Non-local relativistic $\delta$-shell interactions

Abstract

In this paper, new self-adjoint realizations of the Dirac operator in dimension two and three are introduced. It is shown that they may be associated with the formal expression $\mathcal {D}_0+|F\delta _\Sigma \rangle \langle G\delta _\Sigma |$, where $\mathcal {D}_0$ is the free Dirac operator, F and G are matrix valued coefficients, and $\delta _\Sigma$ stands for the single layer distribution supported on a hypersurface $\Sigma$, and that they can be understood as limits of the Dirac operators with scaled non-local potentials. Furthermore, their spectral properties are analysed.

1 Introduction

Let $\Omega$ be an open bounded simply connected subset of $\mathbb {R}^n$, $n=2,\,3$, with the Lipschitz smooth boundary $\Sigma$. In the present paper, we will deal with the n-dimensional Dirac operator perturbed by the formal non-local $\delta$-shell potential

$$\begin{aligned} |F\delta _\Sigma \rangle \langle G\delta _\Sigma |, \end{aligned}$$

(1)

where $F,G\in L^2(\Sigma ;\mathbb {C}^{N\times N})$ are matrix-valued functions with $N:=2^{\lceil \frac{n}{2} \rceil }$ and $\delta _\Sigma$ stands for the single layer distribution supported on $\Sigma$. Identifying the bra-vector with the adjoint of the distribution action and extending naturally the usual operations with distributions, we put, for $\varphi \in \mathscr {D}(\mathbb {R}^n;\mathbb {C}^N)$,

$$\begin{aligned} |F\delta _\Sigma \rangle \langle G\delta _\Sigma |\varphi :=F(\delta _\Sigma ,G^*\varphi )\delta _\Sigma =F\int _\Sigma (G^*\varphi )\,\delta _\Sigma \in \mathscr {D}'(\mathbb {R}^n;\mathbb {C}^N), \end{aligned}$$

(2)

i.e.

$$\begin{aligned} (|F\delta _\Sigma \rangle \langle G\delta _\Sigma |\varphi ,\tilde{\varphi })=\int _\Sigma \sum _{i=1}^N \Big (F\int _\Sigma (G^*\varphi )\Big )_i\tilde{\varphi }_i\in \mathbb {C}\quad (\forall \tilde{\varphi }\in \mathscr {D}(\mathbb {R}^n;\mathbb {C}^N)). \end{aligned}$$

Note that choosing $F=I,\, G=A^*=\text {const.}$, the expression (1) is reduced to

$$\begin{aligned} A|\delta _\Sigma \rangle \langle \delta _\Sigma |, \end{aligned}$$

(3)

which acts as

$$\begin{aligned} A|\delta _\Sigma \rangle \langle \delta _\Sigma |\varphi :=A(\delta _\Sigma ,\varphi )\delta _\Sigma \in \mathscr {D}'(\mathbb {R}^n;\mathbb {C}^N). \end{aligned}$$

(4)

We will show that the Dirac operator with the potential (1) is described by a non-local transmission condition along $\Sigma$ for the functions in the operator domain, which relates the difference of the inner and the outer trace with respect to $\Omega$ and its complement, respectively, to the integral over $\Sigma$ of the their average, see (13). To our best knowledge, such an operator has not been considered before. However, it is quite common to use (3) with $A=A^*$ as a formal expression for the point interaction in the one-dimensional setting (and, in the non-relativistic setting, also for the point interaction in dimensions two and three). Recall that then

$$\begin{aligned} |\delta \rangle \langle \delta |\varphi =(\delta ,\varphi )\delta =\varphi (0)\delta =\varphi \delta =:\delta \varphi , \end{aligned}$$

i.e. the "projection on the Dirac $\delta$-distribution" yields exactly the same result as the "multiplication by the Dirac $\delta$-distribution". This is not true for the single layer distribution, because

$$\begin{aligned} A\delta _\Sigma \varphi :=A\varphi \delta _\Sigma \in \mathscr {D}'(\mathbb {R}^n;\mathbb {C}^N) \end{aligned}$$

(5)

is clearly different from (4). Note that, as far as we know, the first mathematically rigorous treatment of the local $\delta$-shell potential (5) appeared in [1] and [18] in the non-relativistic and relativistic setting, respectively. There, choosing the two-dimensional sphere for $\Sigma$, a reduction to a one-dimensional problem was possible.

The Dirac operator with the formal potential $A\delta _\Sigma$ is described by a local transmission condition along $\Sigma$ and has been studied intensively during recent years, see, e.g. [2, 3, 5, 11]. In particular, an important question of regular approximations was addressed in [6, 14, 23]. It was observed that given a scaled regular potential $V_\varepsilon$ that converges to $\delta _\Sigma$ as $\varepsilon \rightarrow 0+$, the Dirac operator with the potential $A V_\varepsilon$ converges to the Dirac operator with the formal potential $\tilde{A} \delta _\Sigma$, where, except for special cases, $\tilde{A}\ne A$, i.e. the coupling constants have to be renormalized. The same surprising effect occurs in the one-dimensional setting, too [25, 27]. Nevertheless, when non-local approximations of the type $|V_\varepsilon \rangle \langle V_\varepsilon |$ are used, no renormalization is needed [20, 25]. When we tried to show a similar result in a higher dimensional setting, we discovered that the limit operator for the Dirac operator with the potential $A |V_\varepsilon \rangle \langle V_\varepsilon |$ cannot be associated with the formal potential $\tilde{A}\delta _\Sigma$ for any $\tilde{A}$. Instead, we found out that it corresponds to the Dirac operator with the formal potential $A|\delta _\Sigma \rangle \langle \delta _\Sigma |$.

More generally, to approximate potential (1) one may use non-local potentials with separable kernels, cf. Section 4. Note that such potentials were considered before in the non-relativistic, semi-relativistic, and relativistic setting, see, e.g. [19, 28], and [30], respectively. Furthermore, there were also attempts to incorporate non-local interactions into the quantum field theory [15, 29]. While it is widely believed that non-local interactions do not appear in Nature, non-local potential terms are omnipresent in mean-field theories for multi-particle systems. Closest to our setting (see also Remark 3.0.1) is the exchange term of the single particle equation in the Hartree-Fock approximation for the relativistic quantum field model of nuclear matter [13, 21, 24].

The paper is organized as follows. In Sect. 2, we will introduce notations, present basic facts about the free Dirac operator and tubular neighbourhoods of $\Sigma$, and summarize very briefly the theory of generalhyphen usageized boundary triples, that constitutes the cornerstone of our analysis. More concretely, we will rely on the generalized boundary triple for the Dirac operator that has been developed very recently in [7]. After recalling this triple in Sect. 3, the Dirac operator with the potential (1) will be introduced rigorously, the condition on its self-adjointness will be derived, and its spectrum will be investigated. In Sect. 4, approximations of the non-local $\delta$-shell potentials will be constructed by means of scaled non-local finite-rank potentials. We will prove that the resolvents of the respective Dirac operators converge uniformly, which implies the convergence of spectra and eigenfunctions.

2 Preliminaries

2.1 Notations

By $L^2(\mathcal {M};\mathcal {G})$ we denote the space of functions (after the usual factorization) with values in the Banach space $\mathcal {G}$ defined on $\mathcal {M}$ for which the 2nd power of the norm on $\mathcal {G}$ is integrable, where for $\mathcal {M}$ being an open subset of $\mathbb {R}^n$ or a hypersurface embedded in $\mathbb {R}^n$, we integrate with respect to the n-dimensional Lebesgue measure or the surface measure induced by the embedding, respectively. Similarly, for $s\in \mathbb {R}$ we denote by $H^s(\mathcal {M};\mathbb {C}^N)$ the space of $\mathbb {C}^N$-valued functions on $\mathcal {M}$ such that each of their components belongs to the standard $L^2$-based Sobolev space $H^s(\mathcal {M})$. We use the symbol $\langle \cdot ,\cdot \rangle$ for the dot product (conjugate linear in the first argument) on $L^2(\mathbb {R}^n;\mathbb {C}^N)$ and also as a natural abbreviation for the following integrals

$$\begin{aligned} \langle E,\psi \rangle&:= \int _{\mathbb {R}^n} E^*(x)\psi (x)\,\textrm{d}x\in \mathbb {C}^N,\\ \langle E,H\rangle&:= \int _{\mathbb {R}^n} E^*(x)H(x)\,\textrm{d}x\in \mathbb {C}^{N\times N}, \end{aligned}$$

where $\psi \in L^2(\mathbb {R}^n;\mathbb {C}^N)$ and $E,\, H\in L^2(\mathbb {R}^n;\mathbb {C}^{N\times N})$. Then by $|E\rangle \langle H|$ we understand the following finite-rank operator in $L^2(\mathbb {R}^n;\mathbb {C}^N)$,

$$\begin{aligned} |E\rangle \langle H|\psi :=E\langle H,\psi \rangle =E\int _{\mathbb {R}^n}H^*(x)\psi (x)\,\textrm{d}x. \end{aligned}$$

