Self-adjoint Dirac operators on domains in $\mathbb{R}^3$

In this paper the spectral and scattering properties of a family of self-adjoint Dirac operators in $L^2(\Omega; \mathbb{C}^4)$, where $\Omega \subset \mathbb{R}^3$ is either a bounded or an unbounded domain with a compact $C^2$-smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with Robin boundary conditions. Among the Dirac operators treated here is also the so-called MIT bag operator, which has been used by physicists and more recently was discussed in the mathematical literature. Our approach is based on abstract boundary triple techniques from extension theory of symmetric operators and a thorough study of certain classes of (boundary) integral operators, that appear in a Krein-type resolvent formula. The analysis of the perturbation term in this formula leads to a description of the spectrum and a Birman-Schwinger principle, a qualitative understanding of the scattering properties in the case that $\Omega$ is unbounded, and corresponding trace formulas.


Introduction
In recent years the mathematical study of Dirac operators acting on domains Ω ⊂ R d with special boundary conditions that make them self-adjoint gained a lot of attention. The motivation for this arises from several aspects: from the mathematical point of view they can be seen as the relativistic counterpart of Laplacians with Robin type boundary conditions. From the physical point of view Dirac operators with special boundary conditions are used in relativistic quantum mechanics to describe particles that are confined to a predefined area or box. One important model in 3D (dimension three) is the MIT bag model suggested in the 1970s to study confinement of quarks, see [26,27,28,29,37]. In the 2D (dimension two) case Dirac operators with special boundary conditions similar to the MIT bag model are used in the description of graphene; cf. [21,22,25].
To set the stage, let Ω ⊂ R 3 be either a bounded or unbounded domain with a compact C 2 -smooth boundary and let ν be the unit normal vector field at ∂Ω which points outwards of Ω. Choose units such that the Planck constant and the speed of light are both equal to one. Moreover, assume that ϑ : ∂Ω → R is a Hölder continuous function of order a > 1 2 and consider in L 2 (Ω; C 4 ) the operator where α = (α 1 , α 2 , α 3 ) and β are the C 4×4 Dirac matrices defined in (1.4) below and α · x = α 1 x 1 + α 2 x 2 + α 3 x 3 for x = (x 1 , x 2 , x 3 ) ⊤ ∈ R 3 . The time-dependent equation with the Hamiltonian given by A ϑ models the propagation of a relativistic particle subject to the boundary conditions in dom A ϑ with mass m > 0 contained in Ω. The mathematical literature on such types of Dirac operators contains different approaches. In differential geometry there are several articles dealing with selfadjoint Dirac operators on smooth manifolds, see for instance [6,7,51]. In [53] it was shown that the 2D Dirac operator with so-called zigzag boundary conditions (in the massless case) is self-adjoint and zero is an eigenvalue of infinite multiplicity, see also [34]. More recent related publications in the 2D case are [21,22], where the self-adjointness of Dirac operators in bounded C 2 -domains Ω ⊂ R 2 for a wide class of boundary conditions describing quantum dots was shown. Many considerations in [21,22,53] are based on complex analysis techniques, which are not available in the 3D situation. We also refer to [25,41,42,50] for self-adjointness and spectral problems of 2D Dirac operators on different types of domains with special boundary conditions. In contrast to the 2D setting the 3D case is less investigated in the mathematical literature, only the MIT bag operator is well studied. We emphasize [1,49] for the analysis of general properties of the MIT bag operator in 3D and [2,8,23,47], where it is shown that the MIT bag boundary conditions can be interpreted as infinite mass boundary conditions (i.e. Ω is surrounded by a medium with infinite mass).
The main objective of this paper is to develop a systematic approach to the spectral analysis and scattering theory for self-adjoint Dirac operators in the 3D case. Here we are particularly interested in boundary conditions as in (1.1), since these are the 3D analogue of the 2D boundary conditions in [21] which can be used to describe graphene quantum dots (cf. Remark 5.2), and the corresponding Dirac operators can be viewed as the relativistic counterpart of Laplace operators with Robin boundary conditions. We also note that operators of the form (1.1) appear in the treatment of Dirac operators B η,τ = −iα · ∇ + mβ + (ηI 4 + τ β)δ ∂Ω (1.2) in R 3 with singular δ-shell potentials supported on ∂Ω in the confinement (or decoupling) case. In fact, it is well known that for η 2 − τ 2 = −4 the operator B η,τ can be written as the orthogonal sum of operators acting in L 2 (Ω; C 4 ) and L 2 (R 3 \ Ω; C 4 ), respectively, and it turns out that these operators are exactly of the form (1.1), see Section 5.3 for more details. We refer to the recent contributions [3,4,5,9,11,14,15,36,43,44,45,48,49] for a comprehensive study of Dirac operators with singular δ-shell potentials. Hence, A ϑ can be used to describe a relativistic particle actually living in R 3 , but which is confined to Ω for all time, see [4,Section 5]. This is of interest in particle physics.
