# Approximation of Dirac operators with $\boldsymbol{\delta}$-shell potentials
in the norm resolvent sense

Jussi Behrndt  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  Christian Stelzer  Technische
Universität Graz
Institut für Angewandte Mathematik
Steyrergasse 30
8010 Graz, Austria<EMAIL_ADDRESS>
###### Abstract.

In this paper the approximation of Dirac operators with general $\delta$-shell
potentials supported on $C^{2}$-curves in $\mathbb{R}^{2}$ or $C^{2}$-surfaces
in $\mathbb{R}^{3}$, which may be bounded or unbounded, is studied. It is
shown under suitable conditions on the weight of the $\delta$-interaction that
a family of Dirac operators with regular, squeezed potentials converges in the
norm resolvent sense to the Dirac operator with the $\delta$-shell
interaction.

##  1\. Introduction

In mathematical physics, singular potentials supported on a set $\Sigma$ of
measure zero are often used as replacements of potentials that have large
values in a vicinity of $\Sigma$ and small values elsewhere, assuming that
such idealized models have similar (spectral) properties as the original ones.
Nonrelativistic Schrödinger operators with $\delta$-potentials supported on
points were already considered in the early days of quantum mechanics [28,
42], see also the monograph [1], and then later also Schrödinger operators
with $\delta$-potentials supported on curves in $\mathbb{R}^{2}$ or
$\mathbb{R}^{3}$, surfaces in $\mathbb{R}^{3}$, and similar structures in
higher dimensions were investigated, see, e.g., [12, 15, 21] and the
references therein. Concerning relativistic Dirac operators with a singular
$\delta$-potential, the one-dimensional case was first studied in the 1980s
[1, 22], and recently also two and three-dimensional Dirac operators coupled
with $\delta$-shell potentials supported on general curves in $\mathbb{R}^{2}$
and surfaces in $\mathbb{R}^{3}$ were treated; cf. [2, 3, 9, 10, 16] and the
discussion below.

In all of the above situations, it is necessary to justify that the
differential operators with the singular interactions have indeed similar
properties as the original models with regular potentials. One way to do this
is to prove suitable approximation results: One constructs a family of
potentials that converge to the idealized singular interaction and shows that
the associated differential operators also converge in the norm resolvent
sense to the idealized operator – then also the spectral properties of the
approximating and the idealized models are approximately the same [27, 37].
For Schrödinger operators with singular potentials this approximation problem
has been solved under the assumption that the interaction support is a
sufficiently smooth hypersurface in $\mathbb{R}^{\theta}$, $\theta\geq 1$; cf.
[4] and the references therein.

The literature on the approximation of Dirac operators with singular
potentials is less complete than for their nonrelativistic counterparts. The
Dirac operator with a singular $\delta$-potential in $\mathbb{R}^{\theta}$,
$\theta\in\\{1,2,3\\}$, is formally given by

$-i\sum_{j=1}^{\theta}\alpha_{j}\partial_{j}+m\beta+\widetilde{V}\delta_{\Sigma},$
(1.1)

where $\delta_{\Sigma}$ denotes the $\delta$-distribution supported on a point
in $\mathbb{R}$ or the boundary $\Sigma$ of a bounded or unbounded domain in
$\mathbb{R}^{2}$ or $\mathbb{R}^{3}$, $\widetilde{V}$ is a matrix-valued
function on $\Sigma$ modeling the position-dependent strength of the singular
interaction, $m\in\mathbb{R}$, and $\alpha_{1},\dots,\alpha_{\theta},$ and
$\beta$ denote the Dirac matrices defined in (1.15)–(1.16) below (in dimension
$\theta=1$ one can use the same choice as for $\theta=2$). The first
approximation result in dimension one goes back to Šeba [38], where the norm
resolvent convergence for so-called electrostatic (i.e.
$\widetilde{V}=\widetilde{\eta}I_{2}$) and Lorentz scalar (i.e.
$\widetilde{V}=\widetilde{\tau}\beta$) interactions supported on a point was
proved. In [38] it already turned out that one has to renormalize the
interaction strength in order to obtain convergence, i.e. when
$v_{\varepsilon}\rightarrow\eta\delta_{\Sigma}$ in the distributional sense,
then $-i\alpha_{1}\partial_{x}+m\beta+v_{\varepsilon}I_{2}$ does not converge
to $-i\alpha_{1}\partial_{x}+m\beta+\eta I_{2}\delta_{\Sigma}$, but to a
similar operator, where the coefficient of $\delta_{\Sigma}$ depends in a
nonlinear way on $\eta$; for Lorentz scalar potentials a similar effect
appears as well. Šeba suggested that this renormalization is related to
Klein’s paradox for the Dirac equation. Later Hughes proved in [24, 25] strong
resolvent convergence for general point interactions and also determined the
exact form of the renormalization in the one-dimensional situation. Recently,
Tušek proved norm resolvent convergence in the one-dimensional case for Dirac
operators with general point interactions in [43]. In the two and three-
dimensional setting the approximation problem is not as well-studied as in
dimension one and so far there exist only results on strong resolvent
convergence. In dimension three the approximation of Dirac operators with
purely electrostatic or Lorentz scalar $\delta$-shell potentials that fulfil a
certain smallness condition supported on boundaries of bounded domains were
investigated in [29, 30]. The approximation of two-dimensional Dirac operators
with electrostatic, Lorentz scalar, and anomalous magnetic $\delta$-shell
potentials supported on closed bounded curves and on a straight line was
considered in [16] and [11], respectively. We note that also in the two and
three-dimensional setting a renormalization of the interaction strength was
observed in [11, 16, 30].

In this paper we study the approximation problem for two- and three-
dimensional Dirac operators with $\delta$-shell potentials and improve the
present state of art in the following three ways: Under suitable conditions we
show (i) norm resolvent convergence on (ii) bounded and unbounded supports
$\Sigma$ with (iii) general position-dependent interaction strengths
$\widetilde{V}$. More precisely:

  * (i)

Instead of strong resolvent convergence we prove the norm resolvent
convergence of the approximating family, which has not been established in the
multi-dimensional situation so far. This type of convergence ensures that the
spectrum of the limit operator $H_{\widetilde{V}}$ can be completely
characterized by the spectra of the approximating operators and it also
implies the convergence of the related spectral projections.

  * (ii)

Instead of bounded curves in $\mathbb{R}^{2}$, bounded surfaces in
$\mathbb{R}^{3}$, or the straight line in $\mathbb{R}^{2}$ we treat a general
class of bounded and unbounded interaction supports $\Sigma$ which can be
described by finitely many rotated graphs of $C^{2}$-functions with bounded
derivatives (see Hypothesis 2.1) and which includes, in particular, graphs of
$C^{2}$-functions with bounded derivatives and boundaries of bounded
$C^{2}$-domains.

  * (iii)

Instead of considering only electrostatic, Lorentz scalar, and anomalous
magnetic interactions (which can be described by three real-valued functions)
we allow general symmetric $2\times 2$ or $4\times 4$ matrix-valued functions
as interaction strengths in dimensions two or three, respectively, and provide
an explicit formula for the nonlinear renormalization when passing to the
limit.

In the following we explain the approximation procedure and our main result in
more detail. For this some notation needs to be fixed. We denote the space
dimension by $\theta\in\\{2,3\\}$ and set $N=2$ for $\theta=2$ and $N=4$ for
$\theta=3$. Let $\Sigma\subset\mathbb{R}^{\theta}$ be the boundary of a
bounded or unbounded $C^{2}$-domain $\Omega_{+}\subset\mathbb{R}^{\theta}$
that satisfies Hypothesis 2.1. The curve or surface $\Sigma$ splits
$\mathbb{R}^{\theta}$ into two disjoint parts $\Omega_{+}$ and
$\Omega_{-}:=\mathbb{R}^{\theta}\setminus\overline{\Omega_{+}}$, and $\nu$ is
the unit normal vector field at $\Sigma$ pointing outwards of $\Omega_{+}$.
For $u:\mathbb{R}^{\theta}\rightarrow\mathbb{C}^{N}$ we write
$u_{\pm}:=u\upharpoonright\Omega_{\pm}$ and we denote the Dirichlet trace
operator by
$\mathbf{t}_{\Sigma}^{\pm}:H^{1}(\Omega_{\pm};\mathbb{C}^{N})\rightarrow
H^{1/2}(\Sigma;\mathbb{C}^{N})$, where $H^{s}$ are the $L^{2}$-based Sobolev
spaces. Moreover,
$\alpha_{1},\dots,\alpha_{\theta},\beta\in\mathbb{C}^{N\times N}$ are the
Dirac matrices defined in (1.15)–(1.16) and we make use of the notations

$\alpha\cdot\nabla:=\sum_{j=1}^{\theta}\alpha_{j}\partial_{j}\quad\text{and}\quad\alpha\cdot
x:=\sum_{j=1}^{\theta}\alpha_{j}x_{j},\quad
x=(x_{1},\dots,x_{\theta})\in\mathbb{C}^{\theta}.$

We introduce for $m\in\mathbb{R}$ and $\widetilde{V}\in
L^{\infty}(\Sigma;\mathbb{C}^{N\times N})$ such that
$\widetilde{V}(x_{\Sigma})=(\widetilde{V}(x_{\Sigma}))^{*}$ for $\sigma$-a.e.
$x_{\Sigma}\in\Sigma$ in $L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ the
operator

$\begin{split}H_{\widetilde{V}}u&:=\big{(}-i(\alpha\cdot\nabla)+m\beta\big{)}u_{+}\oplus\big{(}-i(\alpha\cdot\nabla)+m\beta\big{)}u_{-},\\\
\textup{dom }H_{\widetilde{V}}&:=\bigg{\\{}u\in
H^{1}(\Omega_{+};\mathbb{C}^{N})\oplus H^{1}(\Omega_{-};\mathbb{C}^{N}):\\\
&\quad\qquad
i(\alpha\cdot\nu)(\boldsymbol{t}_{\Sigma}^{+}u_{+}-\boldsymbol{t}_{\Sigma}^{-}u_{-})+\frac{\widetilde{V}}{2}(\boldsymbol{t}_{\Sigma}^{+}u_{+}+\boldsymbol{t}_{\Sigma}^{-}u_{-})=0\bigg{\\}}.\end{split}$
(1.2)

This is the rigorous definition of the formal operator (1.1), which was
studied under various assumptions on the coefficients $\widetilde{V}$ and the
interaction support $\Sigma$ in [2, 3, 5, 6, 9, 10, 13, 16, 35, 36]. However,
under the general assumptions in this paper even the self-adjointness of
$H_{\widetilde{V}}$ is not yet established, but will follow as a byproduct of
our main result below as $H_{\widetilde{V}}$ is the norm resolvent limit of a
sequence of self-adjoint operators.

It is the main goal of the present paper to show that the operator
$H_{\widetilde{V}}$ in (1.2) can be approximated in the norm resolvent sense
by Dirac operators with strongly localized potentials. To introduce the latter
operators, define the map

$\begin{split}\iota:\Sigma\times\mathbb{R}\to\mathbb{R}^{\theta},\qquad\iota(x_{\Sigma},t):=x_{\Sigma}+t\nu(x_{\Sigma}),\end{split}$
(1.3)

and for $\varepsilon>0$ the set

$\Omega_{\varepsilon}:=\iota(\Sigma\times(-\varepsilon,\varepsilon)),\quad\varepsilon>0.$
(1.4)

We call $\Omega_{\varepsilon}$ tubular neighbourhood of $\Sigma$. Hypothesis
2.1 and Proposition 2.4 imply that there exists $\varepsilon_{1}>0$ such that
$\iota|_{\mathbb{R}\times(-\varepsilon_{1},\varepsilon_{1})}$ is injective
and, hence, all points in $\Omega_{\varepsilon_{1}}$ can be uniquely described
by the map $\iota$. To define the above mentioned strongly localized
potentials choose

$q\in
L^{\infty}((-1,1);\mathbb{R})\quad\text{with}\quad\int_{-1}^{1}q(s)\,ds=1$
(1.5)

and

$V\in W^{1}_{\infty}(\Sigma;\mathbb{C}^{N\times N})\quad\text{such that}\quad
V(x_{\Sigma})=(V(x_{\Sigma}))^{*}\text{ for }\sigma\text{-a.e.
}x_{\Sigma}\in\Sigma,$ (1.6)

where $W^{1}_{\infty}$ denotes the first order $L^{\infty}$-based Sobolev
space. Since $\iota|_{\mathbb{R}\times(-\varepsilon_{1},\varepsilon_{1})}$ is
injective, we can define for $\varepsilon\in(0,\varepsilon_{1})$

$V_{\varepsilon}(x):=\begin{cases}\frac{1}{\varepsilon}V(x_{\Sigma})q\left(\frac{t}{\varepsilon}\right),&\text{if
}x=\iota(x_{\Sigma},t)\in\Omega_{\varepsilon},\\\ 0,&\text{if
}x\notin\Omega_{\varepsilon},\end{cases}$ (1.7)

and for $m\in\mathbb{R}$ and $\varepsilon\in(0,\varepsilon_{1})$ the operator

$\begin{split}H_{\varepsilon}u&:=-i(\alpha\cdot\nabla)u+m\beta
u+V_{\varepsilon}u,\quad\textup{dom
}H_{\varepsilon}:=H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N}).\end{split}$ (1.8)

Note that $H_{\varepsilon}$ is self-adjoint in
$L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$, as $V_{\varepsilon}\in
L^{\infty}(\mathbb{R}^{\theta};\mathbb{C}^{N\times N})$ is symmetric and the
unperturbed Dirac operator $H=H_{\varepsilon}-V_{\varepsilon}$ defined on
$H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ is self-adjoint in
$L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$; cf. Section 2.3.

The sequence $V_{\varepsilon}$ converges to $V\delta_{\Sigma}$ in the sense of
distributions. However, as in [11, 16, 24, 25, 30, 38, 43] the sequence
$H_{\varepsilon}$ does not converge to $H_{V}$, but to a similar operator with
a renormalized interaction strength $\widetilde{V}$. In the present situation
this renormalization is expressed with the help of the matrix-valued function

$S=\textup{sinc}\big{(}\tfrac{(\alpha\cdot\nu)V}{2}\big{)}\cos\big{(}\tfrac{(\alpha\cdot\nu)V}{2}\big{)}^{-1},$
(1.9)

where analytic functions of matrices are defined via the corresponding power
series, or equivalently via the Riesz-Dunford functional calculus. Now we are
prepared to formulate the main result of this paper.

######  Theorem 1.1.

Let $q$ and $V$ be as in (1.5) and (1.6), respectively, and assume that for
some $z\in\mathbb{C}\setminus((-\infty,-|m|]\cup[|m|,\infty))$ the condition

${\left\lVert{V}\right\rVert}_{W^{1}_{\infty}(\Sigma;\mathbb{C}^{N\times
N})}{\left\lVert{q}\right\rVert}_{L^{\infty}(\mathbb{R};\mathbb{R})}<X_{z}$
(1.10)

holds with $X_{z}$ given by (3.28) and that $\varepsilon_{2}>0$ satisfies
(3.11). Furthermore, assume that

$\cos\big{(}\tfrac{(\alpha\cdot\nu)V}{2}\big{)}^{-1}\in
W^{1}_{\infty}(\Sigma;\mathbb{C}^{N\times N})$ (1.11)

and set $\widetilde{V}=VS$, where $S$ is defined as in (1.9). Then the
operator $H_{\widetilde{V}}$ in (1.2) is self-adjoint in
$L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$,
$z\in\rho(H_{\varepsilon})\cap\rho(H_{\widetilde{V}})$ for all
$\varepsilon\in(0,\varepsilon_{2})$, and for any
$r\in\big{(}0,\tfrac{1}{2}\big{)}$ there exists $C>0$ such that

${\left\lVert{(H_{\varepsilon}-z)^{-1}-(H_{\widetilde{V}}-z)^{-1}}\right\rVert}_{L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\leq
C\varepsilon^{1/2-r},\qquad\varepsilon\in(0,\varepsilon_{2}).$

In particular, $H_{\varepsilon}$ converges to $H_{\widetilde{V}}$ in the norm
resolvent sense as $\varepsilon\to 0+$.

The main assumption in Theorem 1.1 is condition (1.10), which means that $V$
has to be small in a suitable sense and which also appears in a similar form
in [30], where the strong resolvent convergence of $H_{\varepsilon}$ for some
special potentials $V$ was proved. The condition for $V$ can be optimized by
choosing $q=\tfrac{1}{2}\chi_{(-1,1)}$, where $\chi_{(-1,1)}$ denotes the
characteristic function of the interval $(-1,1)$, but since for some
applications also more general $q$ can be useful, we keep the more general
formulation of (1.10); a more detailed discussion of (1.10) can be found in
Remark 4.5. We shall show later in Lemma 3.13 that $X_{z}\leq\frac{\pi}{4}$,
if $\Sigma$ is compact and smooth.

In the following, let us consider the special choice

$V=\eta I_{N}+\tau\beta+\lambda i(\alpha\cdot\nu)\beta,$ (1.12)

where $\eta,\tau,\lambda\in W^{1}_{\infty}(\Sigma;\mathbb{R})$ are real
functions (or simply real constants) that model electrostatic, Lorentz scalar,
and anomalous magnetic interactions, respectively. Such potentials $V$ are of
special physical interest and the existing literature on Dirac operators with
$\delta$-shell potentials mostly focuses on this case. It turns out that for
such $V$ the renormalization matrix in (1.9) has the simple form

$S=\tfrac{2\tan(\sqrt{d}/2)}{\sqrt{d}}I_{N},$

where $d=\eta^{2}-\tau^{2}-\lambda^{2}$, and (1.11) simplifies, see (1.13)
below. Note that analogous conditions and renormalizations are given in [16,
Theorem 2.6], [29, Theorem IV.2] and [38, Remarks]. In the present situation
Theorem 1.1 reduces to the following corollary.

######  Corollary 1.2.

Let $q$ and $V$ be as in (1.5) and (1.12), respectively, with
$\eta,\tau,\lambda\in W^{1}_{\infty}(\Sigma;\mathbb{R})$,
$d=\eta^{2}-\tau^{2}-\lambda^{2}$, and assume that for some
$z\in\mathbb{C}\setminus((-\infty,-|m|]\cup[|m|,\infty))$ condition (1.10)
holds with $X_{z}$ given by (3.28) and that $\varepsilon_{2}>0$ satisfies
(3.11). Furthermore, assume

$\inf_{x_{\Sigma}\in\Sigma,k\in\mathbb{N}_{0}}\left\lvert{(2k+1)^{2}\pi^{2}-d(x_{\Sigma})}\right\rvert>0$
(1.13)

and let $\widetilde{V}=\frac{2\tan(\sqrt{d}/2)}{\sqrt{d}}V$. Then,
$H_{\widetilde{V}}$ in (1.2) is self-adjoint in
$L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$,
$z\in\rho(H_{\varepsilon})\cap\rho(H_{\widetilde{V}})$ for all
$\varepsilon\in(0,\varepsilon_{2})$, and for any
$r\in\big{(}0,\tfrac{1}{2}\big{)}$ there exists $C>0$ such that

${\left\lVert{(H_{\varepsilon}-z)^{-1}-(H_{\widetilde{V}}-z)^{-1}}\right\rVert}_{L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\leq
C\varepsilon^{1/2-r},\qquad\varepsilon\in(0,\varepsilon_{2}).$

In particular, $H_{\varepsilon}$ converges to $H_{\widetilde{V}}$ in the norm
resolvent sense as $\varepsilon\to 0+$.

In the previous two results we stated conditions on a given $V$ ensuring that
$H_{\varepsilon}$ converges in the norm resolvent sense to
$H_{\widetilde{V}}$. In the following corollary, we address the inverse
question and state conditions on $\widetilde{V}$ such that $H_{\widetilde{V}}$
is the limit of the operators $H_{\varepsilon}$.

######  Corollary 1.3.

Let $\widetilde{V}\in W_{\infty}^{1}(\Sigma;\mathbb{C}^{N\times N})$ such that
$\widetilde{V}(x_{\Sigma})=(\widetilde{V}(x_{\Sigma}))^{*}$ for $\sigma$-a.e.
$x_{\Sigma}\in\Sigma$, and assume that for some given
$z\in\mathbb{C}\setminus((-\infty,-|m|]\cup[|m|,\infty))$

$\|(\alpha\cdot\nu)\widetilde{V}\|_{W^{1}_{\infty}(\Sigma;\mathbb{C}^{N\times
N})}<2\tanh\big{(}\tfrac{X_{z}}{{\left\lVert{\alpha\cdot\nu}\right\rVert}_{W_{\infty}^{1}(\Sigma;C^{N\times
N})}}\big{)}$ (1.14)

holds and that $\varepsilon_{2}>0$ satisfies (3.11). Moreover, set
$q=\tfrac{1}{2}\chi_{(-1,1)}$ and

$V=2(\alpha\cdot\nu)\arctan\big{(}\tfrac{(\alpha\cdot\nu)\widetilde{V}}{2}\big{)}=2(\alpha\cdot\nu)\sum_{n=0}^{\infty}(-1)^{n}\frac{((\alpha\cdot\nu)\widetilde{V})^{2n+1}}{2^{2n+1}(2n+1)},$

and let $H_{\varepsilon}$ be the operator defined in (1.8) with these specific
choices of $q$ and $V$. Then, the operator $H_{\widetilde{V}}$ in (1.2) is
self-adjoint in $L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$,
$z\in\rho(H_{\varepsilon})\cap\rho(H_{\widetilde{V}})$ for all
$\varepsilon\in(0,\varepsilon_{2})$, and for any
$r\in\big{(}0,\tfrac{1}{2}\big{)}$ there exists $C>0$ such that

${\left\lVert{(H_{\varepsilon}-z)^{-1}-(H_{\widetilde{V}}-z)^{-1}}\right\rVert}_{L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\leq
C\varepsilon^{1/2-r},\qquad\varepsilon\in(0,\varepsilon_{2}).$

In particular, $H_{\varepsilon}$ converges to $H_{\widetilde{V}}$ in the norm
resolvent sense as $\varepsilon\to 0+$.