The action of a distribution f on a test function $\varphi$ is denoted by $(f,\varphi )$. This bracket is linear in both arguments. If K is an integral operator then we write K(x, y) for its kernel. The surface measure on $\Sigma$ is denoted by $\,\textrm{d}\sigma$. We always adopt the convention $(\forall w\in \mathbb {C}{\setminus } [0,+\infty ))(\textrm{Im}\sqrt{w}>0).$ We use the standard definition of the Pauli matrices,

$$\begin{aligned} \sigma _1 = \begin{pmatrix} 0 &{} \quad 1\\ 1 &{} \quad 0 \end{pmatrix},\, \sigma _2 = \begin{pmatrix} 0 &{} \quad -i\\ i &{} \quad 0 \end{pmatrix},\, \sigma _3 = \begin{pmatrix} 1 &{} \quad 0\\ 0 &{} \quad -1 \end{pmatrix}, \end{aligned}$$

and by $I_N$ we denote the $N\times N$ identity matrix.

2.2 Free Dirac operator

Let $m\in \mathbb {R}$, $n\in \{2,3\}$, and $N=2^{\lceil \frac{n}{2} \rceil }$. For $n=2$, we put

$$\begin{aligned} \alpha _1=\sigma _1,\, \alpha _2=\sigma _2,\, \alpha _0=\sigma _3, \end{aligned}$$

whereas for $n=3$, we put

$$\begin{aligned} \alpha _k = \begin{pmatrix} 0 &{} \quad \sigma _k\\ \sigma _k &{} \quad 0 \end{pmatrix}\, (\forall k \in \{1,2,3\}),\,\, \alpha _0 = \begin{pmatrix} I_2 &{} \quad 0\\ 0 &{} \quad -I_2 \end{pmatrix}. \end{aligned}$$

Finally, let $\mathcal {D}_0$ be the differential expression that acts on $\mathbb {C}^N$-valued functions of n-variables as

$$\begin{aligned} \mathcal {D}_0:=-i(\alpha \cdot \nabla )+m\alpha _0=-i\sum _{k=1}^n\alpha _k\frac{\partial }{\partial x_k}+m\alpha _0. \end{aligned}$$

Then the n-dimensional free Dirac operator with the mass term m is the following operator in $L^2(\mathbb {R}^n;\mathbb {C}^N)$,

$$\begin{aligned} \begin{aligned} {{\,\textrm{Dom}\,}}(D_0)&= H^1 (\mathbb {R}^n; \mathbb {C}^N),\\ D_0 \psi&= \mathcal {D}_0 \psi . \end{aligned} \end{aligned}$$

It is well known that the operator $D_0$ is self-adjoint and its spectrum is purely absolutely continuous and consists of

$$\sigma (D_0) = \sigma _{\text {ac}}(D_0) = (-\infty ,-|m|]\cup [|m|,+\infty ),$$

cf. [26]. For $z\in \mathbb {C}\setminus \sigma (D_0)$, the integral kernel of the resolvent $(D_0-z)^{-1}$ may be computed from the integral kernel of the resolvent of the Laplacian, employing the relation

$$(D_0-z)(D_0+z)=I_N(-\Delta +m^2-z^2).$$

Explicitly, it is given by $(D_0-z)^{-1}(x,y)=:R_z(x-y)$ with

$$\begin{aligned} R_z(x)= \frac{k(z)}{2\pi }K_1(-ik(z)|x|)\frac{(\alpha \cdot x)}{|x|}+\frac{1}{2\pi }K_0(-ik(z)|x|)(zI_2 +m\alpha _0) \end{aligned}$$

and

$$\begin{aligned} R_z(x)=\left( zI_4+m\alpha _0+(1-ik(z)|x|)\frac{i(\alpha \cdot x)}{|x|^2}\right) \frac{1}{4\pi |x|}\textrm{e}^{ik(z)|x|} \end{aligned}$$

for the dimension two and three, respectively. Here, $K_j$ stands for the modified Bessel function of the second kind and $k(z):= \sqrt{z^2-m^2}.$

2.3 Tubular neighbourhoods of hypersurfaces

If $\Omega$ is as above but now with $C^2$-smooth boundary $\Sigma$, then we may construct tubular neighbourhoods of $\Sigma$ in a standard way, cf. [22]. Namely, given $\varepsilon >0$, we define the $\varepsilon$-tubular neighbourhood $\Sigma _\varepsilon$ of $\Sigma$ as the image of the mapping

$$\begin{aligned} \mathscr {L}_\varepsilon :\, \Sigma \times (-1,1)\rightarrow \mathbb {R}^n:\, \left\{ (x_\Sigma ,u)\mapsto x_\Sigma +\varepsilon u \nu (x_\Sigma )\right\} , \end{aligned}$$

where $\nu (x_\Sigma )$ stands for the unit normal vector pointing outwards of $\Omega$, i.e.

$$\begin{aligned} \Sigma _\varepsilon :=\mathscr {L}_\varepsilon (\Sigma \times (-1,1)). \end{aligned}$$

(6)

Under our assumptions, the principal curvatures $K_\mu ,\, \mu =1,\ldots , n-1,$ of $\Sigma$ are continuous functions on the compact set $\Sigma$. Therefore, $\mathscr {L}_\varepsilon$ is a local diffeomorphism for all $\varepsilon$ sufficiently small, cf. formula (7) below. Moreover, the definition of a $C^k$-smooth domain combined with the compactness of $\Sigma$ yield that $\mathscr {L}_\varepsilon :\Sigma \times (-1,1)\rightarrow \Sigma _\varepsilon$ is bijective for all $\varepsilon$ below a certain threshold.

We may view $\Sigma$ as a Riemannian manifold equipped with the metric g induced by the embedding into $\mathbb {R}^n$ and $\Sigma _\varepsilon$ as a Riemannian manifold with the metric induced by $\mathscr {L}_\varepsilon$. Then the volume element on $\Sigma _\varepsilon$ obeys

$$\begin{aligned} \,\textrm{d}\Omega _\varepsilon =\varepsilon \left[ 1+\sum _{\mu =1}^{n-1}(-\varepsilon u)^{\mu }\left( {\begin{array}{c}d-1\\ \mu \end{array}}\right) K_{\mu } \right] \, \,\textrm{d}\sigma \wedge \,\textrm{d}u=:\varepsilon w_\varepsilon \, \,\textrm{d}\sigma \wedge \,\textrm{d}u, \end{aligned}$$

(7)

where $\,\textrm{d}\sigma := (\det {g^{-1}})^{1/2} \, \,\textrm{d}x^1 \wedge \dots \wedge \,\textrm{d}x^{n-1}$ with $x^{\mu }$ being, for the moment, local coordinates on $\Sigma$ is the volume element on $\Sigma$. Note that the function $w_\varepsilon =w_\varepsilon (x_\Sigma ,u)$ obeys

$$\begin{aligned} w_\varepsilon =1+\mathcal {O}(\varepsilon ) \end{aligned}$$

(8)

uniformly on $\Sigma \times (-1,1)$ as $\varepsilon \rightarrow 0+$.

2.4 Generalized boundary triple

Below, we will summarize basic definitions and results concerning the quasi and the generalized boundary triples. We will mainly follow [7] in our exposition; original statements with complete proofs may be found in [9, 10, 16, 17]. Throughout this section, S is assumed to be a densely defined closed symmetric operator in a Hilbert space $\mathcal {H}$ and T is a linear operator such that $\overline{T}=S^*$. The quasi/generalized boundary triples provide powerful tools for studying certain restrictions of T (which turn out to be extensions of S).

Later, in our particular setting, S will be a restriction of the free Dirac operator $D_0$ to a subspace of functions that vanish along the hypersurface $\Sigma$ and certain extensions of S constructed using a generalized boundary triple will be identified with the Dirac operators perturbed by the non-local $\delta$-shell interaction (1).