The mathematical treatment of the operators A ϑ in (1.1) is based on the application of a suitable so-called quasi boundary triple. Quasi boundary triples and their Weyl functions are an abstract concept from extension and spectral theory for symmetric and self-adjoint operators which were originally introduced to investigate boundary value problems for elliptic partial differential operators in [16], but proved to be useful in many other situations, see, e.g., [10,18,19]. Quasi boundary triples were also applied more recently in [9,14] to Dirac operators with singular potentials as in (1.2). Once a quasi boundary triple and Weyl function in the present situation are available, they allow to deduce in an efficient way the spectral properties of A ϑ from the properties of certain (boundary) integral operators which are induced by the Green's function of the free Dirac operator in R 3 . The more demanding issue here is to establish the proper mapping properties of these integral operators and, in fact, this analysis covers a great part of the present paper. We would like to point out that this approach is independent of the space dimension.
One of the key features in the quasi boundary triple approach is a Krein-type resolvent formula that relates the resolvent of A ϑ via a perturbation term to the resolvent of a reference operator, which is in our model the MIT bag operator T MIT . Using this correspondence we first conclude that A ϑ in (1.1) is self-adjoint in L 2 (Ω; C 4 ) and we show several spectral and scattering properties of A ϑ , which are different for bounded and for unbounded domains Ω. If Ω is an unbounded domain with a compact C 2 -boundary, then the essential spectrum of A ϑ is given by (−∞, −m] ∪ [m, ∞), and there are at most finitely many discrete eigenvalues in the gap (−m, m) that can be characterized by a Birman-Schwinger principle. It also follows that (A ϑ − λ) −3 − (T MIT − λ) −3 is a trace class operator for any λ ∈ C \ R, which implies the existence and completeness of the wave operators for the scattering pair {A ϑ , T MIT }. If Ω is a bounded domain with a compact C 2 -boundary, then the spectrum of A ϑ is purely discrete and all eigenvalues of A ϑ can be characterized by a modified Birman-Schwinger principle; cf. Section 5.1 for more details. The abovementioned properties are proved under the assumption ϑ(x) 2 = 1 for all x ∈ ∂Ω, which we refer to as the non-critical case. We expect that in the critical case ϑ(x) 2 = 1 for some x ∈ ∂Ω the spectral properties of A ϑ may significantly differ from the non-critical case, e.g., essential spectrum may arise also for bounded domains or in the gap (−m, m). Similar difficulties and effects were observed in the 2D situation in [21] and also in the analysis of Dirac operators with singular interactions in [3,9,14,15,49].
We mention that for some models it is more convenient to consider Dirac operators A [ω] in L 2 (Ω; C 4 ) with boundary conditions of the form where ω : ∂Ω → R is a Hölder continuous function. Comparing with the boundary conditions in (1.1) one formally has ω = ϑ −1 . Note that the particularly interesting MIT bag model corresponds to ω ≡ 0. Using the abstract quasi boundary triple approach the spectral and scattering properties of A [ω] can be studied in the same way as those of A ϑ , and similar results as sketched above for A ϑ follow; cf. Section 5.2. Let us give a short overview on the structure of the paper. In Section 2 we review the definitions of quasi boundary triples and their associated Weyl functions. Then, in Section 3 we recall some knowledge on a minimal and a maximal realization of the Dirac operator in Ω, the MIT bag model, and the properties of several families of integral operators associated to the resolvent of the free Dirac operator. Next, in Section 4 we introduce and study a quasi boundary triple which is suitable to investigate Dirac operators in Ω with boundary conditions. Section 5 contains the main results of the present paper. After defining A ϑ and proving its self-adjointness in Section 5.1 we use the quasi boundary triple from Section 4 to conclude various spectral properties. Section 5.2 is then devoted to the study of the operator A [ω] with the boundary conditions (1.3), while in Section 5.3 we discuss the earlier mentioned connection of A ϑ and A [ω] to the operator B η,τ formally given in (1.2). Finally, in an appendix we collect some material on integral operators and their mapping properties in Sobolev spaces on the boundary ∂Ω, which is applied in the proofs of the main results of this paper.