Let us briefly explain the strategy to prove Theorem 1.1 and the structure of
this paper. First, in Section 2 we collect some preliminary results regarding
the geometry of $\Sigma$, the free Dirac operator, and some associated
potential and boundary integral operators (see also Appendix C). In
Proposition 3.2 we then prove the resolvent formula

$(H_{\varepsilon}-z)^{-1}=(H-z)^{-1}-A_{\varepsilon}(z)Vq(I+B_{\varepsilon}(z)Vq)^{-1}C_{\varepsilon}(z),$

where $H$ is the free Dirac operator (see (2.8)) and
$A_{\varepsilon}(z),B_{\varepsilon}(z),C_{\varepsilon}(z)$ are suitable
integral operators defined in (3.4) that are bounded between various
$L^{2}$-spaces; cf. Proposition 3.1. This is a standard approach that has been
applied in other approximation problems for differential operators with
singular potentials, see, e.g., [1, 4, 30, 38]. Typically, the convergence of
$A_{\varepsilon}(z),B_{\varepsilon}(z),C_{\varepsilon}(z)$ is analyzed in the
corresponding $L^{2}$-setting. In the situation of Dirac operators with
singular potentials, $B_{\varepsilon}(z)$ converges only strongly in $L^{2}$
and hence, via this approach only the strong resolvent convergence of
$H_{\varepsilon}$ can be shown; cf. [30]. Contrary to that, we investigate the
convergence of $A_{\varepsilon}(z),B_{\varepsilon}(z),C_{\varepsilon}(z)$
between Sobolev type spaces. For this, a certain shift operator, which allows
to compare the values of functions in $x_{\Sigma}\in\Sigma$ and in
$x_{\Sigma}+\varepsilon s\nu(x_{\Sigma})$, plays a crucial role; cf.
Proposition 3.3. The condition (1.10) in Theorem 1.1 is needed to ensure that
$(I+B_{\varepsilon}(z)Vq)^{-1}$ is uniformly bounded in $\varepsilon$.
Finally, using the limiting behavior of
$A_{\varepsilon}(z),B_{\varepsilon}(z),C_{\varepsilon}(z)$ we compute in
Section 4 the limit of $(H_{\varepsilon}-z)^{-1}$ for $\varepsilon\to 0+$ and
show that it indeed coincides with the resolvent of $H_{\widetilde{V}}$. In
Appendices A and B we provide some technical calculations that are necessary
for the definition of the tubular neighbourhood of $\Sigma$ and the
convergence analysis.

### Notations

By $\theta\in\\{2,3\\}$ we denote the space dimension and we set $N=2$ for
$\theta=2$ and $N=4$ for $\theta=3$. Recall that the Pauli spin matrices are

$\sigma_{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\sigma_{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\quad\text{and}\quad\sigma_{3}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.$

With their help the Dirac matrices
$\alpha_{1},\dots,\alpha_{\theta},\beta\in\mathbb{C}^{N\times N}$ are defined
for $\theta=2$ by

$\alpha_{1}:=\sigma_{1},\quad\alpha_{2}:=\sigma_{2},\quad\text{and}\quad\beta:=\sigma_{3},$
(1.15)

and for $\theta=3$ by

$\alpha_{j}:=\begin{pmatrix}0&\sigma_{j}\\\ \sigma_{j}&0\end{pmatrix}\text{
for }j=1,2,3\quad\text{and}\quad\beta:=\begin{pmatrix}I_{2}&0\\\
0&-I_{2}\end{pmatrix},$ (1.16)

where $I_{2}$ is the $2\times 2$-identity matrix. We will often make use of
the notations

$\alpha\cdot\nabla:=\sum_{j=1}^{\theta}\alpha_{j}\partial_{j}\quad\text{and}\quad\alpha\cdot
x:=\sum_{j=1}^{\theta}\alpha_{j}x_{j},\quad
x=(x_{1},\dots,x_{\theta})\in\mathbb{C}^{\theta}.$

The letter $C>0$ always denotes a constant which may change in-between lines
and does not depend on the space variables or on $\varepsilon$. The symbol
$|\cdot|$ is used for the absolute value, the Euclidean vector norm, or the
Frobenius norm of a number, a vector, or a matrix, respectively. We write
${\left\langle{\cdot},{\cdot}\right\rangle}$ for the Euclidean scalar product
in $\mathbb{C}^{n}$, $n\in\mathbb{N}$, which is anti-linear in the second
argument. Eventually, the application of an analytic function to a matrix $A$
is defined via the associated power series, whenever it converges. This
implies, in particular, for two holomorphic functions $f,g$ that
$f(A)g(A)=(fg)(A)$. We will also make use of the equivalent definition of
$f(A)$ via the Riesz-Dunford calculus

$f(A)=-\frac{1}{2\pi i}\int_{\partial\Omega}f(z)(A-z)^{-1}dz,$ (1.17)

where $\Omega\subset\mathbb{C}$ such that all eigenvalues of $A$ belong to
$\Omega$, $f$ is holomorphic in a neighborhood of $\Omega$, and the integral
is understood as a complex line integral; cf. [19, Chapter VII. §4] and [20,
Chapter VIII.3.1].

Next, let $\mathcal{H},\mathcal{G}$ be Hilbert spaces and let $A$ be a linear
operator from $\mathcal{H}$ to $\mathcal{G}$. The domain, kernel, and range of
$A$ are denoted by $\textup{dom }A$, $\ker A$, and $\textup{ran }A$,
respectively. If $A$ is bounded and everywhere defined, then we write
$\|A\|_{\mathcal{H}\rightarrow\mathcal{G}}$ for its operator norm. If
$\mathcal{H}=\mathcal{G}$ and $A$ is closed, then the resolvent set and the
spectrum $A$ are denoted by $\rho(A)$ and $\sigma(A)$, respectively.

Following [32, Appendix B], we call $(\mathcal{H}_{0},\mathcal{H}_{1})$ a
compatible pair, if $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$ are two Hilbert
spaces which are continuously embedded in a bigger Hausdorff topological
vector space. In this situation, one can construct with the K-method (or
various other methods, see [17, 18, 26, 32], which yield the same spaces with
equivalent norms) a family of Hilbert spaces
$[\mathcal{H}_{0},\mathcal{H}_{1}]_{\tau}$, $\tau\in(0,1)$, such that
$\mathcal{H}_{0}\cap\mathcal{H}_{1}\subset[\mathcal{H}_{0},\mathcal{H}_{1}]_{\tau}\subset\mathcal{H}_{0}+\mathcal{H}_{1}$
for all $\tau\in(0,1)$. Assume that $(\mathcal{G}_{0},\mathcal{G}_{1})$ is
another compatible pair of Hilbert spaces. Recall that for two bounded
operators $A_{0}:\mathcal{H}_{0}\to\mathcal{G}_{0}$ and
$A_{1}:\mathcal{H}_{1}\to\mathcal{G}_{1}$ such that $A_{0}u=A_{1}u$ for all
$u\in\mathcal{H}_{0}\cap\mathcal{H}_{1}$, there exists by [32, Theorem B.2] a
unique bounded linear operator
$A_{\tau}:[\mathcal{H}_{0},\mathcal{H}_{1}]_{\tau}\to[\mathcal{G}_{0},\mathcal{G}_{1}]_{\tau}$
such that $A_{0}u=A_{1}u=A_{\tau}u$ for all
$u\in\mathcal{H}_{0}\cap\mathcal{H}_{1}$ and

${\left\lVert{A_{\tau}}\right\rVert}_{[\mathcal{H}_{0},\mathcal{H}_{1}]_{\tau}\to[\mathcal{G}_{0},\mathcal{G}_{1}]_{\tau}}\leq{\left\lVert{A_{0}}\right\rVert}_{\mathcal{H}_{0}\to\mathcal{G}_{0}}^{1-\tau}{\left\lVert{A_{1}}\right\rVert}_{\mathcal{H}_{1}\to\mathcal{G}_{1}}^{\tau}.$
(1.18)

Finally, if $k,n\in\mathbb{N}$, $U\subset\mathbb{R}^{n}$ is an open set and
$\mathcal{V}$ denotes the space of real or complex scalars, vectors, or
matrices, then we write $C_{b}^{k}(U;\mathcal{V})$ for the space of all $f\in
C^{k}(U;\mathcal{V})$ such that $f$ and all partial derivatives of $f$ up to
order $k$ are bounded. If $f\in C^{1}(U;\mathbb{R}^{k})$, then the symbol $Df$
is used for the Jacobi matrix of $f$. The usual Bochner Lebesgue space of
$\mathcal{H}$-valued functions are denoted by $L^{2}((-1,1);\mathcal{H})$, cf.
Section 2.2. For $\mathcal{H}=H^{r}(\Sigma;\mathbb{C}^{N})$ we use the symbol
${\left\lVert{\cdot}\right\rVert}_{r}$ for the norm in
$L^{2}((-1,1);H^{r}(\Sigma;\mathbb{C}^{N}))$. In a similar way, we write

$\begin{split}{\left\lVert{\cdot}\right\rVert}_{r_{1}\to
r_{2}}&:={\left\lVert{\cdot}\right\rVert}_{L^{2}((-1,1);H^{r_{1}}(\Sigma;\mathbb{C}^{N}))\to
L^{2}((-1,1);H^{r_{2}}(\Sigma;\mathbb{C}^{N}))},\\\
{\left\lVert{\cdot}\right\rVert}_{r\to\mathcal{H}}&:={\left\lVert{\cdot}\right\rVert}_{L^{2}((-1,1);H^{r}(\Sigma;\mathbb{C}^{N}))\to\mathcal{H}},\\\
{\left\lVert{\cdot}\right\rVert}_{\mathcal{H}\to
r^{\prime}}&:={\left\lVert{\cdot}\right\rVert}_{\mathcal{H}\to
L^{2}((-1,1);H^{r^{\prime}}(\Sigma;\mathbb{C}^{N}))}.\end{split}$

### Acknowledgement.

The authors gratefully acknowledge financial support by the Austrian Science
Fund (FWF): P33568-N. This publication is based upon work from COST Action CA
18232 MAT-DYN-NET, supported by COST (European Cooperation in Science and
Technology), www.cost.eu.

##  2\. Preliminaries

In this section we provide some preliminary material that is needed in the
main part of this paper to prove Theorem 1.1. First, in Subsection 2.1 we
formulate suitable assumptions on the hypersurfaces $\Sigma$ that are
convenient for our approximation procedure, then in Subsection 2.2 we provide
some definitions and fundamental results for Bochner Lebesgue spaces that are
used throughout this paper, and in Subsection 2.3 we introduce the free Dirac
operator and a family of associated potential and boundary integral operators.

###  2.1. Description of the boundary $\boldsymbol{\Sigma}$ and its tubular
neighbourhood

In this section we formulate assumptions on the geometry of $\Sigma$ which
will allow us to construct the tubular neighbourhood of $\Sigma$ in (1.4) in a
similar manner as in [4, 30]. The presentation here differs slightly from [4],
as in the present paper we need better properties of the trace map and the
extension operator. The following assumption will be made throughout the
paper:

######  Hypothesis 2.1.

Assume that $\Omega_{+}\subset\mathbb{R}^{\theta}$, $\theta\in\\{2,3\\}$, is
an open set with boundary $\Sigma:=\partial\Omega_{+}$ and that there exist
open sets $W_{1},\dots,W_{p}\subset\mathbb{R}^{\theta}$, mappings
$\zeta_{1},\dots,\zeta_{p}\in C^{2}_{b}(\mathbb{R}^{\theta-1};\mathbb{R})$,
rotation matrices
$\kappa_{1},\dots,\kappa_{p}\in\mathbb{R}^{\theta\times\theta}$, and
$\varepsilon_{0}>0$ such that

  * (i)

$\Sigma\subset\bigcup_{l=1}^{p}W_{l}$;

  * (ii)

if $x\in\partial\Omega_{+}=\Sigma$, then there exists $l\in\\{1,\dots,p\\}$
such that $B(x,\varepsilon_{0})\subset W_{l}$;

  * (iii)

$W_{l}\cap\Omega_{+}=W_{l}\cap\Omega_{l}$, where
$\Omega_{l}=\\{\kappa_{l}\big{(}x^{\prime},x_{\theta}\big{)}:x_{\theta}<\zeta(x^{\prime}),\,(x^{\prime},x_{\theta})\in\mathbb{R}^{\theta}\\}$,
for $l\in\\{1,\dots,p\\}$.

Furthermore, we set
$\Sigma_{l}:=\partial\Omega_{l}=\\{\kappa_{l}(x^{\prime},\zeta_{l}(x^{\prime})):x^{\prime}\in\mathbb{R}^{\theta-1}\\}$,
$\Omega_{-}:=\mathbb{R}^{\theta}\setminus\overline{\Omega_{+}}$, and denote
the unit normal vector field at $\Sigma$ that is pointing outwards of
$\Omega_{+}$ by $\nu$.

We note that $C_{b}^{2}$-hypographs and the boundaries of compact
$C^{2}$-domains satisfy Hypothesis 2.1. Moreover, the sets $\Sigma_{l}$ are a
cover of $\Sigma$, but in general one has $\Sigma_{l}\not\subset\Sigma$. In
the following we recall the construction of Sobolev spaces on $\Sigma$
satisfying Hypothesis 2.1 and related results which will be useful in this
paper; cf. [31, 32] for similar constructions. First, if
$U\subset\mathbb{R}^{\theta-1}$ is open, then for $r\in[0,2]$ the space
$H^{r}(U;\mathbb{C}^{N})$ and also $W^{1}_{\infty}(U;\mathcal{V})$ with
$\mathcal{V}\in\\{\mathbb{C},\mathbb{C}^{N\times N}\\}$ are defined as in
[32]. To transfer these definitions to $\Sigma$, we choose a partition of
unity $\varphi_{1},\dots,\varphi_{p}\in C^{\infty}(\mathbb{R}^{\theta})$
subordinate to $W_{1},\dots,W_{p}$ such that each $\varphi_{l}$,
$l\in\\{1,\dots,p\\}$, has uniformly bounded derivatives (see Lemma B.2).
Next, we define

$x_{\Sigma_{l}}:\mathbb{R}^{\theta-1}\rightarrow\Sigma_{l},\qquad
x_{\Sigma_{l}}(x^{\prime}):=\kappa_{l}(x^{\prime},\zeta_{l}(x^{\prime})),$
(2.1)

and for $\psi\in L^{2}(\Sigma;\mathbb{C}^{N})$ we write

$\psi_{\Sigma_{l}}(x^{\prime}):=\begin{cases}\varphi_{l}(x_{\Sigma_{l}}(x^{\prime}))\psi(x_{\Sigma_{l}}(x^{\prime})),&\text{if
}x_{\Sigma_{l}}(x^{\prime})\in\Sigma,\\\ 0,&\text{if
}x_{\Sigma_{l}}(x^{\prime})\notin\Sigma.\end{cases}$

Then, $\psi_{\Sigma_{l}}\in L^{2}(\mathbb{R}^{\theta-1};\mathbb{C}^{N})$ and
$\psi(x_{\Sigma})=\sum_{l=1}^{p}\psi_{\Sigma_{l}}(x_{\Sigma_{l}}^{-1}(x_{\Sigma}))$
for $x_{\Sigma}\in\Sigma$. As usual, we introduce for $r\in[0,2]$

$H^{r}(\Sigma;\mathbb{C}^{N}):=\big{\\{}\psi\in
L^{2}(\Sigma;\mathbb{C}^{N}):\psi_{\Sigma_{l}}\in
H^{r}(\mathbb{R}^{\theta-1};\mathbb{C}^{N})\text{ for all
}l=1,\dots,p\big{\\}}$

and endow this space with the inner product

${\left({\phi},{\psi}\right)}_{H^{r}(\Sigma;\mathbb{C}^{N})}=\sum_{l=1}^{p}{\left({\phi_{\Sigma_{l}}},{\psi_{\Sigma_{l}}}\right)}_{H^{r}(\mathbb{R}^{\theta-1};\mathbb{C}^{N})},\qquad\phi,\psi\in
H^{r}(\Sigma;\mathbb{C}^{N}).$

Sobolev spaces $H^{r}(\Sigma;\mathbb{C}^{N})$ with $r\in[-2,0]$ are defined,
as usual, by duality. One can prove in the same manner as in [32, Theorem
B.11] that for $r_{1},r_{2}\in[-2,2]$ and $r=(1-\tau)r_{1}+\tau r_{2}$,
$0<\tau<1$, the interpolation property

$H^{r}(\Sigma;\mathbb{C}^{N})=\left[H^{r_{1}}(\Sigma;\mathbb{C}^{N}),H^{r_{2}}(\Sigma;\mathbb{C}^{N})\right]_{\tau}$
(2.2)

holds with equivalent norms. Eventually, we set
$U_{l}:=x_{\Sigma_{l}}^{-1}(\Sigma\cap W_{l})$ and define for
$\mathcal{V}\in\\{\mathbb{C};\mathbb{C}^{N\times N}\\}$

$W^{1}_{\infty}(\Sigma;\mathcal{V}):=\big{\\{}F\in
L^{\infty}(\Sigma;\mathcal{V}):(F\circ x_{\Sigma_{l}})\upharpoonright U_{l}\in
W^{1}_{\infty}(U_{l};\mathcal{V})\text{ for all }l=1,\dots,p\big{\\}}$

equipped with the norm

${\left\lVert{F}\right\rVert}_{W^{1}_{\infty}(\Sigma;\mathcal{V})}:=\max_{l\in\\{1,\dots,p\\}}{\left\lVert{(F\circ
x_{\Sigma_{l}})\upharpoonright
U_{l}}\right\rVert}_{W^{1}_{\infty}(U_{l};\mathcal{V})},\quad F\in
W^{1}_{\infty}(\Sigma;\mathcal{V}).$

Note that if $F\in W^{1}_{\infty}(\Sigma;\mathcal{V})$ and $r\in[0,1]$, then
by direct calculation ($r\in\\{0,1\\}$) and interpolation ($r\in(0,1)$) one
obtains for all $\phi\in H^{r}(\Sigma;\mathbb{C}^{N})$

$F\phi\in
H^{r}(\Sigma;\mathbb{C}^{N})\quad\textup{and}\quad{\left\lVert{F\phi}\right\rVert}_{H^{r}(\Sigma;\mathbb{C}^{N})}\leq
C{\left\lVert{F}\right\rVert}_{W^{1}_{\infty}(\Sigma;\mathcal{V})}{\left\lVert{\phi}\right\rVert}_{H^{r}(\Sigma;\mathbb{C}^{N})},$
(2.3)

where $C>0$ does not depend on $F$. In particular, $\phi\mapsto F\phi$ induces
a bounded operator in $H^{r}(\Sigma;\mathbb{C}^{N})$ and the operator norm is
bounded by
$C{\left\lVert{F}\right\rVert}_{W^{1}_{\infty}(\Sigma;\mathcal{V})}$. We
identify $F$ with this induced operator.

In the next proposition, we state a variant of the trace theorem suitable for
our situation; cf. [31, Theorem 2].

######  Proposition 2.2.

Assume that $\Omega_{\pm}$ and $\Sigma$ satisfy Hypothesis 2.1 and let
$r\in\big{(}\tfrac{1}{2},1\big{]}$. Then, the following is true:

  * (i)

There exists a unique bounded and surjective operator

$\boldsymbol{t}_{\Sigma}^{\pm}:H^{r}(\Omega_{\pm};\mathbb{C}^{N})\rightarrow
H^{r-1/2}(\Sigma;\mathbb{C}^{N})$

such that $\boldsymbol{t}_{\Sigma}^{\pm}u=u|_{\Sigma}$ for all $u\in
H^{r}(\Omega_{\pm};\mathbb{C}^{N})\cap
C(\overline{\Omega_{\pm}};\mathbb{C}^{N})$.

  * (ii)

There exists a unique bounded and surjective operator

$\boldsymbol{t}_{\Sigma}:H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N})\rightarrow
H^{r-1/2}(\Sigma;\mathbb{C}^{N})$

such that $\boldsymbol{t}_{\Sigma}u=u|_{\Sigma}$ for all $u\in
H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N})\cap
C(\mathbb{R}^{\theta};\mathbb{C}^{N})$.

In the rest of this subsection we summarize some statements on tubular
neighbourhoods of $\Sigma$ given by (1.4) by making use of the results in [4,
Section 2.1]. For this, some notation is necessary. Recall that the embedding
$\iota$ is defined by (1.3) and that the set $\Omega_{\varepsilon}$ for
$\varepsilon>0$ is introduced in (1.4). In the following definition we
introduce the Weingarten map (or shape operator) on $\Sigma$:

######  Definition 2.3.

Let $\Omega_{\pm},\Sigma$ satisfy Hypothesis 2.1 and denote for
$x_{\Sigma}=x_{\Sigma_{l}}(x^{\prime})\in\Sigma$ the tangent hyperplane of
$\Sigma$ in $x_{\Sigma}$ by
$T_{x_{\Sigma}}:=\textup{span}\,\\{\partial_{j}x_{\Sigma_{l}}(x^{\prime}):j=1,\dots,\theta-1\\}$.
The Weingarten map is the linear operator $W(x_{\Sigma}):T_{x_{\Sigma}}\to
T_{x_{\Sigma}}$ defined by

$W(x_{\Sigma})\partial_{j}x_{\Sigma_{l}}(x^{\prime})=-\partial_{j}\nu(x_{\Sigma_{l}}(x^{\prime})),\qquad
j\in\\{1,\dots,\theta-1\\}.$

As mentioned in [4, Definition 2.2] and the text below, $W(x_{\Sigma})$ is
well-defined (i.e. it is independent of the parametrization of $\Sigma$ and
the index $l$). Furthermore, we denote the matrix representation of
$W(x_{\Sigma})$ corresponding to the basis
$\\{\partial_{j}x_{\Sigma_{l}}(x^{\prime}):j=1,\dots,\theta-1\\}$ of
$T_{x_{\Sigma}}$ by $L_{l}(x^{\prime})$. Note that if
$x_{\Sigma}=x_{\Sigma_{l}}(x^{\prime})=x_{\Sigma_{k}}(y^{\prime})\in\Sigma$
with $l,k\in\\{1,\dots,p\\}$ and
$x^{\prime},y^{\prime}\in\mathbb{R}^{\theta-1}$, then the eigenvalues of
$L_{l}(x^{\prime})$ and $L_{k}(y^{\prime})$ coincide, see [4, Definition 2.2]
and the text below. Therefore, the expression
$\det(I-tW(x_{\Sigma})):=\det(I_{\theta-1}-tL_{l}(x^{\prime}))$ for
$t\in\mathbb{R}$ is well-defined. In the next proposition we state important
properties of $\iota$ and $W$, and identify $L^{1}(\Omega_{\varepsilon})$ with
$L^{1}(\Sigma\times(-\varepsilon,\varepsilon))$.