Definition 2.4.1

Let T be such that $\overline{T}=S^*$. A triple $(\mathcal {G},\Gamma _0,\Gamma _1)$ consisting of a Hilbert space $\mathcal {G}$ and linear mappings $\Gamma _0,\Gamma _1:{{\,\textrm{Dom}\,}}T \rightarrow \mathcal {G}$ is called a quasi boundary triple for $S^*$ if the following holds:

• (i) For all $f,g\in {{\,\textrm{Dom}\,}}T$, $\langle Tf,g\rangle _\mathcal {H} - \langle f,Tg\rangle _\mathcal {H}= \langle \Gamma _1 f,\Gamma _0 g\rangle _\mathcal {G} - \langle \Gamma _0 f,\Gamma _1 g\rangle _\mathcal {G}.$

• (ii) The range of $(\Gamma _0,\Gamma _1)$ is dense in $\mathcal {G}\times \mathcal {G}$.

• (iii) The restriction $T_0:=T\restriction {{{\,\textrm{Ker}\,}}\Gamma _0}$ is a self-adjoint operator in $\mathcal {H}$.

If conditions (i) and (iii) hold, and the mapping $\Gamma _0:{{\,\textrm{Dom}\,}}T \rightarrow \mathcal {G}$ is surjective, then $(\mathcal {G},\Gamma _0,\Gamma _1)$ is called generalized boundary triple. Note that [17, Lem. 6.1] implies that every generalized boundary triple is also a quasi boundary triple.

Definition 2.4.2

Let $S,\, T$ be as above, $(\mathcal {G},\Gamma _0,\Gamma _1)$ be a quasi boundary triple for $S^*$, and $T_0=T\restriction {{{\,\textrm{Ker}\,}}\Gamma _0}$. Then the associated $\gamma$-field and the Weyl function M are defined by

$$\rho (T_0)\ni z \mapsto \gamma (z) = (\Gamma _0\restriction {{{\,\textrm{Ker}\,}}(T-z)})^{-1}$$

and

$$\rho (T_0)\ni z \mapsto M(z)= \Gamma _1(\Gamma _0\restriction {{{\,\textrm{Ker}\,}}(T-z)})^{-1}.$$

For a linear operator B in $\mathcal {G}$, we put

$$\begin{aligned} T_B = T\restriction {{{\,\textrm{Ker}\,}}(\Gamma _0+B\Gamma _1)}. \end{aligned}$$

(9)

Since ${{\,\textrm{Dom}\,}}{S}=\ker \Gamma _0\cap \ker \Gamma _1$ by [9, Prop. 2.2], $S\subset T_B$. The following theorem yields an eigenvalue condition for $T_B$, an alternative description of ${{\,\textrm{Ran}\,}}(T_B-z)$, which may be used in the proof of self-adjointness of $T_B$, and a Krein-like formula for the resolvent of $T_B$.

Theorem 2.4.1

Let $S,\, T$ be as above, $(\mathcal {G},\Gamma _0,\Gamma _1)$ be a quasi boundary triple for $S^*$, $T_0=T\restriction {{{\,\textrm{Ker}\,}}\Gamma _0}$, and $\gamma$ and M denote the associated $\gamma$-field and the Weyl function, respectively. Finally, let $T_B$ be given by (9). Then the following holds for all $z\in \rho (T_0)$:

• (i) $z\in \sigma _{\mathrm p} (T_B)$ if and only if $0\in \sigma _{\mathrm p}(I+BM(z))$. Moreover,

  $${{\,\textrm{Ker}\,}}(T_B -z) = \{\gamma (z)\psi \mid \psi \in {{\,\textrm{Ker}\,}}(I+BM(z))\}.$$

• (ii) If $z\notin \sigma _{\mathrm p}(T_B)$, then $g\in {{\,\textrm{Ran}\,}}(T_B-z)$ if and only if $B\gamma (\overline{z})^*g\in {{\,\textrm{Ran}\,}}(I+BM(z)).$

• (iii) If $z\notin \sigma _{\mathrm p}(T_B)$, then

  $$\begin{aligned} (T_B-z)^{-1}g = (T_0-z)^{-1}g-\gamma (z)(I+BM(z))^{-1}B\gamma (\overline{z})^*g \end{aligned}$$

  (10)

  holds for all $g\in {{\,\textrm{Ran}\,}}(T_B-z)$.

3 Non-local relativistic delta shell interactions

Recall that we assume $\Sigma$ to be the Lipschitz smooth boundary of an open bounded simply connected set $\Omega \equiv \Omega _+ \subset \mathbb {R}^n,\, n=2,\,3$. Denote the outer domain $\mathbb {R}^n\setminus \overline{\Omega _+}$ by $\Omega _-$. Then we can write the Euclidean space as the disjoint union $\mathbb {R}^n=\Omega _+ \cup \Sigma \cup \Omega _-$. Also recall that we denote by $\nu (x_\Sigma )$ the unit normal vector at $x_\Sigma \in \Sigma$ pointing outwards of $\Omega _+$. For $s\in [0,1]$, define the space

$$H^s_\alpha (\Omega _\pm ):=\{ \psi _\pm \in H^s(\Omega _\pm ; \mathbb {C}^N)\mid (\alpha \cdot \nabla )\psi _\pm \in L^2(\Omega _\pm ; \mathbb {C}^N)\}.$$

It was shown in [7, Lem. 4.1 and Cor. 4.6] that $\psi \in H^s_\alpha (\Omega _\pm )$ admits Dirichlet traces $\mathcal {T}_\pm$ in $H^{s-\frac{1}{2}}(\Sigma ;\mathbb {C}^N).$ In particular, $\mathcal {T}_\pm \psi _\pm \in L^2(\Sigma ;\mathbb {C}^N)$ for $\psi _\pm \in H^{\frac{1}{2}}_\alpha (\Omega _\pm )$.

Now, it is straightforward to check that the restriction $S:=D_0\restriction H_0^1(\mathbb {R}^n\setminus \Sigma ;\mathbb {C}^N)$ is closed, symmetric, and densely defined. Moreover, it follows that $S^*$ is given by

$$\begin{aligned} {{\,\textrm{Dom}\,}}(S^*)&= \{\psi _-\oplus \psi _+\mid \, \psi _\pm \in H^0_\alpha (\Omega _\pm )\},\\ S^*(\psi _-\oplus \psi _+)&=\mathcal {D}_0\psi _-\oplus \mathcal {D}_0\psi _+, \end{aligned}$$

cf. [4, Prop. 3.1]. It was proved in [7] that $H^\frac{1}{2}_\alpha (\Omega _-)\oplus H^\frac{1}{2}_\alpha (\Omega _+)$ is an operator core of $S^*$ and that, with the choice

$$\begin{aligned} T:=S^*\restriction H^\frac{1}{2}_\alpha (\Omega _-)\oplus H^\frac{1}{2}_\alpha (\Omega _+), \end{aligned}$$

(11)

the triple $(\mathcal {G},\Gamma _0,\Gamma _1)$, where $\mathcal {G}=L^2(\Sigma ;\mathbb {C}^N)$ and

$$\begin{aligned} \begin{aligned} \Gamma _0 \psi&= i(\alpha \cdot \nu )(\mathcal {T}_+\psi _+-\mathcal {T}_-\psi _-) :{{\,\textrm{Dom}\,}}T \rightarrow L^2(\Sigma ;\mathbb {C}^N),\\ \Gamma _1 \psi&= \frac{1}{2}(\mathcal {T}_+\psi _+ +\mathcal {T}_-\psi _-):{{\,\textrm{Dom}\,}}T \rightarrow L^2(\Sigma ;\mathbb {C}^N), \end{aligned} \end{aligned}$$

(12)

is a generalized boundary triple for $S^*$. Here, $\alpha \cdot \nu :=\sum _{k=1}^n \nu _k\alpha _k$. In the same paper, it was shown that, for $z\in \rho (D_0)$, the associated $\gamma$-field and the Weyl function are given by

$$\begin{aligned} \gamma (z)\psi (x) = \int _\Sigma R_{z}(x-y_\Sigma )\psi (y_\Sigma )\,\textrm{d}\sigma (y_\Sigma )\quad (\forall x\in \mathbb {R}^n\setminus \Sigma ) \end{aligned}$$

and

$$\begin{aligned} M(z)\psi (x_\Sigma ) = \lim _{\rho \rightarrow 0+}\int _{\Sigma \setminus B(x_\Sigma ,\rho )}R_{z}(x_\Sigma -y_\Sigma )\psi (y_\Sigma )\,\textrm{d}\sigma (y_\Sigma ) \quad (\forall x_\Sigma \in \Sigma ), \end{aligned}$$

respectively. Moreover, $\gamma (z)$ is a bounded and everywhere defined operator from $L^2(\Sigma ;\mathbb {C}^N)$ to $L^2(\mathbb {R}^n;\mathbb {C}^N)$ with a compact adjoint and M(z) is a bounded and everywhere defined operator in $L^2(\Sigma ;\mathbb {C}^N)$.