Notations. Throughout this paper m is always a positive constant that stands for the mass of a particle. The Dirac matrices α 1 , α 2 , α 3 , β ∈ C 4×4 are defined by α j := 0 σ j σ j 0 and β := where I n is the n × n-identity matrix and σ j , j ∈ {1, 2, 3}, are the Pauli spin matrices The Dirac matrices satisfy the anti-commutation relations α j α k + α k α j = 2δ jk I 4 , and α j β + βα j = 0, j, k ∈ {1, 2, 3}. (1.6) For vectors x = (x 1 , x 2 , x 3 ) ⊤ ∈ R 3 we often use the notation α · x := 3 j=1 α j x j . The upper/lower complex half plane is denoted by C ± . The square root √ · is fixed by Im √ λ > 0 for λ ∈ C \ [0, ∞) and √ λ ≥ 0 for λ ≥ 0. The open ball of radius r > 0 centered at x ∈ R 3 is denoted by B(x, r). For a C 2 -domain Ω ⊂ R 3 we write ∂Ω for its boundary and σ is the 2-dimensional Hausdorff measure on ∂Ω. We shall mostly work with the L 2 -spaces L 2 (Ω; C n ) and L 2 (∂Ω; C n ) of C n -valued square integrable functions, the corresponding inner products being denoted by (·, ·) Ω and (·, ·) ∂Ω , respectively. We write C ∞ 0 (Ω; C n ) for the space of C n -valued smooth functions with compact support in Ω and we set We write H k (R 3 ; C n ) for the usual L 2 (R 3 ; C n )-based Sobolev space of k-times weakly differentiable functions, and similarly H k (Ω; C n ). In addition, H 1 0 (Ω; C n ) denotes the closure of C ∞ 0 (Ω; C n ) in H 1 (Ω; C n ). Sobolev spaces on C l -surfaces ∂Ω, l ∈ N, are denoted by H s (∂Ω; C n ), s ∈ (0, l), and the symbol H −s (∂Ω; C n ) is used for their duals. The corresponding norm for s ∈ (0, 1) is The trace of a function f ∈ H 1 (Ω; C n ), which belongs by the trace theorem to H 1/2 (∂Ω; C n ), is denoted by f | ∂Ω . Eventually, given 0 < a ≤ 1 we denote the Hölder continuous functions on ∂Ω of order a by Lip a (∂Ω) := f : For two Hilbert spaces G and H the space B(G, H) is the set of all bounded and everywhere defined operators from G to H. If G = H, then we simply write B(H). We write S p,∞ (G, H) for the weak Schatten-von Neumann ideal of order p > 0; this is the set of all compact operators K : G → H for which there exists a constant κ such that the singular values s k (K) of K fulfill s k (K) ≤ κk −1/p for all k ∈ N, see [35] or [18, Section 2.1]. Again we use S p,∞ (G) if G = H and sometimes we suppress the spaces and just write S p,∞ .
For a linear operator T : G → H we denote the domain, range, and kernel by dom T , ran T , and ker T , respectively. If T is a self-adjoint operator in H then its resolvent set, spectrum, essential spectrum, discrete, and point spectrum are denoted by ρ(T ), σ(T ), σ ess (T ), σ disc (T ), and σ p (T ), respectively. Next, for a Banach space X we use the notation (·, ·) X ′ ×X for the duality product in X ′ × X which is linear in its first and anti-linear in the second entry. Moreover, for T ∈ B(X, Y ) we denote by T ′ ∈ B(Y ′ , X ′ ) the anti-dual operator, which is uniquely determined by the relation Finally (1.8)

Quasi boundary triples and their Weyl functions
This section is devoted to a short introduction to quasi boundary triples and their Weyl functions; the presentation is chosen such that the results can be applied directly in the main part of this paper. For a more detailed exposition and proofs in a general scenario we refer to [16,17]. Throughout this abstract section H is always a complex Hilbert space with inner product (·, ·) H ; if no confusion arises, we skip the index in the inner product.
Definition 2.1. Let S be a densely defined, closed, symmetric operator in H and assume that T is a linear operator in H such that T = S * . Moreover, let G be a complex Hilbert space and let Γ 0 , Γ 1 : dom T → G be linear mappings. Then {G, Γ 0 , Γ 1 } is called a quasi boundary triple for T ⊂ S * if the following conditions are fulfilled: The concept of quasi boundary triples is a generalization of ordinary and generalized boundary triples; cf. [13,24,30,31]. We note that the operator T in the above definition is not unique if the dimension of G is infinite. Moreover, we remark that a quasi boundary triple exists if and only if dim ker(S * − i) = dim ker(S * + i), that is, if and only if S admits self-adjoint extensions in H.