######  Proposition 2.4.

Let $\Omega_{\pm},\Sigma\subset\mathbb{R}^{\theta}$, $\theta\in\\{2,3\\}$,
satisfy Hypothesis 2.1. Then, there exists $\varepsilon_{1}>0$ such that the
following is true:

  * (i)

$\iota|_{\Sigma\times(-\varepsilon_{1},\varepsilon_{1})}$ is injective.

  * (ii)

There exists $C>0$ such that $\left\lvert{1-\det(I-\varepsilon
W(x_{\Sigma}))}\right\rvert\leq C\varepsilon<1/2$ for all
$x_{\Sigma}\in\Sigma$ and $\varepsilon\in(-\varepsilon_{1},\varepsilon_{1})$.

  * (iii)

For $\varepsilon\in(-\varepsilon_{1},\varepsilon_{1})$ one has $u\in
L^{1}(\Omega_{\varepsilon})$ if and only if
$u\circ\iota|_{\Sigma\times(-\varepsilon,\varepsilon)}\in
L^{1}(\Sigma\times(-\varepsilon,\varepsilon))$ and in this case

$\int_{\Omega_{\varepsilon}}u(y)\,dy=\int_{-\varepsilon}^{\varepsilon}\int_{\Sigma}u(y_{\Sigma}+s\nu(y_{\Sigma}))\det(I-sW(y_{\Sigma}))\,d\sigma(y_{\Sigma})\,ds.$

###### Proof.

Let $\varepsilon_{A}$ be the number specified in Proposition A.2 and let
$\varepsilon_{1}\in(0,\varepsilon_{A})$. Then by Proposition A.2 (ii) there
exists $C_{A,2}>0$ such that

$\left\lvert{\iota(x_{\Sigma},t)-\iota(y_{\Sigma},s)}\right\rvert\geq
C_{A,2}^{-1}(\left\lvert{x_{\Sigma}-y_{\Sigma}}\right\rvert+\left\lvert{t-s}\right\rvert),\quad(x_{\Sigma},t),(y_{\Sigma},s)\in\Sigma\times(-\varepsilon_{1},\varepsilon_{1}).$

Hence, $\iota|_{\Sigma\times(-\varepsilon_{1},\varepsilon_{1})}$ is injective,
i.e. item (i) is true.

Next, we show assertion (ii). For this, we first compute for
$l\in\\{1,\dots,p\\}$ and $x^{\prime}\in\mathbb{R}^{\theta-1}$ such that
$x_{\Sigma_{l}}(x^{\prime})\in\Sigma$ an explicit representation of the matrix
$L_{l}(x^{\prime})$, i.e. the matrix representation of
$W(x_{\Sigma_{l}}(x^{\prime}))$ with respect to the basis
$\\{\partial_{j}x_{\Sigma_{l}}(x^{\prime}):j=1,\dots,\theta-1\\}$. Let
$e_{1},\dots,e_{\theta}$ be the standard basis vectors in
$\mathbb{R}^{\theta}$ and note that

$\nu(x_{\Sigma_{l}}(x^{\prime}))=\frac{\kappa_{l}(-\nabla\zeta_{l}(x^{\prime}),1)}{\sqrt{1+\left\lvert{\nabla\zeta_{l}(x^{\prime})}\right\rvert^{2}}}=-\kappa_{l}\sum_{k=1}^{\theta-1}\frac{\partial_{k}\zeta_{l}(x^{\prime})}{\sqrt{1+\left\lvert{\nabla\zeta_{l}(x^{\prime})}\right\rvert^{2}}}e_{k}+\kappa_{l}\frac{1}{\sqrt{1+\left\lvert{\nabla\zeta_{l}(x^{\prime})}\right\rvert^{2}}}e_{\theta}.$

We compute

$\begin{split}-\partial_{j}\Bigg{(}&-\kappa_{l}\sum_{k=1}^{\theta-1}\frac{\partial_{k}\zeta_{l}(x^{\prime})}{\sqrt{1+\left\lvert{\nabla\zeta_{l}(x^{\prime})}\right\rvert^{2}}}e_{k}\Bigg{)}\\\
&=\sum_{k=1}^{\theta-1}\frac{(1+\left\lvert{\nabla\zeta_{l}(x^{\prime})}\right\rvert^{2})\partial_{j}\partial_{k}\zeta_{l}(x^{\prime})-\big{\langle}\nabla\partial_{j}\zeta_{l}(x^{\prime}),\nabla\zeta_{l}(x^{\prime})\big{\rangle}\partial_{k}\zeta_{l}(x^{\prime})}{(1+\left\lvert{\nabla\zeta_{l}(x^{\prime})}\right\rvert^{2})^{3/2}}\kappa_{l}e_{k}\end{split}$

and

$\begin{split}-\partial_{j}\Bigg{(}&\kappa_{l}\frac{1}{\sqrt{1+\left\lvert{\nabla\zeta_{l}(x^{\prime})}\right\rvert^{2}}}e_{\theta}\Bigg{)}=\kappa_{l}\frac{\big{\langle}\nabla\partial_{j}\zeta_{l}(x^{\prime}),\nabla\zeta_{l}(x^{\prime})\big{\rangle}}{(1+\left\lvert{\nabla\zeta_{l}(x^{\prime})}\right\rvert^{2})^{3/2}}e_{\theta}\\\
&=\frac{\big{\langle}\nabla\partial_{j}\zeta_{l}(x^{\prime}),\nabla\zeta_{l}(x^{\prime})\big{\rangle}(1+\left\lvert{\nabla\zeta_{l}(x^{\prime})}\right\rvert^{2})-\big{\langle}\nabla\partial_{j}\zeta_{l}(x^{\prime}),\nabla\zeta_{l}(x^{\prime})\big{\rangle}\left\lvert{\nabla\zeta_{l}(x^{\prime})}\right\rvert^{2}}{(1+\left\lvert{\nabla\zeta_{l}(x^{\prime})}\right\rvert^{2})^{3/2}}\kappa_{l}e_{\theta}\\\
&=\sum_{k=1}^{\theta-1}\frac{(1+\left\lvert{\nabla\zeta_{l}(x^{\prime})}\right\rvert^{2})\partial_{j}\partial_{k}\zeta_{l}(x^{\prime})-\big{\langle}\nabla\partial_{j}\zeta_{l}(x^{\prime}),\nabla\zeta_{l}(x^{\prime})\big{\rangle}\partial_{k}\zeta_{l}(x^{\prime})}{(1+\left\lvert{\nabla\zeta_{l}(x^{\prime})}\right\rvert^{2})^{3/2}}\kappa_{l}\partial_{k}\zeta_{l}(x^{\prime})e_{\theta}.\end{split}$

By combining the latter two equations with
$\partial_{k}x_{\Sigma_{l}}(x^{\prime})=\kappa_{l}(e_{k}+\partial_{k}\zeta_{l}(x^{\prime})e_{\theta})$,
we conclude

$\begin{split}W(&x_{\Sigma_{l}}(x^{\prime}))\partial_{j}x_{\Sigma_{l}}(x^{\prime})=-\partial_{j}\nu(x_{\Sigma_{l}}(x^{\prime}))\\\
&=\sum_{k=1}^{\theta-1}\frac{(1+\left\lvert{\nabla\zeta_{l}(x^{\prime})}\right\rvert^{2})\partial_{j}\partial_{k}\zeta_{l}(x^{\prime})-\big{\langle}\nabla\partial_{j}\zeta_{l}(x^{\prime}),\nabla\zeta_{l}(x^{\prime})\big{\rangle}\partial_{k}\zeta_{l}(x^{\prime})}{(1+\left\lvert{\nabla\zeta_{l}(x^{\prime})}\right\rvert^{2})^{3/2}}\partial_{k}x_{\Sigma_{l}}(x^{\prime}),\end{split}$

which shows that

$L_{l}(x^{\prime})=\frac{\Big{(}(1+\left\lvert{\nabla\zeta_{l}(x^{\prime})}\right\rvert^{2})\partial_{j}\partial_{k}\zeta_{l}(x^{\prime})-\big{\langle}\nabla\partial_{j}\zeta_{l}(x^{\prime}),\nabla\zeta_{l}(x^{\prime})\big{\rangle}\partial_{k}\zeta_{l}(x^{\prime})\Big{)}_{k,j\in\\{1,\dots,\theta-1\\}}}{(1+\left\lvert{\nabla\zeta_{l}(x^{\prime})}\right\rvert^{2})^{3/2}}.$
(2.4)

Let $x_{\Sigma}=x_{\Sigma_{l}}(x^{\prime})\in\Sigma\cap\Sigma_{l}$ and recall
that $\det(I-\varepsilon W(x_{\Sigma}))=\det(I_{\theta-1}-\varepsilon
L_{l}(x^{\prime}))$. Expressing the determinant as the product of the
eigenvalues one verifies $1-\det(I_{\theta-1}-\varepsilon
L_{l}(x^{\prime}))=\varepsilon P_{l}(\varepsilon)$, where $P_{l}$ is a
polynomial in $\varepsilon$ with coefficients depending continuously on the
entries of $L_{l}(x^{\prime})$. Since $\zeta_{l}\in
C^{2}_{b}(\mathbb{R}^{\theta-1};\mathbb{R})$ by Hypothesis 2.1, equation (2.4)
implies $\sup_{l\in\\{1,\dots,p\\},x^{\prime}\in
x_{\Sigma_{l}}^{-1}(\Sigma)}\left\lvert{L_{l}(x^{\prime})}\right\rvert<\infty.$
This shows that (ii) holds if $\varepsilon_{1}$ is chosen sufficiently small.

Finally, the claim in (iii) follows from [4, Proposition 2.6], since (i),
(ii), and Proposition A.2 (i) show that $\Sigma$ satisfying Hypothesis 2.1
also fulfils [4, Hypothesis 2.3]. ∎

###  2.2. Bochner Lebesgue spaces

In this subsection we summarize some results on Bochner Lebesgue spaces that
will be used throughout this paper; for details we refer to [26, Chapter 1].
We always assume that $\mathcal{H}$ and $\mathcal{G}$ are separable Hilbert
spaces, $\mathcal{O},\mathcal{O}_{1},\mathcal{O}_{2}$ are Borel sets in
$\mathbb{R}^{n}$, $n\in\mathbb{N}$, and $\mathcal{L}(\mathcal{H},\mathcal{G})$
is the set of bounded linear operators from $\mathcal{H}$ to $\mathcal{G}$.

######  Definition 2.5.

We call $f:\mathcal{O}\to\mathcal{H}$ (weakly) measurable, if for all
$\varphi\in\mathcal{H}$ the mapping $\mathcal{O}\ni
t\mapsto{\left({f(t)},{\varphi}\right)}_{\mathcal{H}}$ is measurable with
respect to the Lebesgue measure on $\mathcal{O}$. Furthermore, we call
$F:\mathcal{O}\to\mathcal{L}(\mathcal{H},\mathcal{G})$ measurable, if
$\mathcal{O}\ni t\mapsto F(t)h$ is measurable for all $h\in\mathcal{H}$.

We recall for completeness that a function $f:\mathcal{O}\to\mathcal{H}$ is
(strongly) measurable, if $f$ is the pointwise limit of simple functions, and
that in the present situation both notions of measurability coincide due to
Pettis theorem, see [26, Theorem 1.1.20]. Moreover, if
$f:\mathcal{O}\to\mathcal{H}$ and
$F:\mathcal{O}\to\mathcal{L}(\mathcal{H},\mathcal{G})$ are measurable, then
the function $\mathcal{O}\ni t\mapsto F(t)f(t)\in\mathcal{G}$ is measurable,
see [26, Proposition 1.1.28].

For a measurable function $f:\mathcal{O}\to\mathcal{H}$ such that
$t\mapsto{\left\lVert{f(t)}\right\rVert}_{\mathcal{H}}$ is integrable, the
Bochner integral $\int_{\mathcal{O}}f(t)\,dt\in\mathcal{H}$ is defined in the
standard way, see [26, Definition 1.2.1 and Proposition 1.2.2]. Many standard
results for (usual) Lebesgue integrals admit natural generalizations to
Bochner integrals. In particular, we will make use of Fubini’s theorem for
Bochner integrals, see [26, Proposition 1.2.7]: If
$f:\mathcal{O}_{1}\times\mathcal{O}_{2}\to\mathcal{H}$ is measurable and the
integral
$\int_{\mathcal{O}_{1}\times\mathcal{O}_{2}}{\left\lVert{f(t,s)}\right\rVert}_{\mathcal{H}}\,\,dt\,ds$
is finite, then the Bochner integral $\int_{\mathcal{O}_{2}}f(t,s)\,ds$ exists
for a.e. $t\in\mathcal{O}_{1}$ and the function $\mathcal{O}_{1}\ni
t\mapsto\int_{\mathcal{O}_{2}}f(t,s)$ is measurable and Bochner integrable.

Next, we introduce Bochner Lebesgue spaces.

######  Definition 2.6.

We define $L^{2}(\mathcal{O};\mathcal{H})$ as the space which contains all
(equivalence classes of) measurable functions $f:\mathcal{O}\to\mathcal{H}$
such that

$\int_{\mathcal{O}}{\left\lVert{f(t)}\right\rVert}_{\mathcal{H}}^{2}\,dt<\infty.$

Furthermore, we equip this space with the scalar product

$\int_{\mathcal{O}}{\left({f(t)},{g(t)}\right)}_{\mathcal{H}}\,dt,\qquad
f,g\in L^{2}(\mathcal{O};\mathcal{H}).$

It is not difficult to show that $L^{2}(\mathcal{O};\mathcal{H})$ is a Hilbert
space; cf. the comments below [26, Definition 1.2.15].

Next, duality and interpolation properties of Bochner Lebesgue spaces are
discussed. According to [26, Corollary 1.3.13 and Theorem 1.3.21], the
identification of $f\in L^{2}(\mathcal{O};\mathcal{H}^{*})$ with the
functional defined by

$L^{2}(\mathcal{O};\mathcal{H})\ni
g\mapsto\int_{\mathcal{O}}{{}_{\mathcal{H}^{*}}}{\left\langle{f(t)},{g(t)}\right\rangle}_{\mathcal{H}}\,dt,$

where
${{}_{\mathcal{H}^{*}}}{\left\langle{\cdot},{\cdot}\right\rangle}_{\mathcal{H}}$
denotes the linear dual pairing on $\mathcal{H}^{*}\times\mathcal{H}$, induces
an isometric isomorphism between $L^{2}(\mathcal{O};\mathcal{H}^{*})$ and the
dual space of $L^{2}(\mathcal{O};\mathcal{H})$, i.e.

$L^{2}(\mathcal{O};\mathcal{H})^{*}\simeq L^{2}(\mathcal{O};\mathcal{H}^{*}).$
(2.5)

If $\mathcal{G}$ is a Hilbert space such that $(\mathcal{H},\mathcal{G})$ is a
compatible pair (cf. the notations section), then also
$(L^{2}(\mathcal{O};\mathcal{H}),L^{2}(\mathcal{O};\mathcal{G}))$ is a
compatible pair and

$L^{2}\big{(}\mathcal{O};[\mathcal{H},\mathcal{G}]_{\tau}\big{)}=\big{[}L^{2}\big{(}\mathcal{O};\mathcal{H}\big{)},L^{2}\big{(}\mathcal{O};\mathcal{G}\big{)}\big{]}_{\tau},\quad\tau\in(0,1),$
(2.6)

with equivalent norms, see [26, Theorem 2.2.6 and Corollary C.4.2].

In this paper we are particularly interested in the Bochner Lebesgue spaces
$L^{2}((-1,1);H^{r}(\Sigma;\mathbb{C}^{N}))$, $r\in[-2,2]$, where
$\Sigma\subset\mathbb{R}^{\theta}$, $\theta\in\\{2,3\\}$, is a hypersurface
satisfying Hypothesis 2.1 and $H^{r}(\Sigma;\mathbb{C}^{N})$ is defined as in
Section 2.1. We summarize important properties of these spaces in the
following proposition.

######  Proposition 2.7.

Let $\Omega_{\pm},\Sigma\subset\mathbb{R}^{\theta}$, $\theta\in\\{2,3\\}$,
satisfy Hypothesis 2.1. Then, the following is true:

  * (i)

If $\tau\in(0,1)$, $r_{1},r_{2}\in[-2,2]$ and $r=(1-\tau)r_{1}+\tau r_{2}$,
then

$\begin{split}&L^{2}\big{(}(-1,1);H^{r}(\Sigma;\mathbb{C}^{N})\big{)}\\\
&\hskip
30.0pt=\big{[}L^{2}\big{(}(-1,1);H^{r_{1}}(\Sigma;\mathbb{C}^{N})\big{)},L^{2}\big{(}(-1,1);H^{r_{2}}(\Sigma;\mathbb{C}^{N})\big{)}\big{]}_{\tau}\end{split}$

and the corresponding norms are equivalent.

  * (ii)

For $r\in[0,2]$ there exists an isometric isomorphism between the dual space
of $L^{2}((-1,1);H^{r}(\Sigma;\mathbb{C}^{N}))$ and
$L^{2}((-1,1);H^{-r}(\Sigma;\mathbb{C}^{N}))$, i.e.

$L^{2}\big{(}(-1,1);H^{r}(\Sigma;\mathbb{C}^{N})\big{)}^{*}\simeq
L^{2}\big{(}(-1,1);H^{-r}(\Sigma;\mathbb{C}^{N})\big{)}.$

  * (iii)

The identification of $F\in L^{2}(\Sigma\times(-1,1);\mathbb{C}^{N})$ with the
function $f:t\mapsto F(\cdot,t)$ induces an isometric isomorphism between
$L^{2}(\Sigma\times(-1,1);\mathbb{C}^{N})$ and
$L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$ and

$\bigg{(}\int_{-1}^{1}f(t)\,dt\bigg{)}(x_{\Sigma})=\int_{-1}^{1}f(t)(x_{\Sigma})\,dt\quad\textup{
for }\sigma\textup{-a.e. }x_{\Sigma}\in\Sigma.$

###### Proof.

Items (i) and (ii) follow from (2.6) and (2.5) and the properties of
$H^{r}(\Sigma;\mathbb{C}^{N})$ in Section 2.1 and (2.2). Assertion (iii)
follows from [26, Proposition 1.2.24 and Proposition 1.2.25]. ∎

The following identification turns out to be useful in our considerations: For
$\mathcal{Q}\in L^{\infty}((-1,1);\mathbb{C})$ and a bounded operator
$\mathcal{A}$ in $H^{r}(\Sigma;\mathbb{C}^{N})$, $r\in[-2,2]$, we identify

$\mathcal{M}_{\mathcal{Q}}:L^{2}\big{(}(-1,1);H^{r}(\Sigma;\mathbb{C}^{N})\big{)}\to
L^{2}\big{(}(-1,1);H^{r}(\Sigma;\mathbb{C}^{N})\big{)},\quad\\!\\!(\mathcal{M}_{\mathcal{Q}}f)(t):=\mathcal{Q}(t)f(t),$

and

$\mathcal{M}_{\mathcal{A}}:L^{2}\big{(}(-1,1);H^{r}(\Sigma;\mathbb{C}^{N})\big{)}\to
L^{2}\big{(}(-1,1);H^{r}(\Sigma;\mathbb{C}^{N})\big{)},\quad(\mathcal{M}_{\mathcal{A}}f)(t):=\mathcal{A}(f(t)),$

with $\mathcal{Q}$ and $\mathcal{A}$, respectively. Note that the norms
${\left\lVert{\mathcal{M}_{\mathcal{Q}}}\right\rVert}_{r\to r}$ and
${\left\lVert{\mathcal{M}_{\mathcal{A}}}\right\rVert}_{r\to r}$ are bounded by
${\left\lVert{\mathcal{Q}}\right\rVert}_{L^{\infty}((-1,1);\mathbb{C})}$ and
${\left\lVert{\mathcal{A}}\right\rVert}_{H^{r}(\Sigma;\mathbb{C}^{N})\to
H^{r}(\Sigma;\mathbb{C}^{N})}$, respectively. We will also use the bounded
embedding

$\mathfrak{J}:H^{r}(\Sigma;\mathbb{C}^{N})\to
L^{2}\big{(}(-1,1);H^{r}(\Sigma;\mathbb{C}^{N})\big{)},\quad(\mathfrak{J}\varphi)(t):=\varphi,$
(2.7)

and its adjoint

$\mathfrak{J}^{*}:L^{2}\big{(}(-1,1);H^{r}(\Sigma;\mathbb{C}^{N})\big{)}\to
H^{r}(\Sigma;\mathbb{C}^{N}),\quad\mathfrak{J}^{*}f=\int_{-1}^{1}f(t)\,dt.$

###  2.3. The free Dirac operator and associated integral operators

Let $m\in\mathbb{R}$ and recall that the Dirac matrices
$\alpha_{1},\dots,\alpha_{\theta},\beta\in\mathbb{C}^{N\times N}$ are given by
(1.15)–(1.16). Then the free Dirac operator $H$ is the differential operator
in $L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ given by

$\begin{split}Hu&:=-i(\alpha\cdot\nabla)u+m\beta u,\qquad\textup{dom
}H:=H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N}).\end{split}$ (2.8)

With the help of the Fourier transform one gets that $H$ is self-adjoint in
$L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ and
$\sigma(H)=\left(-\infty,-\left\lvert{m}\right\rvert\right]\cup\left[\left\lvert{m}\right\rvert,\infty\right)$,
see for instance [11, Section 2] for $\theta=2$ and [41, Theorem 1.1] for
$\theta=3$. For $z\notin\sigma(H)$ the resolvent $R_{z}$ is

$R_{z}u(x):=(H-z)^{-1}u(x)=\int_{\mathbb{R}^{\theta}}G_{z}(x-y)u(y)\,dy,\qquad
u\in L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N}),~{}x\in\mathbb{R}^{\theta},$
(2.9)

where $G_{z}$ is given for $\theta=2$ and $x\in\mathbb{R}^{2}\setminus\\{0\\}$
by

$\begin{split}G_{z}(x)&=\frac{\sqrt{z^{2}-m^{2}}}{2\pi}K_{1}\big{(}-i\sqrt{z^{2}-m^{2}}\left\lvert{x}\right\rvert\big{)}\frac{\alpha\cdot
x}{\left\lvert{x}\right\rvert}\\\ &\hskip
100.0pt+\frac{1}{2\pi}K_{0}\big{(}-i\sqrt{z^{2}-m^{2}}\left\lvert{x}\right\rvert\big{)}\left(m\beta+zI_{2}\right)\end{split}$
(2.10)

and for $\theta=3$ and $x\in\mathbb{R}^{3}\setminus\\{0\\}$ by

$G_{z}(x)=\left(zI_{4}+m\beta+i\left(1-i\sqrt{z^{2}-m^{2}}\left\lvert{x}\right\rvert\right)\frac{\alpha\cdot
x}{\left\lvert{x}\right\rvert^{2}}\right)\frac{e^{i\sqrt{z^{2}-m^{2}}\left\lvert{x}\right\rvert}}{4\pi\left\lvert{x}\right\rvert};$
(2.11)

cf., e.g., [9, 10, 41]. Here, $K_{0}$ and $K_{1}$ denote the modified Bessel
functions of the second kind of order zero and one, respectively, and the
branch of the square root is fixed by $\text{Im}\,\sqrt{w}>0$ for
$w\in\mathbb{C}\setminus[0,\infty)$. Note that $R_{z}$ is bounded in
$L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ and it can also be viewed as a
bounded operator from $L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ to
$H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})$. In fact, with the help of the
Fourier transform, it is not difficult to show that $R_{z}$ gives rise to a
bounded operator from $H^{s}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ to
$H^{s+1}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ for all $s\in\mathbb{R}$.

