
  * [4] N. Arrizabalaga, A. Mas, L. Vega, Shell interactions for Dirac operators: on the point spectrum and the confinement, SIAM J. Math. Anal. 47 (2015), 1044–1069.
  * [5] N. Arrizabalaga, A. Mas, L. Vega, An isoperimetric-type inequality for electrostatic shell interactions for Dirac operators, Commun. Math. Phys. 344, (2016), 483–505.
  * [6] J. Behrndt, P. Exner, M. Holzmann, V. Lotoreichik, On the spectral properties of Dirac operators with electrostatic $\delta$-shell interactions, J. Math. Pures Appl. 111 (2018), 47–78.
  * [7] J. Behrndt, P. Exner, M. Holzmann, V. Lotoreichik, On Dirac operators in $\mathbb{R}^{3}$ with electrostatic and Lorentz scalar $\delta$-shell interactions, Quantum Stud. Math. Found. 6 (2019), 295–314.
  * [8] J. Behrndt, M. Holzmann, On Dirac operators with electrostatic $\delta$-shell interactions of critical strength, J. Spectr. Theory 10 (2020), 147–184.
  * [9] J. Behrndt, M. Holzmann, T. Ourmières-Bonafos, K. Pankrashkin, Two-dimensional Dirac operators with singular interactions supported on closed curves, J. Funct. Anal. 279 (2020), 108700.
  * [10] J. Behrndt, M. Holzmann, C. Stelzer, G. Stenzel, Boundary triples and Weyl functions for Dirac operators with singular interactions, arXiv:2211.05191.
  * [11] J. Behrndt, M. Holzmann, M. Tušek, Two-dimensional Dirac operators with general $\delta$-shell interactions supported on a straight line, J. Phys. A 56 (2023), 045201.
  * [12] R. Benguria, S. Fournais, E. Stockmeyer, H. Van Den Bosch, Spectral gaps of Dirac operators describing graphene quantum dots, Math. Phys. Anal. Geom. 20 (2017), 11.
  * [13] M. Sh. Birman, M. Z. Solomjak, Spectral theory of self-adjoint operators in Hilbert space, D. Reidel Publishing Company, Dordrecht, 1987.
  * [14] V. Bonnaillie-Noël, S. Fournais, A. Kachmar, N. Raymond, Discrete spectrum of the magnetic Laplacian on perturbed half-planes, arXiv:2208.13646.
  * [15] B. Cassano, V. Lotoreichik, A. Mas, M. Tušek, General $\delta$-shell interactions for the two-dimensional Dirac operator: self-adjointness and approximation, to appear in Rev. Mat. Iberoam.
  * [16] J. Dittrich, P. Exner, P. Šeba, Dirac operators with a spherically symmetric $\delta$-shell interaction, J. Math. Phys. 30 (1989), 2875–2882.
  * [17] P. Duclos, P. Exner, Curvature-induced bound states in quantum waveguides in two and three dimensions, Rev. Math. Phys. 7 (1995), 73–102.
  * [18] P. Duclos, P. Exner, D. Krejčiřík, Bound states in curved quantum layers, Comm. Math. Phys. 223 (2001), 13–28.
  * [19] P. Exner, M. Holzmann,  Dirac operator spectrum in tubes and layers with a zigzag type boundary,  Lett. Math. Phys. 112 (2022), 102.
  * [20] P. Exner, T. Ichinose, Geometrically induced spectrum in curved leaky wires, J. Phys. A, Math. Gen. 34 (2001), 1439–1450.
  * [21] P. Exner, H. Kovařík,  Quantum Waveguides.  Theoretical and Mathematical Physics. Springer, Cham, 2015.
  * [22] P. Exner, V. Lotoreichik, A. Pérez-Obiol, On the bound states of magnetic Laplacians on wedges, Rep. Math. Phys. 82 (2018), 161–185.
  * [23] P. Exner, A. Minakov, Curvature-induced bound states in Robin waveguides and their asymptotical properties, J. Math. Phys. 55 (2014), 122101\.
  * [24] P. Exner, K. Němcová, Leaky quantum graphs: Approximations by point-interaction Hamiltonians, J. Phys. A, Math. Gen. 36 (2003), 10173–10193.
  * [25] D. Frymark, V. Lotoreichik, Self-adjointness of the 2D Dirac operator with singular interactions supported on star graphs, Ann. Henri Poincaré 24 (2023), 179–221.
  * [26] M. Holzmann, T. Ourmières-Bonafos, K. Pankrashkin, Dirac operators with Lorentz scalar shell interactions, Rev. Math. Phys. 30 (2018), 1850013.
  * [27] D. Krejčiřík, Z. Lu, Location of the essential spectrum in curved quantum layers, J. Math. Phys. 55 (2014), 083520.
  * [28] L. Le Treust, T. Ourmières-Bonafos, Self-adjointness of Dirac operators with infinite mass boundary conditions in sectors, Ann. Henri Poincaré 19 (2018), 1465–1487.
  * [29] V. Lotoreichik, T. Ourmières-Bonafos, A Sharp Upper Bound on the Spectral Gap for Graphene Quantum Dots, Math. Phys. Anal. Geom. 22 (2019), 13.
  * [30] J.-L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972.
  * [31] W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000.
  * [32] K. Pankrashkin, On the discrete spectrum of Robin Laplacians in conical domains, Math. Model. Nat. Phenom. 11 (2016), 100–110.
  * [33] K. Pankrashkin, Variational proof of the existence of eigenvalues for star graphs, in “Functional Analysis and Operator Theory for Quantum Physics. The Pavel Exner Anniversary Volume”, EMS Series of Congress Reports, ed. J. Dittrich, H. Kovašik and A. Laptev, EMS Publishing House, Berlin (2018), 447–458.
  * [34] K. Pankrashkin, M. Vogel, On Schrödinger operators with $\delta^{\prime}$-potentials supported on star graphs, J. Phys. A 55 (2022), 295201.
  * [35] F. Pizzichillo, H. Van Den Bosch, Self-adjointness of two-dimensional Dirac operators on corner domains, J. Spectr. Theory 11 (2021), 1043–1079.
  * [36] K. Schmidt, A remark on boundary value problems for the Dirac operator, Quart. J. Math. Oxford Ser. (2) 46 (1995), 509–516.
  * [37] K. Schmüdgen, Unbounded self-adjoint operators on Hilbert space, Dordrecht, Springer, 2012.