Next, we introduce the γ-field and the Weyl function associated to a given quasi boundary triple. Let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T ⊂ S * and let A 0 := T ↾ ker Γ 0 . The definition of the γ-field and the Weyl function is based on the direct sum decomposition and is formally the same as in the case of ordinary boundary triples, see [13,30]. Note that (2.1) implies, in particular, that Γ 0 ↾ ker(T −λ) is injective for λ ∈ ρ(A 0 ).
Let us now assume that {G, Γ 0 , Γ 1 } is a quasi boundary triple for T ⊂ S * and set A 0 := T ↾ ker Γ 0 . In the following we collect several useful properties of the associated γ-field γ and Weyl function M ; for the proofs see for instance [16, Proposition 2.6] and [17, Propositions 6.13 and 6.14]. First, for any λ ∈ ρ(A 0 ) the mapping γ(λ) is densely defined and bounded from G into H with dom γ(λ) = ran Γ 0 . Using the abstract Green's identity it is not difficult to see that the adjoint γ(λ) * : H → G is given by γ(λ) * = Γ 1 (A 0 − λ) −1 . This implies, in particular, γ(λ) * ∈ B(H, G). In a similar manner we have for any λ ∈ ρ(A 0 ) that the mapping M (λ) is densely defined in G with dom M (λ) = ran Γ 0 and ran M (λ) ⊂ ran Γ 1 . By definition we have M (λ)Γ 0 f λ = Γ 1 f λ for λ ∈ ρ(A 0 ) and f λ ∈ ker(T − λ). Next, for any λ, µ ∈ ρ(A 0 ) and ϕ ∈ ran Γ 0 the identity In the main part of this paper we will use quasi boundary triples to introduce special extensions of a symmetric operator S. Let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T ⊂ S * and let Θ be a symmetric operator in G. Then we define the operator A Θ acting in H by In other words, a vector f ∈ dom T belongs to dom A Θ if Γ 0 f ∈ dom Θ and if it satisfies the abstract boundary condition Γ 1 f = ΘΓ 0 f . It follows immediately from the abstract Green's identity that A Θ is symmetric. Of course, one is typically interested in the self-adjointness of A Θ . However, in general, for quasi boundary triples the self-adjointness of Θ in G does not necessarily imply that A Θ is self-adjoint in H. Nevertheless the next theorem provides an explicit Krein-type resolvent formula which allows to deduce several properties of A Θ from Θ; for a proof of this result see for instance [16,Theorem 2.8].
Theorem 2.3. Let S be a densely defined, closed, symmetric operator in H, let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T ⊂ S * , set A 0 := T ↾ ker Γ 0 , and let γ and M be the associated γ-field and Weyl function, respectively. Moreover, let Θ be a symmetric operator in G and let the associated operator A Θ be defined by (2.2). Then the following statements hold for λ ∈ ρ(A 0 ): .
We point out that assertion (ii) in Theorem 2.3 gives an efficient tool to check the self-adjointness of A Θ . Since A Θ is symmetric by Green's identity, it suffices to show that ran(A Θ − λ ± ) = H for some λ ± ∈ C ± . According to Theorem 2.3 (ii) this is true if ran γ(λ ± ) * ⊂ ran(Θ − M (λ ± )). Furthermore, if λ ∈ ρ(A Θ ) = ∅ then the resolvent formula in Theorem 2.3 (iii) holds for all f ∈ H, that is, for Note that, if {G, Γ 0 , Γ 1 } is a quasi boundary triple for T ⊂ S * , then Theorem 2.3 shows how the eigenvalues of self-adjoint extensions of S, that are contained in ρ(A 0 ), can be characterized by the Weyl function M . If the symmetric operator S is simple, then all eigenvalues can be characterized with the help of M , in particular, also those that are embedded in σ(A 0 ), compare [19,Corollary 3.4]. Note that there are also similar characterizations for the other types of the spectrum available in [19], but in our applications we restrict ourselves to find the eigenvalues. Proposition 2.4. Let S be a densely defined, closed, symmetric operator in H, let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T ⊂ S * , and let M be the associated Weyl function. Moreover, let Θ be a bounded and self-adjoint operator in G and assume that the associated operator A Θ defined by (2.2) is self-adjoint. Assume, in addition, that S is simple. Then ran(M (λ) − Θ) is independent of λ ∈ C \ R, and λ ∈ R is an eigenvalue of A Θ if and only if there exists ϕ ∈ ran(M (λ + iε) − Θ) such that lim In fact, using that Θ is bounded and self-adjoint we deduce from the abstract Green's identity for {G, Γ 0 , Γ 1 } and for f, g ∈ dom T that and hence the abstract Green's identity holds also for the triple {G, Γ Θ 0 , Γ Θ 1 }. Next, the definition of Γ Θ 0 , Γ Θ 1 can be written equivalently as Since Θ is bounded, it follows that B is boundedly invertible with } is a quasi boundary triple for S * . Next, we compute on C \ R the Weyl function M Θ corresponding to the triple {G, Γ Θ 0 , Γ Θ 1 }. For a fixed λ ∈ C\R this is the mapping which is determined uniquely by the relation Note that M (λ) − Θ is invertible by Theorem 2.3, as otherwise the self-adjoint operator A Θ would have the non-real eigenvalue λ. Thus, we conclude In particular, this implies that dom M Θ (λ) = ran(M (λ) − Θ) = ran Γ Θ 0 is independent of λ ∈ ρ(A Θ ).