Now, our aim will be to identify the Dirac operator perturbed by the formal potential (1) with a certain operator of the form (9), where T is given by (11) and $\Gamma _0,\,\Gamma _1$ by (12). First, using integration by parts, one gets

$$\begin{aligned} \mathcal {D}_0(\psi _-\oplus \psi _+)=T(\psi _-\oplus \psi _+)+i(\alpha \cdot \nu )(\mathcal {T}_+ \psi _+ -\mathcal {T}_- \psi _-)\delta _\Sigma \quad (\forall \psi \equiv \psi _-\oplus \psi _+\in {{\,\textrm{Dom}\,}}(T)). \end{aligned}$$

Next, note that the right-hand side of (2) makes sense also for $\varphi \in L^2(\Sigma ;\mathbb {C}^N)$. Therefore, it is natural to extend the action of (1) on $\psi \in {{\,\textrm{Dom}\,}}(T)$ as follows

$$\begin{aligned} |F\delta _\Sigma \rangle \langle G\delta _\Sigma |\psi :=F\int _\Sigma \Big (G^*\frac{1}{2}(\mathcal {T}_+ \psi _+ +\mathcal {T}_- \psi _-)\Big )\,\delta _\Sigma . \end{aligned}$$

We see that, for $\psi \in {{\,\textrm{Dom}\,}}(T)$, $(\mathcal {D}_0+|F\delta _\Sigma \rangle \langle G\delta _\Sigma |)\psi$ belongs to $L^2(\mathbb {R}^n;\mathbb {C}^N)$ if and only if

$$\begin{aligned} i(\alpha \cdot \nu )(\mathcal {T}_+ \psi _+ -\mathcal {T}_- \psi _-)+F\int _\Sigma \Big (G^* \frac{1}{2}(\mathcal {T}_+ \psi _+ +\mathcal {T}_- \psi _-)\Big )=0 \end{aligned}$$

(13)

as an element of $L^2(\Sigma ;\mathbb {C}^N)$. This leads us to the following definition.

Definition 3.0.1

By the Dirac operator with non-local $\delta$-shell interaction of the type $|F\delta _\Sigma \rangle \langle G\delta _\Sigma |$ we mean the linear operator $D_{F,G}$ in $L^2(\mathbb {R}^n;\mathbb {C}^N)$ given by

$$\begin{aligned} \begin{aligned} {{\,\textrm{Dom}\,}}{D_{F,G}}&=\{\psi _-\oplus \psi _+\in H^{\frac{1}{2}}_\alpha (\Omega _-)\oplus H^{\frac{1}{2}}_\alpha (\Omega _+)\mid \, \psi \text { satisfies }(13)\},\\ D_{F,G}(\psi _-\oplus \psi _+)&=\mathcal {D}_0\psi _-\oplus \mathcal {D}_0\psi _+. \end{aligned} \end{aligned}$$

The transmission condition (13) may be rewritten as

$$\begin{aligned} \Gamma _0\psi +B\Gamma _1\psi =0\quad \text {with}\quad B=|F\rangle _\Sigma \langle G|_\Sigma , \end{aligned}$$

(14)

where $|F\rangle _\Sigma \langle G|_\Sigma$ defined by

$$\varphi \mapsto F\int _\Sigma (G^*\varphi )$$

is a finite-rank operator in $L^2(\Sigma ;\mathbb {C}^N)$. Note that

$$\begin{aligned} (|F\rangle _\Sigma \langle G|_\Sigma )^*=|G\rangle _\Sigma \langle F|_\Sigma . \end{aligned}$$

(15)

With B given in (14), $D_{F,G}=T_B$. In particular, $D_{0,0}=T_0=D_0$ is the free Dirac operator. Hence, we may use Theorem 2.4.1 to study $D_{F,G}$ efficiently. We start with a result on self-adjointness.

Theorem 3.0.1

Let $F,\,G\in L^2(\Sigma ;\mathbb {C}^{N\times N})$ be such that

$$\begin{aligned} |F\rangle _\Sigma \langle G|_\Sigma =|G\rangle _\Sigma \langle F|_\Sigma . \end{aligned}$$

(16)

Then $D_{F,G}$ is a self-adjoint operator.

Proof

In view of (15), the condition (16) is equivalent to the hermiticity of $B=|F\rangle _\Sigma \langle G|_\Sigma$. The property (i) from Definition 2.4.1 together with (14) then imply that the operator $D_{F,G}$ is symmetric. Hence, to prove the self-adjointness of $D_{F,G}$ it is sufficent to show that $\forall z\in \mathbb {C}{\setminus }\mathbb {R},\,{{\,\textrm{Ran}\,}}(D_{F,G} - z) = L^2(\mathbb {R}^n;\mathbb {C}^N)$. By the symmetry of $D_{F,G}$, we also have $\sigma _{\mathrm p}(D_{F,G})\subset \mathbb {R}$. Therefore, from the point (i) of Theorem 2.4.1, the operator $(I+B M(z))$ is injective for all $z\in \mathbb {C}\setminus \mathbb {R}$. Furthermore, B is a finite rank operator and thus compact. In addition, M(z) is bounded, and so we deduce that the operator BM(z) is also compact in $L^2(\Sigma ;\mathbb {C}^N)$. On top of that, I is Fredholm operator with index 0 and the same holds true for its compact perturbation $(I+B M(z))$ which implies that the operator $(I+B M(z))$ is also surjective. This yields ${{\,\textrm{Ran}\,}}(D_{F,G}-z) = L^2(\mathbb {R}^n;\mathbb {C}^N)$, due to the point (ii) of Theorem 2.4.1. $\square$

Remark 3.0.1

In the proof of Theorem 3.0.1, only hermiticity and compactness of $|F\rangle _\Sigma \langle G|_\Sigma$ played role. Therefore, one can introduce a wider class of self-adjoint Dirac operators with generally non-local $\delta$-shell potentials described by the transmission condition

$$i(\alpha \cdot \nu )(\mathcal {T}_+ \psi _+ -\mathcal {T}_- \psi _-)+\frac{1}{2}B (\mathcal {T}_+ \psi _+ +\mathcal {T}_- \psi _-)=0,$$

where B is compact and hermitian in $L^2(\Sigma ;\mathbb {C}^N)$. In particular, (1) may be extended to finite sums $\sum _{i=1}|F_i\delta _\Sigma \rangle \langle G_i\delta _\Sigma |$. With this choice, it is straightforward to generalize the results of Sect. 4 on regular approximations.

Remark 3.0.2

The condition (16) is clearly equivalent to

$$\begin{aligned} \Big \langle \int _\Sigma F^*\varphi ,\int _\Sigma G^*\tilde{\varphi }\Big \rangle _{\mathbb {C}^N}=\Big \langle \int _\Sigma G^*\varphi ,\int _\Sigma F^*\tilde{\varphi }\Big \rangle _{\mathbb {C}^N}\quad (\forall \varphi ,\,\tilde{\varphi }\in L^2(\Sigma ;\mathbb {C}^N)) \end{aligned}$$

(17)

that is, in turn, equivalent to the formal symmetry of (1). To see the latter, we note that

$$\begin{aligned} \langle (|F\delta _\Sigma \rangle \langle G\delta _\Sigma |\psi ),\tilde{\psi }\rangle\equiv & {} \Big (\overline{F\int _\Sigma (G^*\psi )}\delta _\Sigma ,\tilde{\psi }\Big )=\int _\Sigma \Big \langle F\int _\Sigma (G^*\psi ),\tilde{\psi }\Big \rangle _{\mathbb {C}^N}\\= & {} \Big \langle \int _\Sigma G^*\psi ,\int _\Sigma F^*\tilde{\psi }\Big \rangle _{\mathbb {C}^N} \end{aligned}$$

and, on the other hand,

$$\begin{aligned} \langle \psi ,(|F\delta _\Sigma \rangle \langle G\delta _\Sigma |\tilde{\psi })\rangle\equiv & {} \overline{\langle (|F\delta _\Sigma \rangle \langle G\delta _\Sigma |\tilde{\psi }),\psi \rangle }=\overline{\Big \langle \int _\Sigma G^*\tilde{\psi },\int _\Sigma F^*\psi \Big \rangle }_{\mathbb {C}^N}\\= & {} \Big \langle \int _\Sigma F^*\psi ,\int _\Sigma G^*\tilde{\psi }\Big \rangle _{\mathbb {C}^N}. \end{aligned}$$

Above, we may consider, e.g. $\psi ,\,\tilde{\psi }\in H^1(\mathbb {R}^n;\mathbb {C}^N)$. In that case, the values of $\psi ,\,\tilde{\psi }$ in the integrals over $\Sigma$ should be understood in the sense of traces. The claim then follows from the facts that the trace mapping maps $H^1(\mathbb {R}^n;\mathbb {C}^N)$ onto $H^{\frac{1}{2}}(\Sigma ;\mathbb {C}^N)$ surjectively and that the latter space is dense in $L^2(\Sigma ;\mathbb {C}^N)$.