We move on to the discussion of potential and boundary integral operators
associated with the free Dirac operator. In the following, let
$z\in\rho(H)=\mathbb{C}\setminus((-\infty,-|m|]\cup[|m|,\infty))$ be fixed and
let $\Omega_{\pm}$ and $\Sigma\subset\mathbb{R}^{\theta}$ satisfy Hypothesis
2.1. First, we introduce the potential operator
$\Phi_{z}:L^{2}(\Sigma;\mathbb{C}^{N})\rightarrow
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ by

$\displaystyle\Phi_{z}\varphi(x)$
$\displaystyle:=\int_{\Sigma}G_{z}(x-y_{\Sigma})\varphi(y_{\Sigma})\,d\sigma(y_{\Sigma}),\qquad\varphi\in
L^{2}(\Sigma;\mathbb{C}^{N}),~{}x\in\mathbb{R}^{\theta}.$ (2.12)

We note that $\Phi_{z}$ is indeed well-defined and bounded, see [2, Lemma
2.1]. Further properties of $\Phi_{z}$ are summarized in the following
proposition. For compact hypersurfaces $\Sigma$, these results are well known,
see, e.g., [10, Theorem 4.3], but for unbounded $\Sigma$ they were not treated
in the literature so far. For completeness, we give a proof of these results
in Appendix C.

######  Proposition 2.8.

Let $z\in\rho(H)=\mathbb{C}\setminus((-\infty,-|m|]\cup[|m|,\infty))$ and let
$\Phi_{z}$ be given by (2.12). Then, the following is true:

  * (i)

For any $r\in\big{[}0,\tfrac{1}{2}\big{]}$ the operator $\Phi_{z}$ gives rise
to a bounded operator

$\begin{split}\Phi_{z}:H^{r}(\Sigma;\mathbb{C}^{N})\to
H^{r+1/2}(\Omega_{+};\mathbb{C}^{N})\oplus
H^{r+1/2}(\Omega_{-};\mathbb{C}^{N}).\end{split}$

  * (ii)

For $\varphi\in H^{1/2}(\Sigma;\mathbb{C}^{N})$ one has
$[(-i(\alpha\cdot\nabla)+m\beta-zI_{N})\Phi_{z}\varphi]_{\pm}=0$.

  * (iii)

The adjoint $\Phi_{z}^{*}:L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
L^{2}(\Sigma;\mathbb{C}^{N})$ of $\Phi_{z}$ acts on $u\in
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ as

$\Phi_{z}^{*}u(x_{\Sigma})=\int_{\mathbb{R}^{\theta}}G_{\overline{z}}(x_{\Sigma}-y)u(y)\,dy=\boldsymbol{t}_{\Sigma}R_{\overline{z}}u(x_{\Sigma}),\qquad
x_{\Sigma}\in\Sigma,$ (2.13)

and $\Phi_{z}^{*}$ gives rise to a bounded operator
$\Phi_{z}^{*}:L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\rightarrow
H^{1/2}(\Sigma;\mathbb{C}^{N})$.

Finally, we introduce a family of boundary integral operators for the Dirac
equation. Let
$z\in\rho(H)=\mathbb{C}\setminus((-\infty,-|m|]\cup[|m|,\infty))$. Then, we
define the map $\mathcal{C}_{z}:H^{1/2}(\Sigma;\mathbb{C}^{N})\rightarrow
H^{1/2}(\Sigma;\mathbb{C}^{N})$ by

$\mathcal{C}_{z}\varphi:=\frac{1}{2}\big{(}\boldsymbol{t}_{\Sigma}^{+}(\Phi_{z}\varphi)_{+}+\boldsymbol{t}_{\Sigma}^{-}(\Phi_{z}\varphi)_{-}\big{)},\qquad\varphi\in
H^{1/2}(\Sigma;\mathbb{C}^{N}).$ (2.14)

We remark that the operator $\mathcal{C}_{z}$ can be represented as a strongly
singular boundary integral operator, see for instance [10, eq. (4.5) and
Proposition 4.4 (ii)] for the case that $\Omega_{+}$ is bounded. However, for
our purposes the representation in (2.14) is more convenient. The basic
properties of $\mathcal{C}_{z}$ are stated in the following proposition.
Again, for compact hypersurfaces $\Sigma$ they are well-known, see, e.g., [10,
Theorem 4.3, Proposition 4.4, and Corollary 4.5]. For general (also unbounded)
hypersurfaces $\Sigma$ satisfying Hypothesis 2.1 we give a proof in Appendix
C.

######  Proposition 2.9.

Let $z\in\rho(H)=\mathbb{C}\setminus((-\infty,-|m|]\cup[|m|,\infty))$ and let
$\mathcal{C}_{z}$ be given by (2.14). Then, the following is true:

  * (i)

For any $r\in\big{[}-\frac{1}{2},\frac{1}{2}\big{]}$ the map $\mathcal{C}_{z}$
has a bounded extension
$\mathcal{C}_{z}:H^{r}(\Sigma;\mathbb{C}^{N})\rightarrow
H^{r}(\Sigma;\mathbb{C}^{N})$.

  * (ii)

For any $r\in\big{(}0,\frac{1}{2}\big{]}$ and $\varphi\in
H^{r}(\Sigma;\mathbb{C}^{N})$ one has

$\mathcal{C}_{z}\varphi=\pm\frac{i}{2}(\alpha\cdot\nu)\varphi+\boldsymbol{t}_{\Sigma}^{\pm}\left(\Phi_{z}\varphi\right)_{\Omega_{\pm}}.$

We note that item (ii) of the previous proposition is a version of the well-
known Plemelj-Sokhotski formula, see for instance [2, Lemma 3.3 (i)], where
$\mathcal{C}_{z}$ is represented as a singular boundary integral operator.

##  3\. Resolvent formula for $H_{\varepsilon}$ and analysis of the associated
integral operators

Throughout this section let $\varepsilon_{1}$ be the number specified in
Proposition 2.4, so that $\iota$ acts as a bijective map from
$\Sigma\times(-\varepsilon_{1},\varepsilon_{1})$ to
$\Omega_{\varepsilon_{1}}$. Moreover, let $\varepsilon\in(0,\varepsilon_{1})$,
$z\in\rho(H)=\mathbb{C}\setminus((-\infty,-|m|]\cup[|m|,\infty))$, $G_{z}$ be
the function defined in (2.10)–(2.11), and $\Sigma$ be a hypersurface
satisfying Hypothesis 2.1 with associated Weingarten map $W$ (see Definition
2.3). This section is devoted to the study of the integral operators which
formally act on $f\in L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$ and $u\in
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ as

$\displaystyle A_{\varepsilon}(z)f(x)$
$\displaystyle=\int_{-1}^{1}\int_{\Sigma}G_{z}(x-y_{\Sigma}-\varepsilon
s\nu(y_{\Sigma}))f(s)(y_{\Sigma})$ (3.1a) $\displaystyle\hskip
120.0pt\cdot\det(I-\varepsilon sW(y_{\Sigma}))\,d\sigma(y_{\Sigma})\,ds,$
$\displaystyle B_{\varepsilon}(z)f(t)(x_{\Sigma})$
$\displaystyle=\int_{-1}^{1}\int_{\Sigma}G_{z}(x_{\Sigma}+\varepsilon
t\nu(x_{\Sigma})-y_{\Sigma}-\varepsilon s\nu(y_{\Sigma}))f(s)(y_{\Sigma})$
(3.1b) $\displaystyle\hskip 120.0pt\cdot\det(I-\varepsilon
sW(y_{\Sigma}))\,d\sigma(y_{\Sigma})\,ds,$ $\displaystyle
C_{\varepsilon}(z)u(t)(x_{\Sigma})$
$\displaystyle=\int_{\mathbb{R}^{\theta}}G_{z}(x_{\Sigma}+\varepsilon
t\nu(x_{\Sigma})-y)u(y)\,dy,$ (3.1c)

for a.e. $x\in\mathbb{R}^{\theta}$, a.e. $t\in(-1,1)$, and $\sigma$-a.e.
$x_{\Sigma}\in\Sigma$. First, in Section 3.1 we define these operators
rigorously and show their relation to the resolvent of the operator
$H_{\varepsilon}$ in (1.8). Then, in Section 3.2 we introduce and investigate
a shift operator which plays an important role in the convergence analysis of
$A_{\varepsilon}(z),B_{\varepsilon}(z)$, and $C_{\varepsilon}(z)$ in Section
3.3. Finally, in Section 3.4 we discuss the convergence of
$\left(I+{B_{\varepsilon}(z)}Vq\right)^{-1}$.

###  3.1. Definition and elementary results on
$A_{\varepsilon}(z),B_{\varepsilon}(z)$, and $C_{\varepsilon}(z)$

First, we rigorously define the operators
$A_{\varepsilon}(z),B_{\varepsilon}(z)$, and $C_{\varepsilon}(z)$ formally
given by (3.1). Recall that $\Omega_{\varepsilon}$ was defined in (1.4) and
introduce the mappings

$\displaystyle\mathcal{I}_{\varepsilon}:L^{2}((-\varepsilon,\varepsilon);L^{2}(\Sigma;\mathbb{C}^{N}))\to
L^{2}(\Omega_{\varepsilon};\mathbb{C}^{N}),$
$\displaystyle\mathcal{I}_{\varepsilon}f(x_{\Sigma}+t\nu(x_{\Sigma})):=f(t)(x_{\Sigma}),$
(3.2)
$\displaystyle\mathcal{I}_{\varepsilon}^{-1}:L^{2}(\Omega_{\varepsilon};\mathbb{C}^{N})\to
L^{2}((-\varepsilon,\varepsilon);L^{2}(\Sigma;\mathbb{C}^{N})),$
$\displaystyle\mathcal{I}_{\varepsilon}^{-1}u(t)(x_{\Sigma}):=u(x_{\Sigma}+t\nu(x_{\Sigma})),$

and

$\displaystyle\mathcal{S}_{\varepsilon}:L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))\to
L^{2}((-\varepsilon,\varepsilon);L^{2}(\Sigma;\mathbb{C}^{N})),$
$\displaystyle\mathcal{S}_{\varepsilon}g(t):=\frac{1}{\sqrt{\varepsilon}}g\left(\frac{t}{\varepsilon}\right),$
(3.3)
$\displaystyle\mathcal{S}_{\varepsilon}^{-1}:L^{2}((-\varepsilon,\varepsilon);L^{2}(\Sigma;\mathbb{C}^{N}))\to
L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N})),$
$\displaystyle\mathcal{S}_{\varepsilon}^{-1}g(t):=\sqrt{\varepsilon}g(\varepsilon
t).$

According to Proposition 2.4 and Proposition 2.7 (iii) for any
$\varepsilon\in(0,\varepsilon_{1})$ these mappings are well-defined, bounded,
invertible, and their inverses have the claimed form, see also [4, equations
(3.6) and (3.7)]. Moreover, set
$u_{\varepsilon}:=\frac{\chi_{\Omega_{\varepsilon}}}{\sqrt{\varepsilon}}$,
where $\chi_{\Omega_{\varepsilon}}$ is the characteristic function for
$\Omega_{\varepsilon}$, and define the operators

$U_{\varepsilon}:L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
L^{2}(\Omega_{\varepsilon};\mathbb{C}^{N})\quad\text{and}\quad
U_{\varepsilon}^{*}:L^{2}(\Omega_{\varepsilon};\mathbb{C}^{N})\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$

acting on $u\in L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ and $v\in
L^{2}(\Omega_{\varepsilon};\mathbb{C}^{N})$ as

$U_{\varepsilon}u=(u_{\varepsilon}u)\upharpoonright\Omega_{\varepsilon}\quad\text{and}\quad
U_{\varepsilon}^{*}v=\begin{cases}u_{\varepsilon}v&\text{ in
}\Omega_{\varepsilon},\\\ 0&\text{ in
}\mathbb{R}^{\theta}\setminus\Omega_{\varepsilon}.\end{cases}$

Recall that we use the notation $R_{z}=(H-z)^{-1}$ for the resolvent of the
free Dirac operator $H$ given by (2.8). Then, we define

$\displaystyle
A_{\varepsilon}(z):=R_{z}U_{\varepsilon}^{*}\mathcal{I}_{\varepsilon}\mathcal{S}_{\varepsilon}$
$\displaystyle:L^{2}\big{(}(-1,1);L^{2}(\Sigma;\mathbb{C}^{N})\big{)}\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N}),$ (3.4) $\displaystyle
B_{\varepsilon}(z):=\mathcal{S}_{\varepsilon}^{-1}\mathcal{I}_{\varepsilon}^{-1}U_{\varepsilon}R_{z}U_{\varepsilon}^{*}\mathcal{I}_{\varepsilon}\mathcal{S}_{\varepsilon}$
$\displaystyle:L^{2}\big{(}(-1,1);L^{2}(\Sigma;\mathbb{C}^{N})\big{)}\to
L^{2}\big{(}(-1,1);L^{2}(\Sigma;\mathbb{C}^{N})\big{)},$ $\displaystyle
C_{\varepsilon}(z):=\mathcal{S}_{\varepsilon}^{-1}\mathcal{I}_{\varepsilon}^{-1}U_{\varepsilon}R_{z}$
$\displaystyle:L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
L^{2}\big{(}(-1,1);L^{2}(\Sigma;\mathbb{C}^{N})\big{)}.$

Note that by definition these operators are well-defined and bounded. In the
next proposition we show that these operators coincide with the formal
expressions in (3.1).

######  Proposition 3.1.

For the operators $A_{\varepsilon}(z)$, $B_{\varepsilon}(z)$, and
$C_{\varepsilon}(z)$ defined in (3.4) the representations in (3.1) hold.

###### Proof.

First, we show the claim for $A_{\varepsilon}(z)$. Let $f\in
L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$. Using (2.9), Proposition 2.4
(iii), (3.2), and (3.3) we find

$\begin{split}A_{\varepsilon}&(z)f(x)=\left(R_{z}U_{\varepsilon}^{*}\mathcal{I}_{\varepsilon}\mathcal{S}_{\varepsilon}f\right)(x)\\\
&=\int_{\mathbb{R}^{\theta}}G_{z}(x-y)\left(U_{\varepsilon}^{*}\mathcal{I}_{\varepsilon}\mathcal{S}_{\varepsilon}f\right)(y)\,dy\\\
&=\int_{\Omega_{\varepsilon}}G_{z}(x-y)u_{\varepsilon}(y)\left(\mathcal{I}_{\varepsilon}\mathcal{S}_{\varepsilon}f\right)(y)\,dy\\\
&=\int_{-\varepsilon}^{\varepsilon}\int_{\Sigma}G_{z}(x-(y_{\Sigma}+s\nu(y_{\Sigma})))\frac{1}{\sqrt{\varepsilon}}\left(\mathcal{I}_{\varepsilon}\mathcal{S}_{\varepsilon}f\right)(y_{\Sigma}+s\nu(y_{\Sigma}))\\\
&\hskip 200.0pt\cdot\det(I-sW(y_{\Sigma}))\,d\sigma(y_{\Sigma})\,ds\\\
&=\int_{-\varepsilon}^{\varepsilon}\int_{\Sigma}G_{z}(x-y_{\Sigma}-s\nu(y_{\Sigma}))\frac{1}{\sqrt{\varepsilon}}(\mathcal{S}_{\varepsilon}f)(s)(y_{\Sigma})\det(I-sW(y_{\Sigma}))\,d\sigma(y_{\Sigma})\,ds\\\
&=\int_{-\varepsilon}^{\varepsilon}\int_{\Sigma}G_{z}(x-y_{\Sigma}-s\nu(y_{\Sigma}))\frac{1}{{\varepsilon}}f\\!\left(\frac{s}{\varepsilon}\right)\\!(y_{\Sigma})\det(I-sW(y_{\Sigma}))\,d\sigma(y_{\Sigma})\,ds\\\
&=\int_{-1}^{1}\int_{\Sigma}G_{z}(x-y_{\Sigma}-\varepsilon
s\nu(y_{\Sigma}))f(s)(y_{\Sigma})\det(I-\varepsilon
sW(y_{\Sigma}))\,d\sigma(y_{\Sigma})\,ds\end{split}$

for a.e. $x\in\mathbb{R}^{\theta}$, which is the claimed identity. Next, to
prove the claim for $C_{\varepsilon}(z)$ we note for $v\in
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$, a.e. $t\in(-1,1)$, and
$\sigma$-a.e. $x_{\Sigma}\in\Sigma$ that

$\begin{split}C_{\varepsilon}(z)v(t)(x_{\Sigma})&=\bigl{(}\mathcal{S}_{\varepsilon}^{-1}\mathcal{I}_{\varepsilon}^{-1}U_{\varepsilon}R_{z}v\bigr{)}(t)(x_{\Sigma})\\\
&=\sqrt{\varepsilon}(U_{\varepsilon}R_{z}v)(x_{\Sigma}+\varepsilon
t\nu(x_{\Sigma}))\\\ &=\int_{\mathbb{R}^{\theta}}G_{z}(x_{\Sigma}+\varepsilon
t\nu(x_{\Sigma})-y)v(y)\,dy.\end{split}$

The representation for $B_{\varepsilon}(z)$ follows by combining the last two
calculations. ∎

Next, we show a resolvent formula for $H_{\varepsilon}$ involving the
operators $A_{\varepsilon}(z),B_{\varepsilon}(z)$, and $C_{\varepsilon}(z)$
which will be useful for the convergence analysis. Using the identifications
in the end of Section 2.2 we regard $Vq$ in the following as a multiplication
operator in $L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$. A similar formula is
shown in [30, Lemma 3.1].

######  Proposition 3.2.

Let $A_{\varepsilon}(z)$, $B_{\varepsilon}(z)$, and $C_{\varepsilon}(z)$ be
given by (3.4), let $q$ and $V$ be given by (1.5) and (1.6), respectively, and
let $H_{\varepsilon}$ be given by (1.8). If $-1\in\rho(B_{\varepsilon}(z)Vq)$,
then $z\in\rho(H_{\varepsilon})$ and

$(H_{\varepsilon}-z)^{-1}=R_{z}-A_{\varepsilon}(z)Vq(I+B_{\varepsilon}(z)Vq)^{-1}C_{\varepsilon}(z).$

###### Proof.

Let $V_{\varepsilon}$ be given by (1.7) and set
$v_{\varepsilon}:=\sqrt{\varepsilon}V_{\varepsilon}$. Then,
$H_{\varepsilon}=H+v_{\varepsilon}u_{\varepsilon}$ and
$v_{\varepsilon}\mathcal{I}_{\varepsilon}\mathcal{S}_{\varepsilon}=u_{\varepsilon}\mathcal{I}_{\varepsilon}\mathcal{S}_{\varepsilon}Vq$.
Hence, due to the invertibility of $I+B_{\varepsilon}(z)Vq$ and (3.4) we
obtain

$\begin{split}(H_{\varepsilon}&-z)\bigl{(}R_{z}-A_{\varepsilon}(z)Vq\left(I+B_{\varepsilon}(z)Vq\right)^{-1}C_{\varepsilon}(z)\bigr{)}\\\
&=(H-z+v_{\varepsilon}u_{\varepsilon})\bigl{(}R_{z}-R_{z}v_{\varepsilon}\left(I+u_{\varepsilon}R_{z}v_{\varepsilon}\right)^{-1}u_{\varepsilon}R_{z}\bigr{)}\\\
&=I+v_{\varepsilon}u_{\varepsilon}R_{z}-v_{\varepsilon}\left(I+u_{\varepsilon}R_{z}v_{\varepsilon}\right)^{-1}u_{\varepsilon}R_{z}-v_{\varepsilon}u_{\varepsilon}R_{z}v_{\varepsilon}\left(I+u_{\varepsilon}R_{z}v_{\varepsilon}\right)^{-1}u_{\varepsilon}R_{z}\\\
&=I+v_{\varepsilon}u_{\varepsilon}R_{z}-v_{\varepsilon}\left(I+u_{\varepsilon}R_{z}v_{\varepsilon}\right)^{-1}u_{\varepsilon}R_{z}+v_{\varepsilon}\left(I+u_{\varepsilon}R_{z}v_{\varepsilon}\right)^{-1}u_{\varepsilon}R_{z}-v_{\varepsilon}u_{\varepsilon}R_{z}\\\
&=I.\end{split}$

A similar calculation shows that

$\begin{split}\bigl{(}R_{z}-A_{\varepsilon}(z)Vq\left(I+B_{\varepsilon}(z)Vq\right)^{-1}C_{\varepsilon}(z)\bigr{)}(H_{\varepsilon}-z)=I\end{split}$

is true. The latter two equations imply the claim of this proposition. ∎

###  3.2. The shift operator

In this subsection we introduce and study a shift operator which turns out to
be useful in the convergence analysis of the maps
$A_{\varepsilon}(z),B_{\varepsilon}(z)$, and $C_{\varepsilon}(z)$ in (3.4).
For that, we first construct a suitable extension of the normal vector field
$\nu$ on $\Sigma$. Let $\varphi_{\nu}\in C^{1}(\mathbb{R};\mathbb{R})$ with
$\varphi_{\nu}(0)=1$ and compact support in
$(-\varepsilon_{1},\varepsilon_{1})$, where $\varepsilon_{1}$ is the number
specified in Proposition 2.4. Since $\Sigma$ is assumed to satisfy Hypothesis
2.1, it is not difficult to show that

$\widetilde{\nu}(x)=\begin{cases}\nu(x_{\Sigma})\varphi_{\nu}(t),&\text{if
}x=x_{\Sigma}+t\nu(x_{\Sigma})\in\Omega_{\varepsilon_{1}}\text{ with
}(x_{\Sigma},t)\in\Sigma\times(-\varepsilon_{1},\varepsilon_{1}),\\\
0,&\text{if }x\notin\Omega_{\varepsilon_{1}},\end{cases}$

is a $C^{1}$ extension of $\nu$ which is supported in
$\Omega_{\varepsilon_{1}}$. For the sake of readability we use also the symbol
$\nu$ when referring to the extension $\widetilde{\nu}$ from now on.