After all these preparations the claim of the proposition follows from [19, Corollary 3.4] applied to the quasi boundary triple {G, Γ Θ 0 , Γ Θ 1 }, as S is simple.
Let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T ⊂ S * and let B be a symmetric operator in G. In some applications it is more convenient to consider ) if and only if 1 ∈ σ p (BM (λ)). Furthermore, one has Note that for λ ∈ ρ(A [B] ) ∩ ρ(A 0 ) the resolvent formula in Theorem 2.5 (iii) reads as Finally, we state the counterpart of Proposition 2.4 for extensions A [B] given by (2.3) to detect all eigenvalues of A [B] . Proposition 2.6. Let S be a densely defined, closed, symmetric operator in H, let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T ⊂ S * , and let M be the associated Weyl function. Moreover, let B be a bounded and self-adjoint operator in G and assume that the associated operator A [B] defined by (2.3) is self-adjoint. Assume, in addition, that S is simple. Then ran(I − BM (λ)) is independent of λ ∈ C \ R, and λ ∈ R is an eigenvalue of A [B] if and only if there exists ϕ ∈ ran(I − BM (λ + iε)) such that lim Proof. The proof is very similar as the one of Proposition 2.4, so we only sketch the main differences here. Define the boundary mappings Γ 1 } for λ ∈ C \ R, then we can apply again [19,Corollary 3.4] to characterize all eigenvalues of A [B] . Let λ ∈ C \ R and f λ ∈ ker(T − λ) be fixed. Note that M (λ) is invertible, as otherwise the symmetric operator T ↾ ker Γ 1 would have the non-real eigenvalue λ, cf. Theorem 2.
Note that I − BM (λ) is invertible by Theorem 2.5, as otherwise the self-adjoint operator A [B] would have the non-real eigenvalue λ. Thus, we conclude This implies, in particular, that dom M [B] (λ) = ran(I − BM (λ)) = ran Γ After all these preparations the claim of the proposition follows from [19, Corollary 3.4] applied to the quasi boundary triple {G, Γ 1 }, as S is simple.
3. The minimal, the maximal, the MIT bag, and some associated integral operators In this section we provide some facts on Dirac operators and associated integral operators. First, we collect some properties of the minimal and the maximal realization of the Dirac operator on a domain Ω ⊂ R 3 . Then we introduce and discuss the MIT bag operator, which is a distinguished self-adjoint realization of the Dirac operator in Ω, and which serves as a reference operator later. Finally, we introduce several families of integral operators which will play a crucial role in Section 4 and Section 5 in the proofs of the main results of this paper. Throughout this section let Ω be a C 2 -domain in R 3 with compact boundary, that is, Ω is either a bounded C 2 -domain or the complement of the closure of such a set. The unit normal vector field at ∂Ω pointing outwards Ω is denoted by ν.
3.1. The minimal and the maximal Dirac operator. We are going to study the following two operators acting in L 2 (Ω; C 4 ): The maximal Dirac operator where the derivatives are understood in the distributional sense, and the minimal Dirac operator T min = T max ↾ H 1 0 (Ω; C 4 ), which is given in a more explicit form by Some basic and well-known properties of T min and T max are collected in the following lemma; cf. [ Observe that for f, g ∈ H 1 (Ω; C 4 ) the integration by parts formula In the next proposition we verify that T min is a simple symmetric operator, that is, there exists no nontrivial invariant subspace for T min in L 2 (Ω; C 4 ) on which T min reduces to a self-adjoint operator. The simplicity of T min is essential in Proposition 5.5 and Proposition 5.11 for the characterization of eigenvalues of self-adjoint extensions of T min which are embedded in the spectrum of the MIT bag operator.