Example 3.0.1

If the columns of F are linearly independent in $L^2(\Sigma ;\mathbb {C}^N)$ then the linear mapping $\langle F|_\Sigma :\, L^2(\Sigma ;\mathbb {C}^N)\rightarrow \mathbb {C}^N$ defined by $\langle F|_\Sigma \varphi :=\int _\Sigma (F^*\varphi )$ is surjective. Therefore, (16) holds true if and only if there exists a constant hermitian $N\times N$ matrix L such that $G=F L$. We then have $|F\rangle _\Sigma \langle G|_\Sigma =|F\rangle _\Sigma \langle FL|_\Sigma =:|F\rangle _\Sigma L \langle F|_\Sigma$.

Now, let us inspect spectral properties of $D_{F,G}$. According to the point (i) of Theorem 2.4.1, $z\in \sigma _{\text {p}}(D_{F,G})\setminus \sigma (D_0)$ if and only if

$$\begin{aligned} (I+|F\rangle _\Sigma \langle G|_\Sigma M(z))\psi =0 \end{aligned}$$

(18)

has a nonzero solution $\psi \in L^2(\Sigma ;\mathbb {C}^N)$. Let $\{f_1,f_2,\ldots ,f_N\}$ be columns of F,

$$\begin{aligned} \mathcal {F}:=\textrm{span}\{f_1,f_2,\ldots ,f_N\}, \end{aligned}$$

and $\{\tilde{f}_1,\tilde{f}_2,\ldots ,\tilde{f}_{\tilde{N}}\}$ a basis of $\mathcal {F}$. Consequently, there exist unique constants $C_{kl}\in \mathbb {C}$ such that

$$\begin{aligned} f_k=\sum _{l=1}^{\tilde{N}} C_{kl}\tilde{f}_l\quad (k=1,2,\ldots ,N). \end{aligned}$$

(19)

Note that $\psi \in \mathcal {F}$ if and only if $(I+|F\rangle _\Sigma \langle G|_\Sigma M(z))\psi \in \mathcal {F}$. In particular, (18) yields that

$$\begin{aligned} \psi =\sum _{l=1}^{\tilde{N}}a_l \tilde{f}_l \end{aligned}$$

for some $a_l\in \mathbb {C}$. Substituting this decomposition back to (18) and using (19), we get

$$\begin{aligned} \sum _{j=1}^{\tilde{N}}\sum _{k=1}^N \Big (\int _\Sigma G^* M(z)\tilde{f}_j\Big )_k C_{kl}\, a_j=-a_l \quad (l=1,2,\ldots , \tilde{N}), \end{aligned}$$

(20)

where the lower index k denotes the kth component of the column vector in the round bracket. Introducing

$$\begin{aligned} \tilde{F}:=\begin{pmatrix} \tilde{f}_1&\tilde{f}_2&\ldots&\tilde{f}_{\tilde{N}}\end{pmatrix}\in L^2(\Sigma ;\mathbb {C}^{N\times \tilde{N}}),\quad C:=(C_{kl})_{k,l=1}^{N,\tilde{N}}\in \mathbb {C}^{N\times \tilde{N}}, \end{aligned}$$

(21)

we see that the existence of a non-trivial solution $(a_1,a_2,\ldots , a_{\tilde{N}})^T$ to (20) is equivalent to the condition

$$\begin{aligned} \det \Big (I_{\tilde{N}}+C^T\int _\Sigma (G^*M(z)\tilde{F})\Big )=0. \end{aligned}$$

(22)

These considerations prove partially the following theorem.

Theorem 3.0.2

Let $F,\,G\in L^2(\Sigma ;\mathbb {C}^{N\times N})$ be such that (16) holds true. Then $\sigma _{\textrm{ess}}(D_{F,G})=(-\infty ,-|m|]\cup [|m|,+\infty )$ and the number of discrete eigenvalues of $D_{F,G}$ counting multiplicities is at most equal to the number of the linearly independent (in $L^2(\Sigma ;\mathbb {C}^N)$) columns of F. Furthermore, for $z\in (-|m|,|m|)$, it holds

$$\begin{aligned} z\in \sigma _{\textrm{p}}(D_{F,G}) \quad \text {if and only if} \quad -1\in \sigma \Big (C^T\int _\Sigma (G^*M(z)\tilde{F})\Big ), \end{aligned}$$

(23)

where $\tilde{F}$ and C are given in (21).

Proof

Firstly, the condition (23) is equivalent to (22) derived above. Next, in the proof of Theorem 3.0.1 it was shown that for $z\in \mathbb {C}\setminus \mathbb {R}$ (in fact, for all $z\in \rho (D_0)\setminus \sigma _{\textrm{p}}(D_{F,G})$), ${{\,\textrm{Ran}\,}}(D_{F,G}-z)=L^2(\mathbb {R}^n;\mathbb {C}^N)$. Therefore, applying (10) with $T_0=D_0,\, T_B=D_{F,G},\,$ and $B=|F\rangle _\Sigma \langle G|_\Sigma$ we see that the difference $(D_{F,G}-z)^{-1}-(D_0-z)^{-1}$ is a finite-rank operator, because B projects on the $\tilde{N}$-dimensional space $\mathcal {F}$ spanned by the columns of F, $(I+BM(z))$ maps $\mathcal {F}$ onto $\mathcal {F}$ bijectively, and $\gamma (z)$ is bounded from $L^2(\Sigma ;\mathbb {C}^N)$ to $L^2(\mathbb {R}^n;\mathbb {C}^N)$. More concretely, the rank of the difference cannot be larger than $\tilde{N}$. The claim about the essential spectrum then follows from the Weyl criterion and the fact that $\sigma _{\textrm{ess}}(D_0)=(-\infty ,-|m|]\cup [|m|,+\infty )$. The bound on the number of discrete eigenvalues is a consequence of [12, Chpt. 9.3, Theo. 3]. $\square$

Remark 3.0.3

It can be easily seen that for F with linearly independent columns we may choose $\tilde{F}=F$ and $C = I_N$. Then, in view of Example 3.0.1, the matrix G can be written as $G=FL$, where L is a constant hermitian matrix. Consequently, the spectral condition (23) reduces to

$$z\in \sigma _{\textrm{p}}(D_{F,G}) \quad \text {if and only if} \quad -1\in \sigma \Big (L\int _\Sigma (F^*M(z) F)\Big ).$$

4 Non-local approximations

In this section, we will show that the operator $D_{F,G}$ may be understood as a limit of the free operator $D_0$ with a scaled finite-rank perturbation. Since we will use the results of Sect. 2.3 to construct the perturbation, let us assume from now on that $\Omega$ has $C^2$-smooth boundary $\Sigma$. First, note that the tubular $\varepsilon$-neighbourhood of $\Sigma$ introduced in (6) obeys

$$\begin{aligned} \Sigma _\varepsilon = \{x_\Sigma + t \nu (x_\Sigma ) \mid x_\Sigma \in \Sigma , \, t \in (-\varepsilon ,\varepsilon ) \}. \end{aligned}$$

(24)

Here, $t=\varepsilon u$ measures the distance of a point $x\in \Sigma _\varepsilon$ from $\Sigma$. Since, for all $\varepsilon$ small enough, the representation of $x\in \Sigma _\varepsilon$ given by the right-hand side of (24) is unique, we will identify x with the pair $(x_\Sigma ,t)$. Next, let $v \in L^{\infty }(\mathbb {R};\mathbb {R})$ be such that ${{\,\textrm{supp}\,}}v \subset (-1,1)$ and $\int _{-1}^{1} v(t)\, \,\textrm{d}t = 1$. For $\varepsilon >0$, we put $v_\varepsilon (t):=\varepsilon ^{-1}v(\varepsilon ^{-1} t)$ and

$$\begin{aligned} F_{\varepsilon }(x) := {\left\{ \begin{array}{ll} F(x_\Sigma )v_\varepsilon (t) \quad &{}\text { for }x\equiv (x_\Sigma ,t)\in \Sigma _\varepsilon \\ 0 \quad &{}\text { away from } \Sigma _{\varepsilon }. \end{array}\right. } \end{aligned}$$