Next, we define for $\delta\in\mathbb{R}$ the shift operator
$\tau_{\delta}:L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ by

$\tau_{\delta}u(x):=u\left(x+\delta\nu(x)\right),\quad
x\in\mathbb{R}^{\theta}.$ (3.5)

In the next proposition we study basic properties of $\tau_{\delta}$.

######  Proposition 3.3.

Let $D\nu$ be the Jacobi matrix of $\nu$ and
$\delta_{0}\in(0,{\left\lVert{D\nu}\right\rVert}_{L^{\infty}(\mathbb{R}^{\theta};\mathbb{R}^{\theta\times\theta})}^{-1})$.
Then, for any $r\in[0,1]$ the operators $\tau_{\delta}$,
$\delta\in[-\delta_{0},\delta_{0}]$, are uniformly bounded in
$H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ and for $r^{\prime}\in[0,r]$

${\left\lVert{\tau_{\delta}-I}\right\rVert}_{H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
H^{r^{\prime}}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\leq
C\left\lvert{\delta}\right\rvert^{r-r^{\prime}}$ (3.6)

holds for all $\delta\in[-\delta_{0},\delta_{0}]$, where $C>0$ is independent
of $\delta$.

###### Proof.

Fix $\delta\in[-\delta_{0},\delta_{0}]$ and observe first that
$I_{\theta}+\delta D\nu(x)$ is invertible for all $x\in\mathbb{R}^{\theta}$
and the norm of the inverse is bounded by
$(1-\left\lvert{\delta_{0}}\right\rvert{\left\lVert{D\nu}\right\rVert}_{L^{\infty}(\mathbb{R}^{\theta};\mathbb{R}^{\theta\times\theta})})^{-1}$.
The same bound holds for the modulus of the eigenvalues of $(I_{\theta}+\delta
D\nu(x))^{-1}$ and hence we conclude

$\left\lvert{\det\big{(}(I_{\theta}+\delta
D\nu(x))^{-1}\big{)}}\right\rvert\leq\frac{1}{\big{(}1-\left\lvert{\delta_{0}}\right\rvert{\left\lVert{D\nu}\right\rVert}_{L^{\infty}(\mathbb{R}^{\theta};\mathbb{R}^{\theta\times\theta})}\big{)}^{\theta}},\quad
x\in\mathbb{R}^{\theta}.$ (3.7)

We start by showing the uniform boundedness of $\tau_{\delta}$ for $r=0$. Let
$u\in L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$. Then, change of variables
and (3.7) lead to

$\begin{split}\int_{\mathbb{R}^{\theta}}&\left\lvert{\tau_{\delta}u(x)}\right\rvert^{2}\,dx=\int_{\mathbb{R}^{\theta}}\left\lvert{u(x+\delta\nu(x))}\right\rvert^{2}\,dx\\\
&=\int_{\mathbb{R}^{\theta}}\left\lvert{u(x+\delta\nu(x))}\right\rvert^{2}\frac{\left\lvert{\det\left(I_{\theta}+\delta
D\nu(x)\right)}\right\rvert}{\left\lvert{\det\left(I_{\theta}+\delta
D\nu(x)\right)}\right\rvert}\,dx\\\
&=\int_{\mathbb{R}^{\theta}}\left\lvert{u(x+\delta\nu(x))}\right\rvert^{2}\left\lvert{\det\left(I_{\theta}+\delta
D\nu(x)\right)}\right\rvert\left\lvert{\det\big{(}(I_{\theta}+\delta
D\nu(x))^{-1}\big{)}}\right\rvert\,dx\\\
&\leq\frac{1}{{\big{(}1-\left\lvert{\delta_{0}}\right\rvert{\left\lVert{D\nu}\right\rVert}_{L^{\infty}(\mathbb{R}^{\theta};\mathbb{R}^{\theta\times\theta})}\big{)}^{\theta}}}\int_{\mathbb{R}^{\theta}}\left\lvert{u(x)}\right\rvert^{2}\,dx,\end{split}$
(3.8)

and it follows that the operators $\tau_{\delta}$,
$\delta\in[-\delta_{0},\delta_{0}]$, are uniformly bounded in
$L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$. To see the uniform boundedness of
the operators $\tau_{\delta}$ in $H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})$,
let $u\in\mathcal{D}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ and compute in a
similar way as above

$\begin{split}\int_{\mathbb{R}^{\theta}}\left\lvert{D(\tau_{\delta}u)(x)}\right\rvert^{2}\,dx&=\int_{\mathbb{R}^{\theta}}\left\lvert{(Du)(x+\delta\nu(x))(I_{\theta}+\delta
D\nu(x))}\right\rvert^{2}\,dx\\\
&\leq\frac{\big{(}1+\delta_{0}{\left\lVert{D\nu}\right\rVert}_{L^{\infty}(\mathbb{R}^{\theta};\mathbb{R}^{\theta\times\theta})}\big{)}^{2}}{{\big{(}1-\left\lvert{\delta_{0}}\right\rvert{\left\lVert{D\nu}\right\rVert}_{L^{\infty}(\mathbb{R}^{\theta};\mathbb{R}^{\theta\times\theta})}\big{)}^{\theta}}}\int_{\mathbb{R}^{\theta}}\left\lvert{Du(x)}\right\rvert^{2}\,dx.\end{split}$
(3.9)

By density this estimate remains valid for $u\in
H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})$. Therefore, the uniform boundedness
of the operators $\tau_{\delta}$ in
$H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ follows from (3.8) and (3.9).
Eventually, using interpolation one concludes that $\tau_{\delta}$ is
uniformly bounded in $H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ for any
$r\in[0,1]$.

It remains to prove (3.6). Since we already have shown that $\tau_{\delta}$ is
uniformly bounded in $H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N})$, the claim in
(3.6) holds for $r=r^{\prime}\in[0,1]$. Next, we show (3.6) for $r^{\prime}=0$
and $r=1$. With the main theorem of calculus and the chain rule we find for
$u\in\mathcal{D}(\mathbb{R}^{\theta};\mathbb{C}^{N})$

$\begin{split}\int_{\mathbb{R}^{\theta}}\left\lvert{\tau_{\delta}u(x)-u(x)}\right\rvert^{2}\,dx&=\int_{\mathbb{R}^{\theta}}\left\lvert{\int_{0}^{\delta}\frac{d}{dt}u(x+t\nu(x))dt}\right\rvert^{2}\,dx\\\
&=\int_{\mathbb{R}^{\theta}}\left\lvert{\int_{0}^{\delta}Du(x+t\nu(x))\nu(x)dt}\right\rvert^{2}\,dx\\\
&\leq\int_{\mathbb{R}^{\theta}}\left(\int_{0}^{\delta}\left\lvert{(\tau_{t}Du)(x)}\right\rvert^{2}\,dt\right)\left(\int_{0}^{\delta}\left\lvert{\nu(x)}\right\rvert^{2}\,dt\right)\,dx\\\
&\leq|\delta|{\left\lVert{\nu}\right\rVert}_{L^{\infty}(\mathbb{R}^{\theta};\mathbb{R}^{\theta})}^{2}\int_{0}^{\delta}{\left\lVert{\tau_{t}Du}\right\rVert}_{L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N\times\theta})}^{2}\,dt\\\
&\leq
C|\delta|\int_{0}^{\delta}{\left\lVert{Du}\right\rVert}_{L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N\times\theta})}^{2}\,dt\\\
&\leq
C\left\lvert{\delta}\right\rvert^{2}{\left\lVert{u}\right\rVert}_{H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})}^{2},\end{split}$

where $\tau_{t}Du$ is understood column-wise. By density this estimate remains
valid for $u\in H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ and hence
${\left\lVert{\tau_{\delta}-I}\right\rVert}_{H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\leq
C\left\lvert{\delta}\right\rvert$. It remains to prove the claim in the case
$0\leq r^{\prime}<r\leq 1$ with $(r^{\prime},r)\neq(0,1)$. We set
$\mu=r-r^{\prime}\in(0,1)$ and
$\upsilon=\frac{r^{\prime}}{1-(r-r^{\prime})}\in[0,1]$. Then,

$r^{\prime}=(1-\mu)\upsilon+\mu 0\quad\text{and}\quad r=(1-\mu)\upsilon+\mu 1$

and consequently [32, Theorem B.7] implies

$H^{r^{\prime}}(\mathbb{R}^{\theta};\mathbb{C}^{N})=[H^{\upsilon}(\mathbb{R}^{\theta};\mathbb{C}^{N}),H^{0}(\mathbb{R}^{\theta};\mathbb{C}^{N})]_{\mu}=[H^{\upsilon}(\mathbb{R}^{\theta};\mathbb{C}^{N}),L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})]_{\mu}$

and

$H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N})=[H^{\upsilon}(\mathbb{R}^{\theta};\mathbb{C}^{N}),H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})]_{\mu}.$

Applying (1.18) yields

$\begin{split}&{\left\lVert{I-\tau_{\delta}}\right\rVert}_{H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
H^{r^{\prime}}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\\\ &\hskip 20.0pt\leq
C{\left\lVert{I-\tau_{\delta}}\right\rVert}_{[H^{\upsilon}(\mathbb{R}^{\theta};\mathbb{C}^{N}),H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})]_{\mu}\to[H^{\upsilon}(\mathbb{R}^{\theta};\mathbb{C}^{N}),L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})]_{\mu}}\\\
&\hskip 20.0pt\leq
C{\left\lVert{I-\tau_{\delta}}\right\rVert}_{H^{\upsilon}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
H^{\upsilon}(\mathbb{R}^{\theta};\mathbb{C}^{N})}^{1-\mu}{\left\lVert{I-\tau_{\delta}}\right\rVert}_{H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})}^{\mu}\\\ &\hskip
20.0pt=C\left\lvert{\delta}\right\rvert^{r-r^{\prime}},\end{split}$

which is exactly (3.6). This finishes the proof of this proposition. ∎

We will also need a variant of the shift operator $\tau_{\delta}$ that acts on
functions defined on $\Omega_{\pm}$. Since $\Omega_{\pm}$ satisfy Hypothesis
2.1 we can make use of Stein’s extension operator
$E:L^{2}(\Omega_{\pm};\mathbb{C}^{N})\rightarrow
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ which has the properties
$(Ef)_{\pm}=f$ for $f\in L^{2}(\Omega_{\pm};\mathbb{C}^{N})$ and which has a
continuous restriction $E:H^{r}(\Omega_{\pm};\mathbb{C}^{N})\rightarrow
H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ for any $r\geq 0$, see [40, Chapter
6, Section 3, Theorem 5]. We then define the shift operator for functions on
$\Omega_{\pm}$ by

$\tau_{\delta}^{\Omega_{\pm}}:=(\tau_{\delta}E(\cdot))_{\pm}:L^{2}(\Omega_{\pm};\mathbb{C}^{N})\to
L^{2}(\Omega_{\pm};\mathbb{C}^{N}).$ (3.10)

The following properties of $\tau_{\delta}^{\Omega_{\pm}}$ follow immediately
from the properties of $E$ and Proposition 3.3.

######  Corollary 3.4.

Let $D\nu$ be the Jacobi matrix of $\nu$ and
$\delta_{0}\in(0,{\left\lVert{D\nu}\right\rVert}_{L^{\infty}(\mathbb{R}^{\theta};\mathbb{R}^{\theta\times\theta})}^{-1})$.
Then, for any $r\in[0,1]$ the operators $\tau_{\delta}^{\Omega_{\pm}}$,
$\delta\in[-\delta_{0},\delta_{0}]$, are uniformly bounded in
$H^{r}(\Omega_{\pm};\mathbb{C}^{N})$ and for $r^{\prime}\in[0,r]$

${\left\lVert{\tau_{\delta}^{\Omega_{\pm}}-I}\right\rVert}_{H^{r}(\Omega_{\pm};\mathbb{C}^{N})\to
H^{r^{\prime}}(\Omega_{\pm};\mathbb{C}^{N})}\leq
C\left\lvert{\delta}\right\rvert^{r-r^{\prime}}$

holds for all $\delta\in[-\delta_{0},\delta_{0}]$, where $C>0$ is independent
of $\delta$.

Eventually, we show that the map $t\mapsto\tau_{t\delta}u$ has a useful
continuity property.

######  Proposition 3.5.

Let $D\nu$ be the Jacobi matrix of $\nu$,
$\delta_{0}\in(0,{\left\lVert{D\nu}\right\rVert}_{L^{\infty}(\mathbb{R}^{\theta};\mathbb{R}^{\theta\times\theta})}^{-1})$,
$\delta\in[-\delta_{0},\delta_{0}]$, $r\in[0,1]$, $u\in
H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ and $v\in
H^{r}(\Omega_{\pm};\mathbb{C}^{N})$. Then, the functions

$\begin{split}f_{u}:(-1,1)&\to
H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N}),\qquad
t\mapsto\tau_{t\delta}u,\end{split}$

and

$\begin{split}f_{v}^{\pm}:(-1,1)&\to
H^{r}(\Omega_{\pm}\;\mathbb{C}^{N}),\qquad
t\mapsto\tau_{t\delta}^{\Omega_{\pm}}v,\end{split}$

are continuous.

###### Proof.

First, consider $u\in\mathcal{D}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ and let
$t_{n},t\in(-1,1)$ such that $t_{n}\rightarrow t$ as $n\rightarrow\infty$.
Then, with dominated convergence one gets

$\lim_{n\to\infty}f_{u}(t_{n})=\lim_{n\to\infty}u((\cdot)+\delta
t_{n}\nu)=u((\cdot)+\delta t\nu)=f_{u}(t)\quad\textup{in
}H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N}).$

Since $H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ is continuously embedded in
$H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N})$, the assertion follows for
$u\in\mathcal{D}(\mathbb{R}^{\theta};\mathbb{C}^{N})$. If $u\in
H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N})$, then there exists a sequence
$(u_{n})_{n\in\mathbb{N}}\subset\mathcal{D}(\mathbb{R}^{\theta};\mathbb{C}^{N})$
such that $u_{n}\to u$ in $H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ as
$n\to\infty$. Applying Proposition 3.3 yields

$\begin{split}{\left\lVert{f_{u}(t)-f_{u_{n}}(t)}\right\rVert}_{H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N})}&={\left\lVert{\tau_{\delta
t}(u-u_{n})}\right\rVert}_{H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\leq
C{\left\lVert{u-u_{n}}\right\rVert}_{H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\end{split}$

for all $n\in\mathbb{N}$ and $t\in(-1,1)$. Hence, $f_{u_{n}}(t)\to f_{u}(t)$
uniformly with respect to $t$ in $H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N})$
as $n\to\infty$. Thus, $f_{u}$ is also continuous.

It remains to verify the claim for $f_{v}^{\pm}$. Let $t_{n},t\in(-1,1)$ such
that $t_{n}\rightarrow t$ as $n\rightarrow\infty$. Using the properties of
Stein’s extension operator $E$ and the above observations, we get that
$f_{Ev}(t_{n})\rightarrow f_{Ev}(t)$ in
$H^{r}(\mathbb{R}^{\theta};\mathbb{C}^{N})$. Moreover, the boundedness of the
restriction mapping gives us that $f_{v}^{\pm}(t_{n})=\left(\tau_{\delta
t_{n}}Ev\right)_{\pm}=\left(f_{Ev}(t_{n})\right)_{\pm}$ converges to
$\left(f_{Ev}(t)\right)_{\pm}=f_{v}^{\pm}(t)$ in
$H^{r}(\Omega_{\pm};\mathbb{C}^{N})$. This shows the continuity of
$f_{v}^{\pm}$. ∎

###  3.3. Convergence of $A_{\varepsilon}(z),B_{\varepsilon}(z)$, and
$C_{\varepsilon}(z)$

This section is devoted to the convergence analysis of the operators
$A_{\varepsilon}(z),B_{\varepsilon}(z)$, and $C_{\varepsilon}(z)$ introduced
in (3.4) for $\varepsilon\rightarrow 0+$. First, in Proposition 3.7 we study
the convergence of $C_{\varepsilon}(z)$, which allows us with a duality
argument to investigate the convergence of $A_{\varepsilon}(z)$ in Proposition
3.8. Eventually, in Proposition 3.10 we consider the convergence of
$B_{\varepsilon}(z)$.

We choose the number $\varepsilon_{2}>0$ such that

$\varepsilon_{2}\leq\min\left\\{\frac{\varepsilon_{1}}{2},\frac{\varepsilon_{0}}{4},\frac{1}{2{\left\lVert{D\nu}\right\rVert}_{L^{\infty}(\mathbb{R}^{\theta};\mathbb{R}^{\theta\times\theta})}}\right\\},$
(3.11)

where $\varepsilon_{0}$ and $\varepsilon_{1}$ are specified in Hypothesis 2.1
and Proposition 2.4, respectively. Let $W$ be the Weingarten map associated
with $\Sigma$ introduced in Definition 2.3. In our analysis, the operator
$M_{\varepsilon}:L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))\to
L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$ acting as

$\begin{split}M_{\varepsilon}f(t)=\det\left(I-t\varepsilon
W\right)f(t)\qquad\text{ for a.e. }t\in(-1,1)\end{split}$ (3.12)

will be useful. In the following lemma, which is an immediate consequence of
Proposition 2.4 (ii), some relevant properties of $M_{\varepsilon}$ are
stated.

######  Lemma 3.6.

For any $\varepsilon\in(0,\varepsilon_{2}]$ the operator $M_{\varepsilon}$ is
bounded, invertible,

${\left\lVert{M_{\varepsilon}}\right\rVert}_{0\to 0}\leq(1+\varepsilon
C)\quad\text{and}\quad{\left\lVert{M_{\varepsilon}-I}\right\rVert}_{0\to
0}\leq\varepsilon C.$

To formulate the result about the convergence of $C_{\varepsilon}(z)$, recall
that the embedding $\mathfrak{J}$ is defined in (2.7) and introduce the
operator

$\begin{split}C_{0}(z):=\mathfrak{J}\Phi_{\overline{z}}^{*}:L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N})).\end{split}$ (3.13)

In fact, the properties of $\mathfrak{J}$ and $\Phi_{\overline{z}}^{*}$, see
(2.7) and Proposition 2.8, imply that $C_{0}(z)$ gives rise also to a bounded
operator from $L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ to
$L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$.

######  Proposition 3.7.

Let $z\in\rho(H)$, $R_{z}=(H-z)^{-1}$, $\tau_{(\cdot)}$ be the shift operator
in (3.5), and $\varepsilon\in(0,\varepsilon_{2})$ with $\varepsilon_{2}$
satisfying (3.11). Then, the relation

$C_{\varepsilon}(z)u(t)=\boldsymbol{t}_{\Sigma}\tau_{\varepsilon
t}R_{z}u\qquad\text{ for a.e. }t\in(-1,1)$ (3.14)

holds in $L^{2}(\Sigma;\mathbb{C}^{N})$ and $\textup{ran
}C_{\varepsilon}(z)\subset L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$.
Moreover, the operators $C_{\varepsilon}(z)$ are uniformly bounded from
$L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ to
$L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$ and for any
$r\in\big{(}0,\tfrac{1}{2}\big{)}$ one has

${\left\lVert{C_{\varepsilon}(z)-C_{0}(z)}\right\rVert}_{L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
0}\leq C\varepsilon^{1/2-r}.$ (3.15)

###### Proof.

First, we show (3.14) for
$u\in\mathcal{D}(\mathbb{R}^{\theta};\mathbb{C}^{N})$. By density and
continuity, this implies (3.14) for all $u\in
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$. Recall that
$R_{z}:H^{s}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
H^{s+1}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ is bounded for $s\in\mathbb{R}$;
cf. Section 2.3. Hence, by the Sobolev embedding theorem $R_{z}u$ is
continuous for $u\in\mathcal{D}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ and the
same is true for $\tau_{\varepsilon t}R_{z}u$. Furthermore, as
$\tau_{\varepsilon t}R_{z}u\in H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ we
conclude with Proposition 3.1 for $t\in(-1,1)$ and $x_{\Sigma}\in\Sigma$ that

$\boldsymbol{t}_{\Sigma}\tau_{\varepsilon
t}R_{z}u(x_{\Sigma})=\tau_{\varepsilon
t}R_{z}u(x_{\Sigma})=\int_{\mathbb{R}^{\theta}}G_{z}(x_{\Sigma}+\varepsilon
t\nu(x_{\Sigma})-y)u(y)dy=(C_{\varepsilon}(z)u)(t)(x_{\Sigma}).$

Hence, (3.14) is true.