Let us show that (T min ) 2 is simple. We define the operator and we claim that A Ω ⊂ ((T min ) 2 ) * . In fact, consider arbitrary f ∈ dom A Ω and let g ∈ dom (T min ) 2 . Then g, (−iα · ∇ + mβ)g ∈ H 1 0 (Ω; C 4 ) and the identity (3.3) shows if Ω is bounded this is essentially a consequence of unique continuation (for details see [13,Section 8.3]) and if Ω is unbounded this fact can be found in [20, and we conclude This implies that (T min ) 2 is simple (see (1.8)). Therefore, H 1 = {0} and hence T min is simple.
3.2. The MIT bag operator. In this subsection we discuss the MIT bag Dirac operator in Ω which will often play the role of a self-adjoint reference operator in this paper. The MIT bag operator is the partial differential operator in L 2 (Ω; C 4 ) defined by In the following proposition we summarize the basic properties of T MIT . For some further results on T MIT , as, e.g., symmetry relations of the spectrum or asymptotics of eigenvalues for large masses m we refer to [1]. Moreover, we note that the orthogonal sum of the MIT bag operator in Ω and R 3 \ Ω is a Dirac operator with a Lorentz scalar δ-shell interaction, see Proposition 5.15. Using this, one can show even some further properties of T MIT ; cf. (5.33).
and the following statements hold: Proof. First, the self-adjointness of T MIT is shown in [49,Theorem 3.2]. The proof of assertion (i) follows similar considerations in [1, Theorem 1.5] for C 3 -domains, but the arguments are basically independent of the smoothness of ∂Ω. Indeed, one can show for f ∈ dom T MIT with the help of (3.3) and (1.6) that holds, where the boundary condition for f ∈ dom T MIT was used in the last step.
In a similar fashion as for the MIT bag model we also state some basic properties of another distinguished self-adjoint realization of the Dirac operator on Ω. This operator has similar boundary conditions as T MIT , but with opposite sign, and is given by The next proposition is the counterpart of Proposition 3.3 for T −MIT . However, in contrast to Proposition 3.3 (i) the interval (−m, m) may also contain spectrum; cf. Theorem 5.4 and Proposition 5.5.
and the following statements hold: Remark 3.5. We will show later in Theorem 5.4 (i) that the inclusion in item (ii) of the above proposition is in fact an equality, i.e.
This holds, as the operator T −MIT corresponds to T ↾ ker Γ 1 defined as in (5.2) with the parameter ϑ = 0. But for our next considerations the above inclusion is sufficient.

Proof of Proposition 3.4. Define the auxiliary operator
and consider the unitary and self-adjoint matrix We claim that To see this we note that Hence, we have shown (3.6). In particular, this implies together with Proposition 3.3 that T is self-adjoint. Since multiplication by mβ is a bounded and self-adjoint operator in L 2 (Ω; C 4 ) we conclude that also T −MIT = T + 2mβ is self-adjoint. Eventually, assertions (i) and (ii) can be shown in exactly the same way as Proposition 3.3 (ii) and (iii); in particular, for the proof of item (ii) the same singular sequence as in Proposition 3.3 (iii) can be used.

Integral operators.
In this section we introduce several families of integral operators that will play an important role in the analysis of Dirac operators on domains. We also summarize some of their well-known properties. Define for Recall that G λ is the integral kernel of the resolvent of the free Dirac operator in R 3 , see [54, Section 1.E]. We introduce the operators Φ λ : and C λ : It is well-known that Φ λ and C λ are bounded and everywhere defined and that holds; cf. [3, Lemmas 2.1 and 3.3] or [9, Proposition 3.4]. Furthermore, the adjoint of Φ λ is given by Φ * λ : 11) and this operator is also bounded when viewed as an operator from L 2 (Ω; C 4 ) to H 1/2 (∂Ω; C 4 ); cf. [11, equation (2.12) and the discussion below]. Hence, we can define the anti-dual of Φ * Since we have for ϕ ∈ L 2 (∂Ω; C 4 ) and f ∈ L 2 (Ω; C 4 ) Some further properties of Φ λ are summarized in the following proposition.
In the next proposition we collect some additional properties of C λ that will be useful in the sequel.
For ϕ ∈ H 1/2 (∂Ω; C 4 ) the trace of Φ λ,1/2 ϕ and the function C λ,1/2 ϕ are closely related. The formula in the next lemma will be useful in the next sections.