A matrix-valued function $G_\varepsilon$ is introduced similarly. It is straightforward to check that, in the sense of distributions, $\lim _{\varepsilon \rightarrow 0+}F_\varepsilon =F\delta _\Sigma$. Therefore, the natural candidate for approximating potential is the following finite-rank operator in $L^2(\mathbb {R}^n;\mathbb {C}^N)$,

$$\begin{aligned} |F_\varepsilon \rangle \langle G_\varepsilon |\psi :=F_\varepsilon \int _{\mathbb {R}^n} (G_\varepsilon ^*\psi )=F_\varepsilon \int _{\Sigma _\varepsilon } (G_\varepsilon ^*\psi ). \end{aligned}$$

Our aim will be to show that

$$D_{F,G}^{\varepsilon }:=D_0+|F_\varepsilon \rangle \langle G_\varepsilon |$$

converges in the norm resolvent sense to $D_{F,G}$ as $\varepsilon \rightarrow 0+$. Note that if (16) is satisfied, then $|F_\varepsilon \rangle \langle G_\varepsilon |$ is hermitian, and hence $D_{F,G}^{\varepsilon }$ is self-adjoint on ${{\,\textrm{Dom}\,}}(D_{F,G}^\varepsilon )={{\,\textrm{Dom}\,}}(D_0)$. For the resolvent of $D_{F,G}^\varepsilon$, we have

$$\begin{aligned} (D_{F,G}^{\varepsilon }-z)^{-1}=R_z(I+|F_\varepsilon \rangle \langle G_\varepsilon | R_z)^{-1}, \end{aligned}$$

(25)

where $R_z$ should be now understood as a shorthand notation for $(D_0-z)^{-1}$. Of course, (25) is only valid for $z\notin \sigma (D_0)$ such that the inverse on the right-hand side exists. Writing

$$(I+|F_\varepsilon \rangle \langle G_\varepsilon | R_z)^{-1}=(I+|F_\varepsilon \rangle \langle G_\varepsilon | R_z)^{-1}((I+|F_\varepsilon \rangle \langle G_\varepsilon | R_z)-|F_\varepsilon \rangle \langle G_\varepsilon | R_z)$$

we get

$$\begin{aligned} (D_{F,G}^{\varepsilon }-z)^{-1}=R_z-R_z(I+|F_\varepsilon \rangle \langle G_\varepsilon | R_z)^{-1}|F_\varepsilon \rangle \langle G_\varepsilon | R_z \end{aligned}$$

(26)

For further calculations we will abandon the bra-ket notation and introduce the operator $\Pi :\, L^2(\mathbb {R}^n;\mathbb {C}^N)\rightarrow \mathbb {C}^N, \, \psi \mapsto \int _{\mathbb {R}^n}\psi$ instead; so, in particular, we have $|F_\varepsilon \rangle \langle G_\varepsilon |=F_\varepsilon \Pi G_\varepsilon ^*$, where the matrix-valued functions $F_\varepsilon$ and $G_\varepsilon ^*$ are identified with multiplication operators from $\mathbb {C}^N$ to $L^2(\mathbb {R}^n;\mathbb {C}^N)$ and in $L^2(\mathbb {R}^n;\mathbb {C}^N)$, respectively. We will adopt an analogous convention for the matrix valued functions defined along $\Sigma$. Furthermore, we define the operator $\Pi _\Sigma :\, L^2(\Sigma ;\mathbb {C}^N)\rightarrow \mathbb {C}^N$ by $\Pi _\Sigma \psi :=\int _{\Sigma }\psi$.

Now, recall (21) and put $\tilde{F}_\varepsilon (x):=(\tilde{f}_1(x_\Sigma ) v_\varepsilon (t)\,\, \tilde{f}_2(x_\Sigma ) v_\varepsilon (t)\,\, \ldots \,\, \tilde{f}_{\tilde{N}}(x_\Sigma ) v_\varepsilon (t))$ for $x\equiv (x_\Sigma ,t)\in \Sigma _\varepsilon$. Away from $\Sigma _\varepsilon$, we extend $\tilde{f}_i v_\varepsilon$ and $\tilde{F}_\varepsilon$ by zero. Then we have

$$\begin{aligned} F_\varepsilon =\tilde{F}_\varepsilon C^T. \end{aligned}$$

(27)

Also define $\mathcal {F}_\varepsilon :=\textrm{span}\{\tilde{f}_1 v_\varepsilon ,\, \tilde{f}_2 v_\varepsilon ,\, \ldots ,\, \tilde{f}_{\tilde{N}} v_\varepsilon \}$ and the mapping $P_\varepsilon : \tilde{f}_i v_\varepsilon \mapsto \tilde{f}_i$, which extends by linearity to an isomorphism from $\mathcal {F}_\varepsilon$ onto $\mathcal {F}$. In the following, assume that the basis $\{\tilde{f}_i\}_{i=1}^{\tilde{N}}$ of $\mathcal {F}$ is orthonormal in $L^2(\Sigma ;\mathbb {C}^N)$. With that choice, the mapping

$$\mathcal {P}:\, \mathcal {F}\rightarrow \mathbb {C}^{\tilde{N}},\quad \psi \mapsto (\langle \tilde{f}_1,\psi \rangle _\Sigma ,\, \langle \tilde{f}_2,\psi \rangle _\Sigma ,\, \ldots ,\, \langle \tilde{f}_{\tilde{N}},\psi \rangle _\Sigma )^T$$

obeys

$$\begin{aligned} \tilde{F} \mathcal {P}=I_{\mathcal {F}},\quad \mathcal {P}\tilde{F}=I_{\mathbb {C}^{\tilde{N}}},\quad \tilde{F}_\varepsilon \mathcal {P}P_\varepsilon =I_{\mathcal {F}_\varepsilon },\quad \mathcal {P}P_\varepsilon \tilde{F}_\varepsilon =I_{\mathbb {C}^{\tilde{N}}}. \end{aligned}$$

(28)

Using the latter two equalities and (27) we obtain

$$\begin{aligned} (I_{\mathcal {F}_\varepsilon }+F_\varepsilon \Pi G_\varepsilon ^* R_z)^{-1}F_\varepsilon= & {} \tilde{F}_\varepsilon \mathcal {P}P_\varepsilon (\tilde{F}_\varepsilon (I_{\mathbb {C}^{\tilde{N}}}+ C^T\Pi G_\varepsilon ^* R_z\tilde{F}_\varepsilon )\mathcal {P}P_\varepsilon )^{-1}\tilde{F}_\varepsilon C^T \nonumber \\= & {} \tilde{F}_\varepsilon (I_{\mathbb {C}^{\tilde{N}}}+ C^T\Pi G_\varepsilon ^* R_z\tilde{F}_\varepsilon )^{-1} C^T \end{aligned}$$

(29)

on $\mathbb {C}^N$. Since $(I+|F_\varepsilon \rangle \langle G_\varepsilon |R_z)$ maps into $\mathcal {F}_\varepsilon$ exactly those vectors that belong to $\mathcal {F}_\varepsilon$, we may combine (26) and (29) to get

$$\begin{aligned} (D_{F,G}^{\varepsilon }-z)^{-1}=R_z-R_z\tilde{F}_\varepsilon (I_{\mathbb {C}^{\tilde{N}}}+ C^T\Pi G_\varepsilon ^* R_z\tilde{F}_\varepsilon )^{-1} C^T\Pi G_\varepsilon ^* R_z. \end{aligned}$$

(30)

The formula holds for all $z\notin \sigma (D_0)$ such that the $\tilde{N}\times \tilde{N}$-matrix $I_{\mathbb {C}^{\tilde{N}}}+ C^T\Pi G_\varepsilon ^* R_z\tilde{F}_\varepsilon$ is invertible. Note that this implies that $z\in (-|m|,|m|)$ belongs to the discrete spectrum of $D_{F,G}^\varepsilon$ if and only if

$$\begin{aligned} \det (I_{\mathbb {C}^{\tilde{N}}}+ C^T\Pi G_\varepsilon ^* R_z\tilde{F}_\varepsilon )=0. \end{aligned}$$

(31)

To find the resolvent of $D_{F,G}$, we use (10) with $B=F\Pi _\Sigma G^*$ which yields

$$\begin{aligned} (D_{F,G}-z)^{-1}=R_z-\gamma (z)(I+F\Pi _\Sigma G^* M(z))^{-1}F\Pi _\Sigma G^*\gamma (\bar{z})^*. \end{aligned}$$

With the help of the first two equalities in (28) together with $F=\tilde{F} C^T$ we may use similar manipulations as in (29) to rewrite this as follows:

$$\begin{aligned} (D_{F,G}-z)^{-1}=R_z-\gamma (z)\tilde{F}(I_{\mathbb {C}^{\tilde{N}}}+C^T\Pi _\Sigma G^* M(z)\tilde{F})^{-1}C^T\Pi _\Sigma G^*\gamma (\bar{z})^*. \end{aligned}$$

(32)

We are now prepared to state and prove the main result of this section.