Next, we show the inclusion $\textup{ran }C_{\varepsilon}(z)\subset
L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$. Assume that $u\in
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$. Then, by Proposition 3.5 and the
boundedness of the trace
$\boldsymbol{t}_{\Sigma}:H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
H^{1/2}(\Sigma;\mathbb{C}^{N})$ it follows that the function
$\boldsymbol{t}_{\Sigma}\tau_{(\cdot)\varepsilon}R_{z}u$ is continuous as a
mapping from $(-1,1)$ to $H^{1/2}(\Sigma;\mathbb{C}^{N})$. In particular,
$\boldsymbol{t}_{\Sigma}\tau_{(\cdot)\varepsilon}R_{z}u$ is measurable as a
mapping from $(-1,1)$ to $H^{1/2}(\Sigma;\mathbb{C}^{N})$. Using again the
boundedness of
$\boldsymbol{t}_{\Sigma}:H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
H^{1/2}(\Sigma;\mathbb{C}^{N})$ and the uniform boundedness of the shift
operator in $H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})$, see Propositions 2.2
and 3.3, respectively, we conclude

$\int_{-1}^{1}{\left\lVert{\boldsymbol{t}_{\Sigma}\tau_{\varepsilon
t}R_{z}u}\right\rVert}^{2}_{H^{1/2}(\Sigma;\mathbb{C}^{N})}\,dt\leq\int_{-1}^{1}C{\left\lVert{R_{z}u}\right\rVert}^{2}_{H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\,dt\leq
C{\left\lVert{u}\right\rVert}_{L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})}^{2}$

and therefore $\boldsymbol{t}_{\Sigma}\tau_{(\cdot)\varepsilon}R_{z}u\in
L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$. Moreover, this also shows that
$C_{\varepsilon}(z)$ is uniformly bounded from
$L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ to
$L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$.

Eventually, with Proposition 3.3 and the identity
$C_{0}(z)=\mathfrak{J}\Phi_{\overline{z}}^{*}=\mathfrak{J}\boldsymbol{t}_{\Sigma}R_{z}$,
see Proposition 2.8 (iii), we have for $r\in\big{(}0,\tfrac{1}{2}\big{)}$ and
$u\in L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$

$\begin{split}{\left\lVert{C_{\varepsilon}(z)u-C_{0}(z)u}\right\rVert}_{0}^{2}&={\left\lVert{\boldsymbol{t}_{\Sigma}\tau_{\varepsilon(\cdot)}R_{z}u-\mathfrak{J}\boldsymbol{t}_{\Sigma}R_{z}u}\right\rVert}_{0}^{2}\\\
&=\int_{-1}^{1}{\left\lVert{\boldsymbol{t}_{\Sigma}(\tau_{\varepsilon
t}-I)R_{z}u}\right\rVert}_{L^{2}(\Sigma;\mathbb{C}^{N})}^{2}\,dt\\\ &\leq
C\int_{-1}^{1}{\left\lVert{(\tau_{\varepsilon
t}-I)R_{z}u}\right\rVert}_{H^{r+1/2}(\mathbb{R}^{\theta};\mathbb{C}^{N})}^{2}\,dt\\\
&\leq C\int_{-1}^{1}\left\lvert{\varepsilon
t}\right\rvert^{1-2r}{\left\lVert{R_{z}u}\right\rVert}_{H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})}^{2}\,dt\\\
&\leq
C\int_{-1}^{1}\varepsilon^{1-2r}{\left\lVert{u}\right\rVert}_{L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})}^{2}\,dt\\\
&\leq
C\varepsilon^{1-2r}{\left\lVert{u}\right\rVert}_{L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})}^{2},\end{split}$

which leads to (3.15). Therefore, all claims are shown. ∎

Using the convergence of $C_{\varepsilon}(z)$, it is not difficult to show the
convergence of $A_{\varepsilon}(z)$. We define the natural candidate for the
limit operator by

$A_{0}(z):=\Phi_{z}\mathfrak{J}^{*}:L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N}).$

######  Proposition 3.8.

Let $z\in\rho(H)$ and $\varepsilon\in(0,\varepsilon_{2})$ with
$\varepsilon_{2}$ satisfying (3.11). Then, for any $r\in(0,\tfrac{1}{2})$ one
has

${\left\lVert{A_{\varepsilon}(z)-A_{0}(z)}\right\rVert}_{0\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\leq C\varepsilon^{1/2-r}$

and, in particular, the operators
$A_{\varepsilon}(z):L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ are uniformly bounded.

###### Proof.

Let $\mathcal{I}_{\varepsilon}$, $S_{\varepsilon}$, and $M_{\varepsilon}$ be
the operators given by (3.2), (3.3), and (3.12), respectively. One verifies by
a direct calculation using Proposition 2.4 (iii), (3.2), and (3.3) that
$(\mathcal{I}_{\varepsilon}\mathcal{S}_{\varepsilon})^{*}=M_{\varepsilon}\mathcal{S}_{\varepsilon}^{-1}\mathcal{I}_{\varepsilon}^{-1}$.
Using this relation we conclude from (3.4) that

$\begin{split}(A_{\varepsilon}(z)M_{\varepsilon}^{-1})^{*}&=M_{\varepsilon}^{-1}(A_{\varepsilon}(z))^{*}=M_{\varepsilon}^{-1}(R_{z}U_{\varepsilon}^{*}\mathcal{I}_{\varepsilon}\mathcal{S}_{\varepsilon})^{*}\\\
&=M_{\varepsilon}^{-1}M_{\varepsilon}\mathcal{S}_{\varepsilon}^{-1}\mathcal{I}_{\varepsilon}^{-1}U_{\varepsilon}R_{\overline{z}}=\mathcal{S}_{\varepsilon}^{-1}\mathcal{I}_{\varepsilon}^{-1}U_{\varepsilon}R_{\overline{z}}=C_{\varepsilon}(\overline{z}).\end{split}$

Moreover,
$(A_{0}(z))^{*}=(\Phi_{z}\mathfrak{J}^{*})^{*}=\mathfrak{J}\Phi_{z}^{*}=C_{0}(\overline{z})$.
Hence, Lemma 3.6 and Proposition 3.7 yield

$\begin{split}\big{\|}A_{\varepsilon}(z)&-A_{0}(z)\big{\|}_{0\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\\\
&={\left\lVert{A_{\varepsilon}(z)M_{\varepsilon}^{-1}(M_{\varepsilon}-I)+A_{\varepsilon}(z)M_{\varepsilon}^{-1}-A_{0}(z)}\right\rVert}_{0\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\\\ &\leq
C\varepsilon{\left\lVert{C_{\varepsilon}(\overline{z})}\right\rVert}_{L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
0}+{\left\lVert{C_{\varepsilon}(\overline{z})-C_{0}(\overline{z})}\right\rVert}_{L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
0}\\\ &\leq C\varepsilon^{1/2-r},\end{split}$

which is the claimed estimate. ∎

######  Remark 3.9.

For completeness we note that one can follow the proof of Proposition 3.7 to
show

${\left\lVert{C_{\varepsilon}(z)-C_{0}(z)}\right\rVert}_{L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
r}\leq C\varepsilon^{1/2-r}$

for $r\in(0,\tfrac{1}{2})$. Moreover, if $\Sigma$ is $C^{3}$ smooth, one can
extend the result of Lemma 3.6 to spaces
$L^{2}((-1,1);H^{t}(\Sigma;\mathbb{C}^{N}))$, $t\in[-1,1]$, and use this to
verify

${\left\lVert{A_{\varepsilon}(z)-A_{0}(z)}\right\rVert}_{-r\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\leq C\varepsilon^{1/2-r}$

for $r\in(0,\tfrac{1}{2})$ in a similar way as in the proof of Proposition
3.8.

Next, we study the convergence of the operators $B_{\varepsilon}(z)$. Define
the limit operator by $B_{0}(z):L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))\to
L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$ which acts on $f\in
L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$ evaluated for a.e. $t\in(-1,1)$ as

$\begin{split}B_{0}(z)f(t)&:=\frac{i}{2}(\alpha\cdot\nu)\int_{-1}^{1}\textup{sign}(t-s)f(s)\,ds+\mathcal{C}_{z}\int_{-1}^{1}f(s)\,ds,\end{split}$
(3.16)

where $\mathcal{C}_{z}:L^{2}(\Sigma;\mathbb{C}^{N})\to
L^{2}(\Sigma;\mathbb{C}^{N})$ is the extension of the operator defined in
(2.14) from Proposition 2.9. Using the mapping properties of $\mathcal{C}_{z}$
in Proposition 2.9 (i) it follows that $B_{0}(z)$ can also be regarded as an
operator in $L^{2}((-1,1);H^{r}(\Sigma;\mathbb{C}^{N}))$ for any
$r\in\bigl{[}-\frac{1}{2},\frac{1}{2}\bigr{]}$. In the following proposition
we show that $B_{\varepsilon}(z)$ converges to $B_{0}(z)$. The proof of this
result is more complicated as the proofs of Proposition 3.7 and Proposition
3.8, and therefore some of the more technical calculations are shifted to
Appendix B.

######  Proposition 3.10.

Let $z\in\rho(H)$ and $\varepsilon\in(0,\varepsilon_{2})$ with
$\varepsilon_{2}$ satisfying (3.11). Then, the operators $B_{\varepsilon}(z)$
are uniformly bounded in $L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$ and for
any $r\in(0,\tfrac{1}{2})$ one has

${\left\lVert{B_{\varepsilon}(z)-B_{0}(z)}\right\rVert}_{1/2\to 0}\leq
C\varepsilon^{1/2-r}.$

###### Proof.

The proof is split into several steps. Let $\Phi_{z}$ be as in (2.12) and let
$\tau_{(\cdot)}^{\Omega_{\pm}}$ be defined by (3.10). We introduce the
auxiliary operators

$\widetilde{B}_{\varepsilon}(z):=B_{\varepsilon}(z)M_{\varepsilon}^{-1}:L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))\to
L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N})),$ (3.17)

which are, due to the properties of $B_{\varepsilon}(z)$ and $M_{\varepsilon}$
in (3.4) and (3.12), bounded and act on $f\in
L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$ evaluated at $t\in(-1,1)$ and
$x_{\Sigma}\in\Sigma$ as

$\widetilde{B}_{\varepsilon}(z)f(t)(x_{\Sigma})=\int_{-1}^{1}\int_{\Sigma}G_{z}(x_{\Sigma}+\varepsilon
t\nu(x_{\Sigma})-y_{\Sigma}-\varepsilon
s\nu(y_{\Sigma}))f(s)(y_{\Sigma})\,d\sigma(y_{\Sigma})\,ds.$ (3.18)

Moreover, we define

$\overline{B}_{\varepsilon}(z):L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))\to
L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$

acting on $f\in L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$ for a.e.
$t\in(-1,1)$ as

$\overline{B}_{\varepsilon}(z)f(t)=\int_{-1}^{t}\boldsymbol{t}_{\Sigma}^{-}\tau_{\varepsilon(t-s)}^{\Omega_{-}}(\Phi_{z}f(s))_{-}ds+\int_{t}^{1}\boldsymbol{t}_{\Sigma}^{+}\tau_{\varepsilon(t-s)}^{\Omega_{+}}(\Phi_{z}f(s))_{+}\,ds.$
(3.19)

First, in Step 1 we show that $\overline{B}_{\varepsilon}(z)$ is bounded and
converges to $B_{0}(z)$, then in Step 2 we verify an alternative
representation of $\overline{B}_{\varepsilon}(z)$. In Step 3 we use Appendix B
to compare $\overline{B}_{\varepsilon}(z)$ and
$\widetilde{B}_{\varepsilon}(z)$, and show that
$\widetilde{B}_{\varepsilon}(z)$ is uniformly bounded in $\varepsilon$. In
Step 4 we combine the results from Step 1 to Step 3 to conclude the claim of
this proposition.

Step 1. First, we note that, due to Definition 2.5 and Proposition 3.5, the
function
$(-1,1)^{2}\ni(t,s)\mapsto\Theta(\mp(t-s))\tau_{\varepsilon(t-s)}^{\Omega_{\pm}}\in\mathcal{L}(H^{1}(\Omega_{\pm}),H^{1}(\Omega_{\pm}))$
is measurable, where $\Theta$ is the Heaviside function. Hence, it follows
with Definition 2.5 that the integrands in (3.19) are measurable with respect
to $(t,s)\in(-1,1)^{2}$. Moreover, by the mapping properties of
$\boldsymbol{t}_{\Sigma}^{\pm}$, $\tau_{(\cdot)}^{\Omega_{\pm}}$, and
$\Phi_{z}$ in Proposition 2.2, Corollary 3.4, and Proposition 2.8,
respectively, the integrands are bounded by
$C{\left\lVert{f(s)}\right\rVert}_{H^{1/2}(\Sigma;\mathbb{C}^{N})}$ for
$(t,s)\in(-1,1)^{2}$. In particular, we conclude that for $f\in
L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$

$\int_{-1}^{1}\int_{-1}^{1}\big{\|}\Theta(t-s)\boldsymbol{t}_{\Sigma}^{-}\tau_{\varepsilon(t-s)}^{\Omega_{-}}(\Phi_{z}f(s))_{-}+\Theta(s-t)\boldsymbol{t}_{\Sigma}^{+}\tau_{\varepsilon(t-s)}^{\Omega_{+}}(\Phi_{z}f(s))_{+}\big{\|}^{2}_{H^{1/2}(\Sigma;\mathbb{C}^{N})}dtds$

is finite. Thus, Fubini’s theorem for Bochner integrals, cf. Section 2.2,
yields the integrability of the integrands in (3.19) with respect to
$s\in(-1,1)$ and the measurability of
$t\mapsto\overline{B}_{\varepsilon}(z)f(t)$. Furthermore, the bound for the
integrands implies also
${\left\lVert{\overline{B}_{\varepsilon}(z)f}\right\rVert}_{1/2}\leq
C{\left\lVert{f}\right\rVert}_{1/2}$ for $f\in
L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$. Hence,
$\overline{B}_{\varepsilon}(z)$ is well defined and uniformly bounded in
$L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$. We claim that

${\left\lVert{\overline{B}_{\varepsilon}(z)-{B}_{0}(z)}\right\rVert}_{1/2\to
0}\leq C{\varepsilon}^{1/2-r}.$ (3.20)

To see this we remark that with Proposition 2.9 (ii) we have the pointwise
representation

$B_{0}(z)f(t)=\int_{-1}^{t}\boldsymbol{t}_{\Sigma}^{-}\left(\Phi_{z}f(s)\right)_{-}\,ds+\int_{t}^{1}\boldsymbol{t}_{\Sigma}^{+}\left(\Phi_{z}f(s)\right)_{+}\,ds$
(3.21)

for a.e. $t\in(-1,1)$ and $f\in L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$.
Thus, $r\in\big{(}0,\tfrac{1}{2}\big{)}$ and direct estimates show

$\begin{split}{\left\lVert{\overline{B}_{\varepsilon}(z)f-{B}_{0}(z)f}\right\rVert}_{0}^{2}&=\int_{-1}^{1}\left\|\int_{-1}^{t}\boldsymbol{t}_{\Sigma}^{-}\bigl{(}\tau_{\varepsilon(t-s)}^{\Omega_{-}}-I\bigr{)}(\Phi_{z}f(s))_{-}ds\right.\\\
&\qquad\qquad+\left.\int_{t}^{1}\boldsymbol{t}_{\Sigma}^{+}\bigl{(}\tau_{\varepsilon(t-s)}^{\Omega_{+}}-I\bigr{)}(\Phi_{z}f(s))_{+}ds\,\right\|_{L^{2}(\Sigma;\mathbb{C}^{N})}^{2}\,dt\\\
&\leq\int_{-1}^{1}\left(\int_{-1}^{t}{\left\lVert{\boldsymbol{t}_{\Sigma}^{-}\bigl{(}\tau_{\varepsilon(t-s)}^{\Omega_{-}}-I\bigr{)}(\Phi_{z}f(s))_{-}}\right\rVert}_{H^{r}(\Sigma;\mathbb{C}^{N})}ds\right.\\\
&\left.+\int_{t}^{1}{\left\lVert{\boldsymbol{t}_{\Sigma}^{+}\bigl{(}\tau_{\varepsilon(t-s)}^{\Omega_{+}}-I\bigr{)}(\Phi_{z}f(s))_{+}}\right\rVert}_{H^{r}(\Sigma;\mathbb{C}^{N})}ds\right)^{2}\,dt.\end{split}$
(3.22)

Employing Proposition 2.2, Corollary 3.4, and Proposition 2.8 yields for all
$s,t\in(-1,1)$

${\left\lVert{\boldsymbol{t}_{\Sigma}^{\pm}\big{(}\tau_{\varepsilon(t-s)}^{\Omega_{\pm}}-I\big{)}(\Phi_{z}f(s))_{\pm}}\right\rVert}_{H^{r}(\Sigma;\mathbb{C}^{N})}\leq
C\varepsilon^{1/2-r}{\left\lVert{f(s)}\right\rVert}_{H^{1/2}(\Sigma;\mathbb{C}^{N})}.$

Plugging this into (3.22) we obtain

$\begin{split}\big{\|}\overline{B}_{\varepsilon}&(z)f-{B}_{0}(z)f\big{\|}_{0}^{2}\\\
&\leq
C\varepsilon^{2(1/2-r)}\int_{-1}^{1}\left(\int_{-1}^{t}{\left\lVert{f(s)}\right\rVert}_{H^{1/2}(\Sigma;\mathbb{C}^{N})}\,ds+\int_{t}^{1}{\left\lVert{f(s)}\right\rVert}_{H^{1/2}(\Sigma;\mathbb{C}^{N})}\,ds\right)^{2}\,dt\\\
&\leq
C\varepsilon^{2(1/2-r)}\int_{-1}^{1}{\left\lVert{f(s)}\right\rVert}_{H^{1/2}(\Sigma;\mathbb{C}^{N})}^{2}\,ds=C\varepsilon^{2(1/2-r)}{\left\lVert{f}\right\rVert}_{1/2}^{2},\end{split}$

which implies (3.20).

Step 2. We show that the operator $\overline{B}_{\varepsilon}(z)$ in (3.19)
has the alternative representation

$\overline{B}_{\varepsilon}(z)f(t)(x_{\Sigma})=\int_{-1}^{1}\int_{\Sigma}G_{z}(x_{\Sigma}+\varepsilon(t-s)\nu(x_{\Sigma})-y_{\Sigma})f(s)(y_{\Sigma})\,d\sigma(y_{\Sigma})\,ds.$
(3.23)

Let $f\in L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$ and $t,s\in(-1,1)$ be
fixed such that $t>s$. Note that the choice of $\varepsilon_{1}$ in
Proposition 2.4 implies
$\varepsilon_{2}\leq\frac{\varepsilon_{1}}{2}<\frac{\varepsilon_{A}}{2}$, cf.
also Proposition A.2. Hence, by Corollary A.3 we have
$x_{\Sigma}+\varepsilon(t-s)\nu(x_{\Sigma})\in\Omega_{-}$ for all
$x_{\Sigma}\in\Sigma$. Moreover, we conclude from the representation of
$\Phi_{z}$ given in (2.12) and the form of the integral kernel $G_{z}$, see
(2.10)–(2.11), that $\Phi_{z}f(s)$ is continuous away from $\Sigma$. Thus, we
have for $\sigma$-a.e. $x_{\Sigma}\in\Sigma$

$\begin{split}\boldsymbol{t}_{\Sigma}^{-}\tau_{\varepsilon(t-s)}^{\Omega_{-}}(\Phi_{z}f(s))_{-}(x_{\Sigma})&=(\Phi_{z}f(s))(x_{\Sigma}+\varepsilon(t-s)\nu(x_{\Sigma}))\\\
&=\int_{\Sigma}G_{z}(x_{\Sigma}+\varepsilon(t-s)\nu(x_{\Sigma})-y_{\Sigma})f(s)(y_{\Sigma})\,d\sigma(y_{\Sigma}).\end{split}$

Analogously, for $t<s$ and $\sigma$-a.e. $x_{\Sigma}\in\Sigma$

$\begin{split}\boldsymbol{t}_{\Sigma}^{+}\tau_{\varepsilon(t-s)}^{\Omega_{+}}(\Phi_{z}f(s))_{+}(x_{\Sigma})&=(\Phi_{z}f(s))(x_{\Sigma}+\varepsilon(t-s)\nu(x_{\Sigma}))\\\
&=\int_{\Sigma}G_{z}(x_{\Sigma}+\varepsilon(t-s)\nu(x_{\Sigma})-y_{\Sigma})f(s)(y_{\Sigma})\,d\sigma(y_{\Sigma}).\end{split}$

Combining the previous two equations yields

$\begin{split}&\int_{-1}^{t}\boldsymbol{t}_{\Sigma}^{-}\tau_{\varepsilon(t-s)}^{\Omega_{-}}(\Phi_{z}f(s))_{-}(x_{\Sigma})\,ds+\int_{t}^{1}\boldsymbol{t}_{\Sigma}^{+}\tau_{\varepsilon(t-s)}^{\Omega_{+}}(\Phi_{z}f(s))_{+}(x_{\Sigma})\,ds\\\
&\hskip
80.0pt=\int_{-1}^{1}\int_{\Sigma}G_{z}(x_{\Sigma}+\varepsilon(t-s)\nu(x_{\Sigma})-y_{\Sigma})f(s)(y_{\Sigma})\,d\sigma(y_{\Sigma})\,ds.\end{split}$
(3.24)

Moreover, as the integrands on the right hand side in (3.19) are Bochner
integrable (cf. Step 1), Proposition 2.7 (iii) shows that the pointwise
evaluation of the Bochner integrals in the definition of
$\overline{B}_{\varepsilon}(z)$ in (3.19) coincides with (3.24), i.e.