In the following proposition we show that the commutator of the singular integral operator C λ and a Hölder continuous function of order a > 0 is bounded from L 2 (∂Ω; C 4 ) to H s (∂Ω; C 4 ), s ∈ [0, a), and hence compact in H s (∂Ω; C 4 ) for a > s ≥ 0. This has important consequences for the analysis of self-adjoint Dirac operators on domains and will be used in the proofs of many of the main results in this paper. The proof of the next result relies on the properties of integral operators established in Appendix A. (3.9), and assume that ϑ ∈ Lip a (∂Ω) for some a ∈ (0, 1]. Then the commutator Proof. To prove the claimed mapping properties we show that each component k jl , j, l ∈ {1, . . . , 4}, of the matrix-valued integral kernel of C λ ϑ − ϑC λ satisfies the estimates in (A.4). As a > s the claim of this proposition follows from Theorem A.3 applied to the scalar integral operators with integral kernels k jl , j, l ∈ {1, . . . , 4}. Since the embedding ι s : H s (∂Ω; is compact for s ∈ (0, a), this implies then also the compactness of C λ,s ϑ − ϑC λ,s in H s (∂Ω; C 4 ). Indeed, for s > 0 it follows that C λ,s ϑ − ϑC λ,s = (C λ ϑ − ϑC λ )ι s is compact in H s (∂Ω; C 4 ), and for s = 0 one can choose r ∈ (0, a) and sees with the result of this proposition that C λ ϑ − ϑC λ : In the sequel we use the matrix norm Throughout the proof of this proposition C is a generic constant with different values on different places. First, due to the Hölder continuity of ϑ we conclude immediately from the definition of G λ in (3.7) that 14) and show that each of the two terms on the right hand side of (3.14) fulfills this growth condition. Clearly, using the Hölder continuity of ϑ and the definition of G λ from (3.7) we find first that To get an estimate for the second term in (3.14) we note first that the Hölder continuity of ϑ, the triangle inequality, and |x − y| In the next calculation we use for ξ, ζ ∈ R 3 \ {0} the notation Then we deduce with the main theorem of calculus applied to each entry of the (3.17) Together with (3.16) this leads to It follows now easily from (3.14), the triangle inequality, (3.15), and the last estimate that the components of K also satisfy the second condition in (A.4). This completes the proof of this proposition.
We now provide some useful anti-commutator properties of C λ and the Dirac matrices. These facts are also main ingredients to prove the self-adjointness of Dirac operators on domains later. (ii) The anti-commutator B λ := C λ β + βC λ can be extended to a bounded operator B λ : H −1/2 (∂Ω; C 4 ) → H 1/2 (∂Ω; C 4 ).
Next, we state a result on the invertibility of ± 1 2 β +C λ,s which will be used in the construction of the γ-field and the Weyl function associated to a quasi boundary triple for a Dirac operator. To formulate the result we recall the definitions of T MIT and T −MIT from (3.4) and (3.5), respectively, and denote by T Ω c MIT and T Ω c −MIT the Dirac operator in L 2 (R 3 \ Ω; C 4 ) with the same boundary conditions on ∂Ω as T MIT and T −MIT , respectively. ). For this it suffices to verify that 1 2 β + C λ,1/2 is bijective in H 1/2 (∂Ω; C 4 ). Then the claim for s = − 1 2 follows by duality, as C λ,−1/2 = C ′ λ,1/2 , and for s ∈ (− 1 2 , 1 2 ) by interpolation.
The proof of the item (ii) follows the lines of assertion (i), one just has to note that for ϕ ∈ ker − 1 2 β+C λ,1/2 the functions f Ω and f Ω c defined by (3.19) and (3.20) belong to dom T −MIT and dom T Ω c −MIT , respectively.
Finally, observe that Therefore T ↾ ker Γ 0 coincides with the MIT bag Dirac operator T MIT and T ↾ ker Γ 1 coincides with T −MIT , which shows (4.6). Note that both operators are self-adjoint; cf. Proposition 3.3 and Proposition 3.4. Thus {G Ω , Γ 0 , Γ 1 } is a quasi boundary triple for T ⊂ T max . Now we compute the γ-field and the Weyl function associated to the quasi boundary triple in Theorem 4.1. It turns out that these operators are closely related to the integral operators Φ λ and C λ defined in Section 3.3. For the next proposition recall that 1 2 β + C λ,1/2 admits a bounded and everywhere defined inverse in H 1/2 (∂Ω; C 4 ) for λ ∈ C \ ((−∞, −m] ∪ [m, ∞)), see Proposition 3.12. (i) The value of the γ-field corresponding to {G Ω , Γ 0 , Γ 1 } is given by Each γ(λ) is a densely defined bounded operator from G Ω to L 2 (Ω; C 4 ), and a bounded and everywhere defined operator from G Each γ(λ) * is bounded from L 2 (Ω; C 4 ) to G For the proof of item (i) consider ϕ ∈ ran Γ 0 = G 1/2 Ω and recall that γ(λ)ϕ is the unique solution of the boundary value problem (T − λ)f = 0 and Γ 0 f = ϕ. (4.11) We set Then, due to the mapping properties of Φ λ,1/2 and 1 2 β + C λ,1/2 −1 , see Propositions 3.6 and Proposition 3.12, we have f λ ∈ H 1 (Ω; C 4 ) = dom T . For (4.9) it suffices to check that f λ solves the boundary value problem (4.11). In fact, by Proposition 3.6 (ii) we have (T − λ)f λ = 0 and using Lemma 3.9 we get Using that ϕ ∈ G Ω , (1.6), and P + P − = 0 we obtain then Hence, f λ is the unique solution of the boundary value problem (4.11). This implies γ(λ)ϕ = f λ and leads to the representation (4.9). It remains to check the mapping properties of γ(λ) in (i). From the definition of the γ-field it is clear that γ(λ) is a densely defined bounded operator from G Ω to L 2 (Ω; C 4 ). Moreover, from Proposition 3.6 (i) and Proposition 3.12 (i) it also follows that γ(λ) is a bounded and everywhere defined operator from G 1/2 Ω to H 1 (Ω; C 4 ).