Theorem 4.0.1

Let $\Omega$ have $C^2$-smooth boundary and $F,G\in L^2(\Sigma ;\mathbb {C}^{N\times N})$ be such that (16) holds true. Then for every $z\notin \sigma (D_{F,G})$ there exists $\varepsilon _z>0$ such that for all $\varepsilon \in (0,\varepsilon _z)$, $z\notin \sigma (D_{F,G}^\varepsilon )$ and

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0+}\Vert (D_{F,G}-z)^{-1}-(D_{F,G}^\varepsilon -z)^{-1}\Vert =0. \end{aligned}$$

Proof

Using (30), (32), and the joint continuity of the operator composition we get

$$\begin{aligned} \Vert (D_{F,G}-z)^{-1}-(D_{F,G}^\varepsilon -z)^{-1}\Vert \le const.\,( C_1(\varepsilon )+C_2(\varepsilon )+C_3(\varepsilon )), \end{aligned}$$

whenever the functions

$$\begin{aligned}&C_1(\varepsilon ):=\Vert \gamma (z)\tilde{F}-R_z\tilde{F}_\varepsilon \Vert _{\mathbb {C}^{\tilde{N}}\rightarrow L^2(\mathbb {R}^n;\mathbb {C}^N)},\\&C_2(\varepsilon ):=\Vert (I_{\mathbb {C}^{\tilde{N}}}+C^T\Pi _\Sigma G^* M(z)\tilde{F})^{-1}-(I_{\mathbb {C}^{\tilde{N}}}+ C^T\Pi G_\varepsilon ^* R_z\tilde{F}_\varepsilon )^{-1}\Vert _{\mathbb {C}^{\tilde{N}}\rightarrow \mathbb {C}^{\tilde{N}}},\\&C_3(\varepsilon ):=\Vert C^T(\Pi _\Sigma G^*\gamma (\bar{z})^*-\Pi G_\varepsilon ^* R_z)\Vert _{L^2(\mathbb {R}^n;\mathbb {C}^N)\rightarrow \mathbb {C}^{\tilde{N}}} \end{aligned}$$

are bounded on a right neighbourhood of $\varepsilon =0$. We are going to show that, for $i\in \{1,2,3\}$, $\lim _{\varepsilon \rightarrow 0+}C_i(\varepsilon )=0$. Note that it is sufficient for $C_2(\varepsilon )$ and $C_3(\varepsilon )$ to converge to zero as $\varepsilon \rightarrow 0+$ that

$$\begin{aligned} \tilde{C}_2(\varepsilon ):=\Vert \Pi _\Sigma G^* M(z)\tilde{F}-\Pi G_\varepsilon ^* R_z\tilde{F}_\varepsilon \Vert _{\mathbb {C}^{\tilde{N}}\rightarrow \mathbb {C}^N} \end{aligned}$$

and

$$\begin{aligned} \tilde{C}_3(\varepsilon ):=\Vert \Pi _\Sigma G^*\gamma (\bar{z})^*-\Pi G_\varepsilon ^* R_z\Vert _{L^2(\mathbb {R}^n;\mathbb {C}^N)\rightarrow \mathbb {C}^N}, \end{aligned}$$

tend to zero as $\varepsilon \rightarrow 0+$, respectively. Moreover, if $\lim _{\varepsilon \rightarrow 0+}\tilde{C}_2(\varepsilon )=0$ then the spectral conditions (31) and (23) together with the continuity of the determinant with respect to any matrix norm yield the first statement of the theorem.

First, we will investigate the term $C_1(\varepsilon )$. Since $\int _{-1}^1 v=1$, we have

$$\begin{aligned} (\gamma (z)\tilde{F})(x)=\int _{-1}^{1}\int _\Sigma R_z(x-y_\Sigma )\tilde{F}(y_\Sigma )v(u)\,\textrm{d}\sigma (y_\Sigma )\,\textrm{d}u \quad (\forall x\in \mathbb {R}^n\setminus \Sigma ). \end{aligned}$$

Using ${{\,\textrm{supp}\,}}(\tilde{F}_\varepsilon )\subset \overline{\Sigma _\varepsilon }$, the parallel coordinates $(x_\Sigma ,u)$ introduced in Sect. 2.3, and $t=\varepsilon u$, we get

$$\begin{aligned} (R_z\tilde{F}_\varepsilon )(x)=\int _{-1}^1\int _\Sigma R_z(x-y_\Sigma -\varepsilon u \nu (y_\Sigma ))\tilde{F}(y_\Sigma )v(u) w_\varepsilon (y_\Sigma ,u)\,\textrm{d}\sigma (y_\Sigma )\,\textrm{d}u \end{aligned}$$

for a.e. $x\in \mathbb {R}^n$. Given $a\in \mathbb {C}^{\tilde{N}}$, we may estimate as follows

$$\begin{aligned} \Vert (\gamma (z)\tilde{F}-R_z\tilde{F}_\varepsilon )a\Vert _{L^2(\mathbb {R}^n;\mathbb {C}^N)}&=\Vert (A_0-A_\varepsilon )\left( \sum _{i=1}^{\tilde{N}}a_i\tilde{f}_i v\right) \Vert _{L^2(\mathbb {R}^n;\mathbb {C}^N)}\nonumber \\&\le \Vert (A_0-A_\varepsilon )\Vert _{\mathscr {H}\rightarrow L^2(\mathbb {R}^n;\mathbb {C}^N)}\tilde{N}\max _{i}\Vert \tilde{f}_i v\Vert _\mathscr {H}\max _{j}|a_j|, \end{aligned}$$

(33)

where $\mathscr {H}:=L^2(\Sigma \times (-1,1),\,\textrm{d}\sigma \,\textrm{d}u;\mathbb {C}^N)$ and, for $\psi \in \mathscr {H}$, the operators $A_0$ and $A_\varepsilon$ are defined by

$$\begin{aligned}&(A_0\psi )(x):=\int _{-1}^{1}\int _\Sigma R_z(x-y_\Sigma )\psi (y_\Sigma ,u)\,\textrm{d}\sigma (y_\Sigma )\,\textrm{d}u, \\&(A_\varepsilon \psi )(x):=\int _{-1}^1\int _\Sigma R_z(x-y_\Sigma -\varepsilon u \nu (y_\Sigma ))\psi (y_\Sigma ,u) w_\varepsilon (y_\Sigma ,u)\,\textrm{d}\sigma (y_\Sigma )\,\textrm{d}u. \end{aligned}$$

It was proved in [6, Prop. 3.8] that $\lim _{\varepsilon \rightarrow 0+}\Vert (A_0-A_\varepsilon )\Vert _{\mathscr {H}\rightarrow L^2(\mathbb {R}^n;\mathbb {C}^N)}=0$. In view of (33), we conclude that $\lim _{\varepsilon \rightarrow 0+}C_1(\varepsilon )=0$.

Next, we will look at the term $\tilde{C}_3(\varepsilon )$. Since, for all $\psi \in L^2(\mathbb {R}^n;\mathbb {C}^N)$ and a.e. $x_\Sigma \in \Sigma$,

$$\begin{aligned} (\gamma (\bar{z})^*\psi )(x_\Sigma )=\int _{\mathbb {R}^n}R_z(x_\Sigma -y)\psi (y)\,\textrm{d}y, \end{aligned}$$

and $\int _{-1}^{1}v=1$, we deduce that

$$\begin{aligned} \Pi _\Sigma G^*\gamma (\bar{z})^*\psi =\int _{-1}^1\int _{\Sigma }G^*(x_\Sigma )v(u)\int _{\mathbb {R}^n}R_z(x_\Sigma -y)\psi (y)\,\textrm{d}y\,\textrm{d}\sigma (x_\Sigma )\,\textrm{d}u. \end{aligned}$$

Furthermore, using a similar reasoning as in the previous paragraph, we obtain

$$\begin{aligned}&\Pi G_\varepsilon ^* R_z\psi \\&\quad =\int _{-1}^1\int _{\Sigma }G^*(x_\Sigma )v(u)\int _{\mathbb {R}^n}R_z(x_\Sigma +\varepsilon u \nu (x_\Sigma )-y)\psi (y)\,\textrm{d}y \,w_\varepsilon (x_\Sigma ,u)\,\textrm{d}\sigma (x_\Sigma )\,\textrm{d}u. \end{aligned}$$