$\begin{split}&\bigg{(}\int_{-1}^{t}\boldsymbol{t}_{\Sigma}^{-}\tau_{\varepsilon(t-s)}^{\Omega_{-}}\left(\Phi_{z}f(s)\right)_{-}\,ds+\int_{t}^{1}\boldsymbol{t}_{\Sigma}^{+}\tau_{\varepsilon(t-s)}^{\Omega_{+}}\tau_{\varepsilon(t-s)}^{\Omega_{-}}\left(\Phi_{z}f(s)\right)_{+}\,ds\bigg{)}(x_{\Sigma})\\\
&\hskip
40.0pt=\int_{-1}^{t}\boldsymbol{t}_{\Sigma}^{-}\tau_{\varepsilon(t-s)}^{\Omega_{-}}(\Phi_{z}f(s))_{-}(x_{\Sigma})\,ds+\int_{t}^{1}\boldsymbol{t}_{\Sigma}^{+}\tau_{\varepsilon(t-s)}^{\Omega_{+}}(\Phi_{z}f(s))_{+}(x_{\Sigma})\,ds.\end{split}$

This is exactly the claimed formula in (3.23).

Step 3. By the results in Appendix B the map
$\widetilde{B}_{\varepsilon}(z)-\overline{B}_{\varepsilon}(z)$ admits an
extension to a bounded operator from
$L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$ to
$L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$ and

$\big{\|}\widetilde{B}_{\varepsilon}(z)-\overline{B}_{\varepsilon}(z)\big{\|}_{0\to
1/2}\leq C(\varepsilon\left\lvert{\log(\varepsilon)}\right\rvert)^{1/2}.$
(3.25)

Moreover, we claim that $\widetilde{B}_{\varepsilon}(z)$ is uniformly bounded
in $L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$. To see this, observe first
that

$\begin{split}\big{\|}\widetilde{B}_{\varepsilon}(z)\big{\|}_{1/2\to
1/2}&\leq\big{\|}\widetilde{B}_{\varepsilon}(z)-\overline{B}_{\varepsilon}(z)\big{\|}_{1/2\to
1/2}+\big{\|}\overline{B}_{\varepsilon}(z)\big{\|}_{1/2\to 1/2}\\\
&\leq\big{\|}\widetilde{B}_{\varepsilon}(z)-\overline{B}_{\varepsilon}(z)\big{\|}_{0\to
1/2}+\big{\|}\overline{B}_{\varepsilon}(z)\big{\|}_{1/2\to 1/2}.\end{split}$
(3.26)

Therefore, the estimate (3.25) and the uniform boundedness of the operators
$\overline{B}_{\varepsilon}(z)$ in
$L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$ shown in Step 1 imply that
$\widetilde{B}_{\varepsilon}(z)$ is also uniformly bounded in
$L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$. The same is true for
$\widetilde{B}_{\varepsilon}(\overline{z})$ and the hence also the anti-dual

$(\widetilde{B}_{\varepsilon}(\overline{z}))^{\prime}:L^{2}\bigl{(}(-1,1);H^{-1/2}(\Sigma;\mathbb{C}^{N})\bigr{)}\to
L^{2}\bigl{(}(-1,1);H^{-1/2}(\Sigma;\mathbb{C}^{N})\bigr{)}$

is uniformly bounded. We claim that
$(\widetilde{B}_{\varepsilon}(\overline{z}))^{\prime}$ is an extension of
$\widetilde{B}_{\varepsilon}(z)$, that is,

$\widetilde{B}_{\varepsilon}(z)f=(\widetilde{B}_{\varepsilon}(\overline{z}))^{\prime}f,\qquad
f\in L^{2}\bigl{(}(-1,1);L^{2}(\Sigma;\mathbb{C}^{N})\bigr{)}.$ (3.27)

In fact, the identities
$\widetilde{B}_{\varepsilon}(\overline{z})=B_{\varepsilon}(\overline{z})M_{\varepsilon}^{-1}$,
(3.4), and
$(\mathcal{I}_{\varepsilon}\mathcal{S}_{\varepsilon})^{*}=M_{\varepsilon}\mathcal{S}_{\varepsilon}^{-1}\mathcal{I}_{\varepsilon}^{-1}$,
yield for the adjoint of $\widetilde{B}_{\varepsilon}(\overline{z})$ in
$L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$

$(\widetilde{B}_{\varepsilon}(\overline{z}))^{*}=(\mathcal{S}_{\varepsilon}^{-1}\mathcal{I}_{\varepsilon}^{-1}U_{\varepsilon}R_{\overline{z}}U_{\varepsilon}^{*}\mathcal{I}_{\varepsilon}\mathcal{S}_{\varepsilon}M_{\varepsilon}^{-1})^{*}=\mathcal{S}_{\varepsilon}^{-1}\mathcal{I}_{\varepsilon}^{-1}U_{\varepsilon}R_{z}U_{\varepsilon}^{*}\mathcal{I}_{\varepsilon}\mathcal{S}_{\varepsilon}M_{\varepsilon}^{-1}=\widetilde{B}_{\varepsilon}(z)$

and hence Proposition 2.7 (ii) implies for $f\in
L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$ and $g\in
L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$

$\begin{split}\big{\langle}(\widetilde{B}_{\varepsilon}(\overline{z}))^{\prime}f&,g\big{\rangle}_{L^{2}((-1,1);H^{-1/2}(\Sigma;\mathbb{C}^{N}))\times
L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))}\\\
&=\big{\langle}f,\widetilde{B}_{\varepsilon}(\overline{z})g\big{\rangle}_{L^{2}((-1,1);H^{-1/2}(\Sigma;\mathbb{C}^{N}))\times
L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))}\\\
&=\big{(}f,\widetilde{B}_{\varepsilon}(\overline{z})g\big{)}_{L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))}\\\
&=\big{(}\widetilde{B}_{\varepsilon}(z)f,g\big{)}_{L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))}\\\
&=\big{\langle}\widetilde{B}_{\varepsilon}(z)f,g\big{\rangle}_{L^{2}((-1,1);H^{-1/2}(\Sigma;\mathbb{C}^{N}))\times
L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))},\end{split}$

where
$\langle\cdot,\cdot\rangle_{L^{2}((-1,1);H^{-1/2}(\Sigma;\mathbb{C}^{N}))\times
L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))}$ denotes the sesquilinear
duality product, which is anti-linear in the second argument. This implies
(3.27) and since $(\widetilde{B}_{\varepsilon}(\overline{z}))^{\prime}$ and
$\widetilde{B}_{\varepsilon}(z)$ are both uniformly bounded in
$L^{2}((-1,1);H^{-1/2}(\Sigma;\mathbb{C}^{N}))$ and
$L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$, respectively, an interpolation
argument leads to the uniform boundedness of $\widetilde{B}_{\varepsilon}(z)$
in $L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$.

Step 4. Using the results from Step 1 to Step 3 we will now complete the proof
of Proposition 3.10. Since
$B_{\varepsilon}(z)=\widetilde{B}_{\varepsilon}(z)M_{\varepsilon}$, Lemma 3.6
and the uniform boundedness of $\widetilde{B}_{\varepsilon}(z)$ shown in Step
3 imply the uniform boundedness of $B_{\varepsilon}(z)$ in
$L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$, proving the first claim of this
proposition. Moreover, by applying Lemma 3.6, (3.25), (3.20), and the uniform
boundedness of $\widetilde{B}_{\varepsilon}(z)$ in the space
$L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$ we obtain

$\begin{split}\big{\|}B_{\varepsilon}&(z)-B_{0}(z)\big{\|}_{1/2\to
0}={\left\lVert{\widetilde{B}_{\varepsilon}(z)M_{\varepsilon}-{B}_{0}(z)}\right\rVert}_{1/2\to
0}\\\
&\leq{\left\lVert{\widetilde{B}_{\varepsilon}(z)(M_{\varepsilon}-I)}\right\rVert}_{1/2\to
0}+{\left\lVert{\widetilde{B}_{\varepsilon}(z)-\overline{B}_{\varepsilon}(z)}\right\rVert}_{1/2\to
0}+{\left\lVert{\overline{B}_{\varepsilon}(z)-{B}_{0}(z)}\right\rVert}_{1/2\to
0}\\\
&\leq{\left\lVert{\widetilde{B}_{\varepsilon}(z)(M_{\varepsilon}-I)}\right\rVert}_{0\to
0}+{\left\lVert{\widetilde{B}_{\varepsilon}(z)-\overline{B}_{\varepsilon}(z)}\right\rVert}_{0\to
1/2}+{\left\lVert{\overline{B}_{\varepsilon}(z)-{B}_{0}(z)}\right\rVert}_{1/2\to
0}\\\ &\leq
C\bigl{(}\varepsilon+(\varepsilon\left\lvert{\log(\varepsilon)}\right\rvert)^{1/2}+\varepsilon^{1/2-r}\bigr{)}\\\
&\leq C\varepsilon^{1/2-r}.\end{split}$

This is the claimed norm estimate and finishes the proof of this proposition.
∎

###  3.4. Convergence of $\left(I+{B_{\varepsilon}(z)}Vq\right)^{-1}$

Recall that $B_{\varepsilon}(z)$ is defined in (3.4) and let $q$ and $V$ be as
in (1.5) and (1.6), respectively. In this section we treat the convergence of
$\left(I+{B_{\varepsilon}(z)}Vq\right)^{-1}$ under suitable assumptions on $q$
and $V$. The crucial assumption here, which is also needed in our main result
Theorem 1.1, is that $Vq$ is bounded by

$X_{z}:=\frac{1}{\max\\{\sup_{\varepsilon\in(0,\varepsilon_{2})}{\left\lVert{B_{\varepsilon}(z)}\right\rVert}_{0\to
0},{\left\lVert{B_{0}(z)}\right\rVert}_{1/2\to 1/2}\\}},$ (3.28)

where $\varepsilon_{2}$ is chosen such that (3.11) is satisfied and
$z\in\rho(H)$ is fixed. After the convergence result in the next proposition
we provide more information on $X_{z}$ in Lemma 3.13.

######  Proposition 3.11.

Let $z\in\rho(H)$ and $\varepsilon\in(0,\varepsilon_{2})$ with
$\varepsilon_{2}$ satisfying (3.11), and let $q$ and $V$ be as in (1.5) and
(1.6), respectively, such that

${\left\lVert{V}\right\rVert}_{W^{1}_{\infty}(\Sigma;\mathbb{C}^{N\times
N})}{\left\lVert{q}\right\rVert}_{L^{\infty}(\mathbb{R};\mathbb{R})}<X_{z}.$
(3.29)

Then, $(I+B_{\varepsilon}(z)Vq)^{-1}$ and $(I+B_{0}(z)Vq)^{-1}$ give rise to
bounded operators from $L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$ to
$L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$ and for any
$r\in(0,\tfrac{1}{2})$ one has

${\left\lVert{(I+B_{\varepsilon}(z)Vq)^{-1}-(I+B_{0}(z)Vq)^{-1}}\right\rVert}_{1/2\to
0}\leq C\varepsilon^{1/2-r}.$

######  Remark 3.12.

Note that assumption (3.29) implies the following:

  * (i)

The operators $I+B_{\varepsilon}(z)Vq$ are bijective in
$L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$ for
$\varepsilon\in(0,\varepsilon_{2})$ and their inverses are uniformly bounded.

  * (ii)

The operator $I+B_{0}(z)Vq$ is bijective in
$L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$.

The statement of Proposition 3.11 remains true, if (3.29) is replaced by the
weaker assumptions (i) and (ii).

###### Proof of Proposition 3.11.

Since $V$ and $q$ fulfil (3.29) we have

$\begin{split}\big{(}(I+B_{0}(z)Vq)^{-1}&-(I+B_{\varepsilon}(z)Vq)^{-1}\big{)}f\\\
&=(I+B_{\varepsilon}(z)Vq)^{-1}(B_{\varepsilon}(z)-B_{0}(z))Vq(I+B_{0}(z)Vq)^{-1}f\end{split}$

for $f\in L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$; cf. Remark 3.12.
Thus, with Proposition 3.10 and the fact that $Vq$ induces a bounded operators
in $L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$, see (2.3) and Section 2.2,
we find

$\begin{split}&{\left\lVert{(I+B_{0}(z)Vq)^{-1}-(I+B_{\varepsilon}(z)Vq)^{-1}}\right\rVert}_{1/2\to
0}\\\ &\hskip
5.0pt\leq{\left\lVert{(I+B_{\varepsilon}(z)Vq)^{-1}}\right\rVert}_{0\to
0}{\left\lVert{(B_{0}(z)-B_{\varepsilon}(z))Vq}\right\rVert}_{1/2\to
0}{\left\lVert{(I+B_{0}(z)Vq)^{-1}}\right\rVert}_{1/2\to 1/2}\\\ &\hskip
5.0pt\leq C{\left\lVert{B_{0}(z)-B_{\varepsilon}(z)}\right\rVert}_{1/2\to
0}\\\ &\hskip 5.0pt\leq C\varepsilon^{1/2-r},\end{split}$

which is the claimed result. ∎

In the following lemma we give more explicit estimates for the constant
$X_{z}$ in (3.28). Recall that $E$ denotes Stein’s extension operator.
Moreover, we introduce $C_{1/2,-1/2}:=\frac{C_{2}}{C_{1}}\geq 1$, where
$C_{2}\geq C_{1}>0$ are chosen such that

$C_{1}{\left\lVert{f}\right\rVert}_{0}\leq{\left\lVert{f}\right\rVert}_{[L^{2}((-1,1);H^{-1/2}(\Sigma;\mathbb{C}^{N})),L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))]_{1/2}}\leq
C_{2}{\left\lVert{f}\right\rVert}_{0}$

holds for $f\in L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$; note that
according to Proposition 2.7 (i) such a choice is possible. If $B$ is a
bounded operator in $L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$ that admits a
bounded extension to $L^{2}((-1,1);H^{-1/2}(\Sigma;\mathbb{C}^{N}))$ and a
bounded restriction to $L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$, then
Proposition 2.7 (i) and (1.18) imply

$\|B\|_{0\rightarrow 0}\leq C_{1/2,-1/2}\|B\|_{1/2\rightarrow
1/2}^{1/2}\|B\|_{-1/2\rightarrow-1/2}^{1/2}.$ (3.30)

######  Lemma 3.13.

Let $z\in\rho(H)$ and let $X_{z}$ be as in (3.28). Then, the following
assertions hold.

  * (i)

$X_{z}\geq\frac{1}{C_{\boldsymbol{t}_{\Sigma},E,z}}+\mathcal{O}(\sqrt{\left\lvert{\log\varepsilon_{2}}\right\rvert\varepsilon_{2}})$,
where the constant

$\begin{split}C_{\boldsymbol{t}_{\Sigma},E,z}&=2C_{1/2,-1/2}\max_{(\textup{sign},\mu)\in\\{+,-\\}\times\\{z,\overline{z}\\}}\big{\|}\boldsymbol{t}_{\Sigma}^{\textup{sign}}\big{\|}_{H^{1}(\Omega_{\textup{sign}};\mathbb{C}^{N})\to
H^{1/2}(\Sigma;\mathbb{C}^{N})}\\\
&\qquad\quad\cdot{\left\lVert{E}\right\rVert}_{H^{1}(\Omega_{\textup{sign}};\mathbb{C}^{N})\to
H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})}{\left\lVert{\Phi_{\mu}}\right\rVert}_{H^{1/2}(\Sigma;\mathbb{C}^{N})\to
H^{1}(\Omega_{\textup{sign}};\mathbb{C}^{N})}\end{split}$

depends only on the geometry of $\Sigma$ and $z\in\rho(H)$.

  * (ii)

If $\Sigma$ is $C^{\infty}$-smooth and compact and $m>0$, then
$X_{z}\leq\frac{\pi}{4}$.

In particular, if
${\left\lVert{V}\right\rVert}_{W^{1}_{\infty}(\Sigma;\mathbb{C}^{N\times
N})}{\left\lVert{q}\right\rVert}_{L^{\infty}(\mathbb{R};\mathbb{R})}<\frac{1}{C_{\boldsymbol{t}_{\Sigma},E,z}}$
and $\varepsilon_{2}$ is chosen sufficiently small, then the claim in
Proposition 3.11 is true.

###### Proof.

(i) First, by the definition of $\overline{B}_{\varepsilon}(z)$,
$\tau_{\delta}^{\Omega_{\pm}}$ in (3.19) and (3.10), respectively, and the
Cauchy-Schwarz inequality we have for $f\in
L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$ and
$\varepsilon\in(0,\varepsilon_{2})$

$\begin{split}&{\left\lVert{\overline{B}_{\varepsilon}(z)f}\right\rVert}_{1/2}^{2}\leq\int_{-1}^{1}\left(\int_{-1}^{t}\big{\|}\boldsymbol{t}_{\Sigma}^{-}\tau_{\varepsilon(t-s)}^{\Omega_{-}}(\Phi_{z}f(s))_{-}\big{\|}_{H^{1/2}(\Sigma;\mathbb{C}^{N})}ds\right.\\\
&\hskip
100.0pt\left.+\int_{t}^{1}\big{\|}\boldsymbol{t}_{\Sigma}^{+}\tau_{\varepsilon(t-s)}^{\Omega_{+}}(\Phi_{z}f(s))_{+}\big{\|}_{H^{1/2}(\Sigma;\mathbb{C}^{N})}ds\right)^{2}\,dt\\\
&\leq\\!\int_{-1}^{1}\\!\left(\int_{-1}^{1}\frac{C_{\boldsymbol{t}_{\Sigma},E,z}}{2C_{1/2,-1/2}}\\!\sup_{\delta\in(-2\varepsilon_{2},2\varepsilon_{2})}\\!\\!{\left\lVert{\tau_{\delta}}\right\rVert}_{H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})}{\left\lVert{f(s)}\right\rVert}_{H^{1/2}(\Sigma;\mathbb{C}^{N})}ds\right)^{2}\\!dt\\\
&\leq\left(\frac{C_{\boldsymbol{t}_{\Sigma},E,z}}{C_{1/2,-1/2}}\sup_{\delta\in(-2\varepsilon_{2},2\varepsilon_{2})}{\left\lVert{\tau_{\delta}}\right\rVert}_{H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})}{\left\lVert{f}\right\rVert}_{1/2}\right)^{2}.\end{split}$

Hence,

$\sup_{\varepsilon\in(0,\varepsilon_{2})}{\left\lVert{\overline{B}_{\varepsilon}(z)}\right\rVert}_{1/2\to
1/2}\leq\frac{C_{\boldsymbol{t}_{\Sigma},E,z}}{C_{1/2,-1/2}}\sup_{\delta\in(-2\varepsilon_{2},2\varepsilon_{2})}{\left\lVert{\tau_{\delta}}\right\rVert}_{H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})}$

and a similar calculation using (3.21), the estimate
${\left\lVert{E}\right\rVert}_{H^{1}(\Omega_{\textup{sign}};\mathbb{C}^{N})\to
H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\geq 1$, and $C_{1/2,-1/2}\geq 1$
leads to

${\left\lVert{\overline{B}_{0}(z)}\right\rVert}_{1/2\to 1/2}\leq
C_{\boldsymbol{t}_{\Sigma},E,z}.$ (3.31)

Let us proceed with the estimate for $\overline{B}_{\varepsilon}(z)$. Since by
(3.8) and (3.9)

$\sup_{\delta\in(-2\varepsilon_{2},2\varepsilon_{2})}{\left\lVert{\tau_{\delta}}\right\rVert}_{H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
H^{1}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\leq\frac{1+2\varepsilon_{2}{\left\lVert{D\nu}\right\rVert}_{L^{\infty}(\mathbb{R}^{\theta};\mathbb{R}^{\theta\times\theta})}}{{\big{(}1-2\varepsilon_{2}{\left\lVert{D\nu}\right\rVert}_{L^{\infty}(\mathbb{R}^{\theta};\mathbb{R}^{\theta\times\theta})}\big{)}^{\theta/2}}}=1+\mathcal{O}(\varepsilon_{2}),$

the results from (3.25)–(3.26) and Appendix B imply

$\sup_{\varepsilon\in(0,\varepsilon_{2})}\big{\|}\widetilde{B}_{\varepsilon}(z)\big{\|}_{1/2\to
1/2}\leq\frac{C_{\boldsymbol{t}_{\Sigma},E,z}}{C_{1/2,-1/2}}+\mathcal{O}(\sqrt{\left\lvert{\log\varepsilon_{2}}\right\rvert\varepsilon_{2}}).$
(3.32)

It is clear that the estimates (3.31) and (3.32) hold also with $z$ replaced
by $\overline{z}$. As in Step 3 of the proof of Proposition 3.10 we use again
that the anti-dual $(\widetilde{B}_{\varepsilon}(\overline{z}))^{\prime}$ of
$\widetilde{B}_{\varepsilon}(\overline{z})$ in
$L^{2}((-1,1)H^{-1/2}(\Sigma;\mathbb{C}^{N}))$ is an extension of
$\widetilde{B}_{\varepsilon}(z)$ that is bounded by
$\|\widetilde{B}_{\varepsilon}(\overline{z})\|_{1/2\to 1/2}$. Hence, with
(3.30) one gets

$\begin{split}\sup_{\varepsilon\in(0,\varepsilon_{2})}\big{\|}\widetilde{B}_{\varepsilon}(z)\big{\|}_{0\to
0}&\leq\sup_{\varepsilon\in(0,\varepsilon_{2})}C_{1/2,-1/2}\|\widetilde{B}_{\varepsilon}(z)\|_{1/2\to
1/2}^{1/2}\|(\widetilde{B}_{\varepsilon}(\overline{z}))^{\prime}\|_{-1/2\to-1/2}^{1/2}\\\
&=\sup_{\varepsilon\in(0,\varepsilon_{2})}C_{1/2,-1/2}\|\widetilde{B}_{\varepsilon}(z)\|_{1/2\to
1/2}^{1/2}\|\widetilde{B}_{\varepsilon}(\overline{z})\|_{1/2\to 1/2}^{1/2}\\\
&\leq
C_{\boldsymbol{t}_{\Sigma},E,z}+\mathcal{O}(\sqrt{\left\lvert{\log\varepsilon_{2}}\right\rvert\varepsilon_{2}}).\end{split}$

Thus, it follows from the definition of $\widetilde{B}_{\varepsilon}(z)$ in
(3.17) and Lemma 3.6 that

$\sup_{\varepsilon\in(0,\varepsilon_{2})}{\left\lVert{B_{\varepsilon}(z)}\right\rVert}_{0\to
0}\leq
C_{\boldsymbol{t}_{\Sigma},E,z}+\mathcal{O}(\sqrt{\left\lvert{\log\varepsilon_{2}}\right\rvert\varepsilon_{2}}).$

Together with (3.31) this yields the claim in (i).