If we introduce bounded operators $C_0,C_\varepsilon :\, L^2(\mathbb {R}^n;\mathbb {C}^N)\rightarrow \mathscr {H}$ as follows

$$\begin{aligned}&(C_0\psi )(x_\Sigma ,u):=\int _{\mathbb {R}^n}R_z(x_\Sigma -y)\psi (y)\,\textrm{d}y,\\&(C_\varepsilon \psi )(x_\Sigma ,u):=\int _{\mathbb {R}^n}R_z(x_\Sigma +\varepsilon u \nu (x_\Sigma )-y)\psi (y)\,\textrm{d}y, \end{aligned}$$

then for the ith component of $(\Pi _\Sigma G^*\gamma (\bar{z})^*-\Pi G_\varepsilon ^* R_z)\psi$ we get

$$\begin{aligned} \hspace{-10.0pt}\left( (\Pi _\Sigma G^*\gamma (\bar{z})^*-\Pi G_\varepsilon ^* R_z)\psi \right) _i= & {} \int _{-1}^1\int _\Sigma \bar{g_i}^T(x_\Sigma )v(u)(C_0\psi -w_\varepsilon C_\varepsilon \psi )(x_\Sigma ,u)\,\textrm{d}\sigma (x_\Sigma )\,\textrm{d}t\\= & {} \int _{-1}^1\int _\Sigma \bar{g_i}^T(x_\Sigma )v(u)w_\varepsilon (x_\Sigma ,u)(C_0\psi -C_\varepsilon \psi )(x_\Sigma ,u)\,\textrm{d}\sigma (x_\Sigma )\,\textrm{d}t\\{} & {} \quad +\int _{-1}^1\int _\Sigma (1-w_\varepsilon (x_\Sigma ,u))\bar{g_i}^T(x_\Sigma )v(u)C_0\psi (x_\Sigma ,u)\,\textrm{d}\sigma ( x_\Sigma )\,\textrm{d}t, \end{aligned}$$

where $g_i$ stands for the ith column of the matrix G. Applying the Cauchy-Schwarz inequality together with (8), we deduce that

$$\begin{aligned}{} & {} |\left( (\Pi _\Sigma G^*\gamma (\bar{z})^*-\Pi G_\varepsilon ^* R_z)\psi \right) _i|\\{} & {} \quad \le (1+\mathcal {O}(\varepsilon ))\Vert g_i v\Vert _{\mathscr {H}}\Vert C_0-C_\varepsilon \Vert _{ L^2(\mathbb {R}^n;\mathbb {C}^N)\rightarrow \mathscr {H}}\Vert \psi \Vert _{L^2(\mathbb {R}^n;\mathbb {C}^N)}\\{} & {} \qquad \quad +\mathcal {O}(\varepsilon )\Vert g_i v\Vert _{\mathscr {H}}\Vert C_0\Vert _{ L^2(\mathbb {R}^n;\mathbb {C}^N)\rightarrow \mathscr {H}}\Vert \psi \Vert _{L^2(\mathbb {R}^n;\mathbb {N})}. \end{aligned}$$

By [6, Prop. 3.7], $\lim _{\varepsilon \rightarrow 0+}\Vert C_0-C_\varepsilon \Vert _{ L^2(\mathbb {R}^n;\mathbb {C}^N)\rightarrow \mathscr {H}}=0$. Consequently, $\tilde{C}_3(\varepsilon )$ also converges to zero as $\varepsilon \rightarrow 0+$.

Finally, we will be concerned with the term $\tilde{C}_2(\varepsilon )$. For the (i, j)th element of the matrix $\Pi G_\varepsilon ^* R_z\tilde{F}_\varepsilon$ we find that

$$\begin{aligned} (\Pi G_\varepsilon ^* R_z\tilde{F}_\varepsilon )_{ij}=\int _{-1}^1\int _\Sigma \bar{g_i}^T(x_\Sigma )v(u) (B_\varepsilon (\tilde{f}_j v))(x_\Sigma ,u)w_\varepsilon (x_\Sigma ,u)\,\textrm{d}\sigma (x_\Sigma )\,\textrm{d}u, \end{aligned}$$

where

$$\begin{aligned} (B_\varepsilon \psi )(x_\Sigma ,u):=\int _{-1}^1\int _\Sigma R_z(x_\Sigma +\varepsilon u\nu (x_\Sigma )-y_\Sigma -\varepsilon s\nu (y_\Sigma ))\psi (y_\Sigma ,s)w_\varepsilon (y_\Sigma ,s)\,\textrm{d}\sigma (y_\Sigma )\,\textrm{d}s \end{aligned}$$

is a bounded operator in $\mathscr {H}$. It follows from [6, Prop. 3.10] that $B_\varepsilon$ converges to $B_0$ in the space of bounded operators from $\mathscr {H}_{1/2}:=L^2((-1,1);H^{1/2}(\Sigma ;\mathbb {C}^N))$ to $\mathscr {H}$, where $B_0$ defined by

$$\begin{aligned} (B_0\psi )(\cdot , u):=\frac{i}{2}(\alpha \cdot \nu )\int _{-1}^1\textrm{sgn}(u-s)\psi (\cdot ,s)\,\textrm{d}s+M(z)\int _{-1}^1\psi (\cdot ,s)\,\textrm{d}s \end{aligned}$$

is a bounded operator in $\mathscr {H}$. By density, for every $\psi \in \mathscr {H}$ we find a sequence $(\psi _n)\subset \mathscr {H}_{1/2}$ that converges to $\psi$ in $\mathscr {H}$. Therefore, we have

$$\begin{aligned}&\Vert (B_\varepsilon -B_0)\psi \Vert _{\mathscr {H}}\le \Vert (B_\varepsilon -B_0)\psi _n\Vert _{\mathscr {H}}+\Vert (B_\varepsilon -B_0)(\psi -\psi _n)\Vert _{\mathscr {H}}\\&\quad \le \Vert B_\varepsilon -B_0\Vert _{\mathscr {H}_{1/2}\rightarrow \mathscr {H}}\Vert \psi _n\Vert _{\mathscr {H}_{1/2}}+(\Vert B_\varepsilon \Vert _{\mathscr {H}\rightarrow \mathscr {H}}+\Vert B_0\Vert _{\mathscr {H}\rightarrow \mathscr {H}})\Vert \psi -\psi _n\Vert _{\mathscr {H}}. \end{aligned}$$

Since again by [6, Prop. 3.10] the operators $B_\varepsilon$ are uniformly bounded in $\mathscr {H}$, we infer that $B_\varepsilon$ converges to $B_0$ strongly in $\mathscr {H}$ as $\varepsilon \rightarrow 0+$. Furthermore, due to (8), we see that $\lim _{\varepsilon \rightarrow 0+}(g_i v w_\varepsilon )=g_i v$ in $\mathscr {H}$. Using these two results together with the joint continuity of the dot product and the fact that $\int _{-1}^1 v=1$, we arrive at

$$\begin{aligned}&\lim _{\varepsilon \rightarrow 0+}(\Pi G_\varepsilon ^* R_z\tilde{F}_\varepsilon )_{ij}=\int _{-1}^1\int _\Sigma \bar{g_i}^T(x_\Sigma )v(u) (B_0(\tilde{f}_j v))(x_\Sigma ,u)\,\textrm{d}\sigma (x_\Sigma )\,\textrm{d}u\\&\quad =\frac{i}{2}\int _{-1}^1\int _{-1}^1v(u)\textrm{sgn}(u-s)v(s)\,\textrm{d}u\,\textrm{d}s \int _\Sigma \bar{g_i}^T(x_\Sigma )(\alpha \cdot \nu (x_\Sigma ))\tilde{f}_j(x_\Sigma )\,\textrm{d}\sigma (x_\Sigma )\\&\qquad +\int _\Sigma \bar{g_i}^T(x_\Sigma ) (M(z)\tilde{f}_j)(x_\Sigma )\,\textrm{d}\sigma (x_\Sigma ). \end{aligned}$$

The first term on the right-hand side is clearly zero, whereas the second one is just the integral representation of the (i, j)th element of $\Pi _\Sigma G^*M(z)\tilde{F}$. Hence, we conclude that $\lim _{\varepsilon \rightarrow 0+}\tilde{C}_2(\varepsilon )=0$. $\square$

Remark 4.0.1

Essentially the same operators as $A_\varepsilon ,\, B_\varepsilon ,$ and $C_\varepsilon$ from the proof of Theorem 4.0.1 were originally studied in [23] for $n=3$. Although the convergence results obtained there are weaker than the results from the recent preprint [6], they are still strong enough to support our proof. In fact, they may be generalized to the dimension $n=2$ in a rather straightforward way and then used in our proof, too.