(ii) First, note that there exist $\varphi_{n}\in
H^{1/2}(\Sigma;\mathbb{C}^{N})$ such that

${\left\lVert{\varphi_{n}}\right\rVert}_{H^{1/2}(\Sigma;\mathbb{C}^{N})}=1\quad\text{and}\quad\left\|\left(-\frac{1}{2}I_{N}+\mathcal{C}_{z}\right)\varphi_{n}\right\|_{H^{1/2}(\Sigma;\mathbb{C}^{N})}\rightarrow
0,\quad\text{as }n\rightarrow\infty;$ (3.33)

for $z\in(-m,m)$ this follows from [23, Corollaries 3.8 and 4.12], while for
$z\in\mathbb{C}\setminus\mathbb{R}$ one may additionally use that [10,
Proposition 4.4 (iv)] implies that $\mathcal{C}_{z}-\mathcal{C}_{0}$ is
compact in $H^{1/2}(\Sigma;\mathbb{C}^{N})$. Define the functions $f_{n}\in
L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$ by
$f_{n}(t):=\exp(i(\alpha\cdot\nu)\pi t/4)\varphi_{n}$ and compute with
Proposition 2.7 (iii)

$\begin{split}\frac{i}{2}(\alpha\cdot\nu)&\int_{-1}^{1}\textup{sign}(t-s)f_{n}(s)ds\\\
&=\frac{2}{\pi}\left(\int_{-1}^{t}\frac{d}{ds}\exp\big{(}i(\alpha\cdot\nu)\tfrac{\pi}{4}s\big{)}\varphi_{n}ds-\int_{t}^{1}\frac{d}{ds}\exp\big{(}i(\alpha\cdot\nu)\tfrac{\pi}{4}s\big{)}\varphi_{n}ds\right)\\\
&=\frac{4}{\pi}\left(\exp\big{(}i(\alpha\cdot\nu)\tfrac{\pi}{4}t\big{)}-\cos\big{(}(\alpha\cdot\nu)\tfrac{\pi}{4}\big{)}\right)\varphi_{n}\\\
&=\frac{4}{\pi}\left(f_{n}(t)-\frac{\sqrt{2}}{2}I_{N}\varphi_{n}\right),\end{split}$

where in the last step the identity
$\cos((\alpha\cdot\nu)\pi/4)=\cos(\pi/4)I_{N}=\frac{\sqrt{2}}{2}I_{N}$, which
can be shown via the power series representation of the cosine function and
$(\alpha\cdot\nu)^{2}=I_{N}$, was used. With a similar argument, one verifies
that
$(\alpha\cdot\nu)\sin((\alpha\cdot\nu)\pi/4)=\sin(\pi/4)I_{N}=\frac{\sqrt{2}}{2}I_{N}$,
which yields

$\begin{split}\int_{-1}^{1}f_{n}(s)ds&=\int_{-1}^{1}\exp\big{(}i(\alpha\cdot\nu)\tfrac{\pi}{4}s\big{)}\varphi_{n}ds=\frac{8}{\pi}(\alpha\cdot\nu)\sin\big{(}(\alpha\cdot\nu)\tfrac{\pi}{4}\big{)}\varphi_{n}=\frac{4\sqrt{2}}{\pi}\varphi_{n}.\end{split}$

Combining the last two displayed formulas with the representation of
$B_{0}(z)$ in (3.16), we conclude

$\begin{split}B_{0}(z)f_{n}(t)&=\frac{i}{2}(\alpha\cdot\nu)\int_{-1}^{1}\textup{sign}(t-s)f(s)\,ds+\mathcal{C}_{z}\int_{-1}^{1}f(s)\,ds\\\
&=\frac{4}{\pi}\left(f_{n}(t)+\sqrt{2}\left(-\frac{1}{2}I_{N}+\mathcal{C}_{z}\right)\varphi_{n}\right).\end{split}$

Hence, using (3.33), we find that

$\|B_{0}(z)\|_{1/2\to
1/2}\geq\lim_{n\rightarrow\infty}\frac{\|B_{0}(z)f_{n}\|_{1/2}}{\|f_{n}\|_{1/2}}=\frac{4}{\pi}$

and thus $X_{z}\leq\tfrac{\pi}{4}$. ∎

##  4\. Proof of Theorem 1.1

In this section we prove the main results of this paper. For this we show in
Proposition 4.1 that the resolvent $(H_{\varepsilon}-z)^{-1}$ of the self-
adjoint operator $H_{\varepsilon}$ in (1.8) converges to the limit operator

$\mathcal{L}(z):=R_{z}-A_{0}(z)Vq(I+B_{0}(z)Vq)^{-1}C_{0}(z)$ (4.1)

whenever $q$ and $V$ in (1.5) and (1.6), respectively, satisfy the condition
(3.29) for some
$z\in\rho(H)=\mathbb{C}\setminus((-\infty,-|m|]\cup[|m|,\infty))$. Lemma 4.2
and Lemma 4.3 collect some auxiliary considerations that are needed in
Proposition 4.4, where it is shown for the limit operator that
$\mathcal{L}(z)=(H_{\widetilde{V}}-z)^{-1}$ with $H_{\widetilde{V}}$ defined
by (1.2). After these preparations we complete the proofs of Theorem 1.1,
Corollary 1.2, and Corollary 1.3.

######  Proposition 4.1.

Let $z\in\rho(H)$ and $\varepsilon\in(0,\varepsilon_{2})$ with
$\varepsilon_{2}$ satisfying (3.11), and let $q$ and $V$ be as in (1.5) and
(1.6), respectively, such that (3.29) holds. Then, $z\in\rho(H_{\varepsilon})$
and for any $r\in(0,\tfrac{1}{2})$ one has

$\begin{split}&\|(H_{\varepsilon}-z)^{-1}-\mathcal{L}(z)\|_{L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\leq
C\varepsilon^{1/2-r},\end{split}$

where the operator $\mathcal{L}(z)$ is given by (4.1). In particular,
$(H_{\varepsilon}-z)^{-1}$ converges to $\mathcal{L}(z)$ in the operator norm
as $\varepsilon\to 0+$.

###### Proof.

By Proposition 3.2 and (3.29), see also Remark 3.12, we have
$z\in\rho(H_{\varepsilon})$ and

$\begin{split}(H_{\varepsilon}-&z)^{-1}-\mathcal{L}(z)=(H_{\varepsilon}-z)^{-1}-R_{z}+A_{0}(z)Vq(I+B_{0}(z)Vq)^{-1}C_{0}(z)\\\
=&-A_{\varepsilon}(z)Vq(I+B_{\varepsilon}(z)Vq)^{-1}C_{\varepsilon}(z)+A_{0}(z)Vq(I+B_{0}(z)Vq)^{-1}C_{0}(z)\\\
=&-A_{\varepsilon}(z)Vq(I+B_{\varepsilon}(z)Vq)^{-1}(C_{\varepsilon}(z)-C_{0}(z))\\\
&-A_{\varepsilon}(z)Vq\big{(}(I+B_{\varepsilon}(z)Vq)^{-1}-(I+B_{0}(z)Vq)^{-1}\big{)}C_{0}(z)\\\
&-(A_{\varepsilon}(z)-A_{0}(z))Vq(I+B_{0}(z)Vq)^{-1}C_{0}(z).\end{split}$
(4.2)

In the following we use that
$A_{\varepsilon}(z):L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})$ and
$(I+B_{\varepsilon}(z)Vq)^{-1}:L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))\to
L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$ are uniformly bounded; cf.
Proposition 3.8 and Remark 3.12 (i). Employing this and Proposition 3.7 we see
that

$\begin{split}\bigl{\|}A_{\varepsilon}(z)Vq(I+B_{\varepsilon}(z)&Vq)^{-1}(C_{\varepsilon}(z)-C_{0}(z))\bigr{\|}_{{L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})}}\\\ &\quad\leq
C{\left\lVert{C_{\varepsilon}(z)-C_{0}(z)}\right\rVert}_{L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
0}\\\ &\quad\leq C\varepsilon^{1/2-r}.\end{split}$ (4.3)

Since $C_{0}(z):L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
L^{2}((-1,1);H^{1/2}(\Sigma;\mathbb{C}^{N}))$ in (3.13) is bounded,
Proposition 3.11 yields

$\begin{split}&\big{\|}A_{\varepsilon}(z)Vq\left((I+B_{\varepsilon}(z)Vq)^{-1}-(I+B_{0}(z)Vq)^{-1}\right)C_{0}(z)\big{\|}_{L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\\\ &\quad\quad\leq
C{\left\lVert{(I+B_{\varepsilon}(z)Vq)^{-1}-(I+B_{0}(z)Vq)^{-1}}\right\rVert}_{1/2\to
0}\\\ &\quad\quad\leq C\varepsilon^{1/2-r}.\end{split}$ (4.4)

Eventually, in a similar way as in (4.3) we find with Proposition 3.8 that

$\begin{split}&{\left\lVert{(A_{\varepsilon}(z)-A_{0}(z))Vq(I+B_{0}(z)Vq)^{-1}C_{0}(z)}\right\rVert}_{L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\\\ &\leq
C{\left\lVert{A_{\varepsilon}(z)-A_{0}(z)}\right\rVert}_{0\to
L^{2}(\mathbb{R}^{\theta};\mathbb{C}^{N})}\\\ &\leq
C\varepsilon^{1/2-r}.\end{split}$ (4.5)

Combining (4.3)–(4.5) with (4.2) shows the claim of this proposition. ∎

The next goal is to show that the limit operator $\mathcal{L}(z)$ in (4.1) is
the resolvent of $H_{\widetilde{V}}$ defined in (1.2). This requires some
technical preparations and we first introduce the operator

$\begin{split}T&:L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))\to
L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N})),\\\
Tf(t)&:=\frac{i}{2}\int_{-1}^{1}\textup{sign}(t-s)f(s)\,ds,\end{split}$ (4.6)

and the function

$Q(t):=-\frac{1}{2}+\int_{-1}^{t}q(s)ds,\quad t\in[-1,1].$ (4.7)

Note that $Q$ satisfies $Q^{\prime}=q$, $Q(-1)=-\frac{1}{2}$, and by (1.5)
also $Q(1)=\frac{1}{2}$. Moreover, for $r\in[0,\frac{1}{2}]$ the map $T$ gives
rise to a bounded operator in $L^{2}((-1,1);H^{r}(\Sigma;\mathbb{C}^{N}))$.

######  Lemma 4.2.

Let $q$ and $V$ be as in (1.5) and (1.6), respectively, let
$r\in[0,\frac{1}{2}]$, and assume that
$\cos\big{(}\tfrac{(\alpha\cdot\nu)V}{2}\big{)}^{-1}\in
W^{1}_{\infty}(\Sigma;\mathbb{C}^{N\times N})$. Then, the following is true:

  * (i)

$I+T(\alpha\cdot\nu)Vq$ is boundedly invertible in
$L^{2}((-1,1);H^{r}(\Sigma;\mathbb{C}^{N}))$ and its inverse is given by
(4.10).

  * (ii)

If $f\in\textup{ran }\mathfrak{J}$ (that is, $f$ is independent of
$t\in(-1,1)$), then

$(I+T(\alpha\cdot\nu)Vq)^{-1}f(t)=\cos\big{(}\tfrac{(\alpha\cdot\nu)V}{2}\big{)}^{-1}\exp(-i(\alpha\cdot\nu)VQ(t))f(t)$
(4.8)

holds for a.e. $t\in(-1,1)$.

###### Proof.

(i) It will be shown that the operator defined in (4.10) below is the inverse
of $I+T(\alpha\cdot\nu)Vq$. Fix $r\in[0,\frac{1}{2}]$ and define the operators

$\begin{split}\Xi&:L^{2}((-1,1);H^{r}(\Sigma;\mathbb{C}^{N}))\rightarrow
H^{r}(\Sigma;\mathbb{C}^{N}),\\\ \Xi
f&=\frac{1}{2}\cos\big{(}\tfrac{(\alpha\cdot\nu)V}{2}\big{)}^{-1}i(\alpha\cdot\nu)V\int_{-1}^{1}\exp\big{(}i(\alpha\cdot\nu)V\big{(}Q(s)-\tfrac{1}{2}\big{)}\big{)}q(s)f(s)\,ds,\end{split}$
(4.9)

and

$\begin{split}O&:L^{2}((-1,1);H^{r}(\Sigma;\mathbb{C}^{N}))\to
L^{2}((-1,1);H^{r}(\Sigma;\mathbb{C}^{N}))\\\
Of(t)&:=f(t)+\exp(-i(\alpha\cdot\nu)VQ(t))\Xi f\\\ &\hskip
37.0pt-i(\alpha\cdot\nu)V\int_{-1}^{t}\exp(i(\alpha\cdot\nu)V(Q(s)-Q(t)))q(s)f(s)\,ds.\end{split}$
(4.10)

We will show that $\Xi$ and $O$ are bounded and that
$O=(I+T(\alpha\cdot\nu)Vq)^{-1}$. First, we verify that $\Xi$ is well-defined
and bounded. Let $f\in L^{2}((-1,1);H^{r}(\Sigma;\mathbb{C}^{N}))$. Then the
integrand in (4.9) is measurable as a function from $(-1,1)$ to
$H^{r}(\Sigma;\mathbb{C}^{N})$ since
${\left({\exp\big{(}i(\alpha\cdot\nu)V\big{(}Q(\cdot)-\tfrac{1}{2}\big{)}\big{)}q(\cdot)f(\cdot)},{\psi}\right)}_{H^{r}(\Sigma;\mathbb{C}^{N})}$
is measurable for all $\psi\in H^{r}(\Sigma;\mathbb{C}^{N})$, see Definition
2.5. In fact, the latter function is the pointwise limit of the sequence of
measurable functions

$\begin{split}&t\mapsto\sum_{k=0}^{n}\frac{\big{(}\big{(}i(\alpha\cdot\nu)V\big{(}Q(t)-\tfrac{1}{2}\big{)}\big{)}^{k}q(t)f(t),\psi\big{)}_{H^{r}(\Sigma;\mathbb{C}^{N})}}{k!}\\\
&\hskip
100.0pt=\sum_{k=0}^{n}\frac{q(t)\big{(}Q(t)-\tfrac{1}{2}\big{)}^{k}{\left({f(t)},{(-iV(\alpha\cdot\nu))^{k}\psi}\right)}_{H^{r}(\Sigma;\mathbb{C}^{N})}}{k!}.\end{split}$

Moreover, as
$\cos\big{(}\tfrac{(\alpha\cdot\nu)V}{2}\big{)}^{-1},\alpha\cdot\nu,V\in
W^{1}_{\infty}(\Sigma;\mathbb{C}^{N\times N})$ it follows that

$\begin{split}\|&\Xi f\|_{H^{r}(\Sigma;\mathbb{C}^{N})}\\\
&=\frac{1}{2}{\left\lVert{\cos\big{(}\tfrac{(\alpha\cdot\nu)V}{2}\big{)}^{-1}i(\alpha\cdot\nu)V\int_{-1}^{1}\exp\big{(}i(\alpha\cdot\nu)V\big{(}Q(s)-\tfrac{1}{2}\big{)}\big{)}q(s)f(s)\,ds}\right\rVert}_{H^{r}(\Sigma;\mathbb{C}^{N})}\\\
&\leq
C\int_{-1}^{1}{\left\lVert{\exp\big{(}i(\alpha\cdot\nu)V\big{(}Q(s)-\tfrac{1}{2}\big{)}\big{)}q(s)f(s)}\right\rVert}_{H^{r}(\Sigma;\mathbb{C}^{N})}\,ds\\\
\end{split}$

and $\alpha\cdot\nu,V\in W^{1}_{\infty}(\Sigma;\mathbb{C}^{N\times N})$ also
imply $\exp(i(\alpha\cdot\nu)V(Q(s)-\tfrac{1}{2}))\in
W^{1}_{\infty}(\Sigma;\mathbb{C}^{N\times N})$ via the power series of the
exponential function. Using $q\in L^{\infty}((-1,1);\mathbb{R})$ we conclude

$\begin{split}\|&\Xi f\|_{H^{r}(\Sigma;\mathbb{C}^{N})}\\\ &\leq
C\int_{-1}^{1}{\left\lVert{\exp\big{(}(\alpha\cdot\nu)V(Q(s)-\tfrac{1}{2})\big{)}}\right\rVert}_{W^{1}_{\infty}(\Sigma;\mathbb{C}^{N\times
N})}{\left\lVert{q}\right\rVert}_{L^{\infty}((-1,1);\mathbb{R})}{\left\lVert{f(s)}\right\rVert}_{H^{r}(\Sigma;\mathbb{C}^{N})}\,ds\\\
&\leq
C\int_{-1}^{1}{\left\lVert{f(s)}\right\rVert}_{H^{r}(\Sigma;\mathbb{C}^{N})}\\\
&\leq C{\left\lVert{f}\right\rVert}_{r}.\end{split}$

This shows that $\Xi$ is well-defined and bounded. Analogously one can check
that $O$ is well-defined and bounded. Hence, in order to show (i) it suffices
to prove

$(I+T(\alpha\cdot\nu)Vq)Of=O(I+T(\alpha\cdot\nu)Vq)f=f$ (4.11)

for all $f\in L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$. By Proposition 2.7
(iii) and [26, Proposition 1.2.24] this is true, if for $\sigma$-a.e.
$x_{\Sigma}\in\Sigma$ the relation

$(I+T(\alpha\cdot\nu)Vq)Of(\cdot)(x_{\Sigma})=O(I+T(\alpha\cdot\nu)Vq)f(\cdot)(x_{\Sigma})=f(\cdot)(x_{\Sigma})$

holds a.e. on $(-1,1)$. Let $f\in L^{2}((-1,1);L^{2}(\Sigma;\mathbb{C}^{N}))$
and $x_{\Sigma}\in\Sigma$ be fixed such that
$\varphi=\varphi(x_{\Sigma}):=f(\cdot)(x_{\Sigma})\in
L^{2}((-1,1);\mathbb{C}^{N})$ and set
$A=A(x_{\Sigma}):=(\alpha\cdot\nu(x_{\Sigma}))V(x_{\Sigma})$. Then, we have
for a.e. $t\in(-1,1)$

$\begin{split}&(I+T(\alpha\cdot\nu)Vq)Of(t)(x_{\Sigma})\\\
&=\varphi(t)+\exp(-iAQ(t))\Xi
f(x_{\Sigma})-iA\int_{-1}^{t}\exp(iA(Q(s)-Q(t)))q(s)\varphi(s)\,ds\\\
&\,\,+\frac{i}{2}\int_{-1}^{1}\textup{sign}(t-s)Aq(s)\bigg{(}\varphi(s)+\exp(-iAQ(s))\Xi
f(x_{\Sigma})\\\ &\qquad\qquad\qquad\qquad\qquad\quad-
iA\int_{-1}^{s}\exp(iA(Q(r)-Q(s)))q(r)\varphi(r)\,dr\bigg{)}\,ds.\end{split}$
(4.12)

With a direct calculation we find that

$\begin{split}\frac{i}{2}&\int_{-1}^{1}\textup{sign}(t-s)A\exp(-iAQ(s))q(s)\Xi
f(x_{\Sigma})\,ds\\\
&=\frac{1}{2}\bigg{(}\int_{-1}^{t}iA\exp(-iAQ(s))q(s)\,ds-\int_{t}^{1}iA\exp(-iAQ(s))q(s)\,ds\bigg{)}\Xi
f(x_{\Sigma})\\\ &=-\exp(-iAQ(t))\Xi
f(x_{\Sigma})+\frac{1}{2}\big{(}\exp\big{(}-i\tfrac{A}{2}\big{)}+\exp\big{(}i\tfrac{A}{2}\big{)}\big{)}\Xi
f(x_{\Sigma})\\\ &=-\exp(-iAQ(t))\Xi
f(x_{\Sigma})+\cos\big{(}\tfrac{A}{2}\big{)}\Xi f(x_{\Sigma}).\end{split}$

Furthermore, integration by parts gives us

$\begin{split}-&\frac{i}{2}\int_{-1}^{1}\textup{sign}(t-s)Aq(s)iA\int_{-1}^{s}\exp(iA(Q(r)-Q(s)))q(r)\varphi(r)\,dr\,ds\\\
&\quad=\frac{i}{2}A\int_{-1}^{t}\frac{d}{ds}\big{(}\exp(-iAQ(s))\big{)}\int_{-1}^{s}\exp(iAQ(r))q(r)\varphi(r)\,dr\,ds\\\
&\qquad-\frac{i}{2}A\int_{t}^{1}\frac{d}{ds}\big{(}\exp(-iAQ(s))\big{)}\int_{-1}^{s}\exp(iAQ(r))q(r)\varphi(r)\,dr\,ds\\\
&\quad=iA\exp(-iAQ(t))\int_{-1}^{t}\exp(iAQ(r))q(r)\varphi(r)\,dr\\\
&\qquad-\frac{i}{2}A\int_{-1}^{1}\exp(iA(Q(r)-\tfrac{1}{2}))q(r)\varphi(r)\,dr-\frac{i}{2}A\int_{-1}^{1}\textup{sign}(t-s)q(s)\varphi(s)\,ds\\\
&\quad=iA\exp(-iAQ(t))\int_{-1}^{t}\exp(iAQ(r))q(r)\varphi(r)\,dr\\\
&\qquad-\frac{i}{2}A\int_{-1}^{1}\textup{sign}(t-s)q(s)\varphi(s)\,ds-\cos\big{(}\tfrac{A}{2}\big{)}\Xi
f(x_{\Sigma}).\end{split}$

A combination of the last two calculations with (4.12) yields