It is a consequence of mapping properties of ( 1 2 β + C λ,1/2 ) −1 from Proposition 3.12 (i) that each M (λ) is a densely defined and bounded operator in G Ω , and a bounded and everywhere defined operator in G is a quasi boundary triple for T ⊂ T max such that T ↾ ker Γ 0 = T −MIT . In fact, using that {G Ω , Γ 0 , Γ 1 } is quasi boundary triple it follows that the abstract Green identity is satisfied by the boundary mappings in (4.13) and that the range of ( Γ 0 , Γ 1 ) ⊤ is dense. Moreover, T −MIT = T ↾ ker Γ 0 is a self-adjoint operator by Proposition 3.4. Note that Weyl function M corresponding to the quasi boundary triple {G Ω , Γ 0 , Γ 1 } is given for our choice of λ ∈ ρ( Thus it remains to compute the value of the Weyl function M (λ). We first show the explicit formula for the γ-field corresponding to the quasi boundary triple {G Ω , Γ 0 , Γ 1 } using a similar argument as in the proof of Proposition 4.2. In fact, it is clear that dom γ(λ) = ran Γ 0 = G 1/2 Ω . Next, consider ϕ ∈ dom γ(λ) and recall that γ(λ)ϕ is the unique solution of the boundary value problem (T − λ) = 0 and Γ 0 f λ = ϕ.
We set Then we have (T − λ)f λ = 0 and using P + P − = 0 and ϕ ∈ G Ω we obtain in a similar way as in the proof of Proposition 4.2 that which leads to (4.14).
Let us now compute the Weyl function M . Using (4.14) and Lemma 3.9 we find Thanks to (4.2), P 2 + = P + , and that ϕ ∈ G Ω we finally obtain which gives (4.12).
Finally, we state a lemma on the invertibility of ϑ−M (λ) for a Hölder continuous function ϑ. This result will be needed in the proofs of several results of this paper.
Recall that the closure M (λ) of the Weyl function corresponding to the triple {G Ω , Γ 0 , Γ 1 } is bounded in G Ω , see Proposition 4.2.

Dirac operators on domains -definition and basic spectral properties
This section contains the main results of this paper. First, in Section 5.1 we introduce Dirac operators A ϑ on Ω with boundary conditions of the form for a real-valued Hölder continuous function ϑ : ∂Ω → R of order a > 1 2 and P + given by (4.1). Using the quasi boundary triple {G Ω , Γ 0 , Γ 1 } from Theorem 4.1 we show that A ϑ is self-adjoint if |ϑ(x)| = 1 for all x ∈ ∂Ω. We also obtain a Krein type resolvent formula and some qualitative spectral properties of A ϑ . In Section 5.2 we sketch how Dirac operators A [ω] with boundary conditions of the form for a real-valued Hölder continuous function ω : ∂Ω → R of order a > 1 2 can be handled with similar arguments. Finally, in Section 5.3 we relate the operators A ϑ to Dirac operators B η,τ with singular δ-shell potentials of the form (1.2). This relation allows to translate results for B η,τ to A ϑ , and vice versa.
Throughout this section let Ω be a bounded or unbounded domain in R 3 with a compact C 2 -smooth boundary, and denote by ν the normal vector field at ∂Ω pointing outwards of Ω. 5.1. Self-adjointness and spectral properties of A ϑ . We start with the rigorous mathematical definition of the Dirac operator A ϑ with boundary conditions (5.1). We shall use the quasi boundary triple {G Ω , Γ 0 , Γ 1 } from Theorem 4.1 in the next definition.
Definition 5.1. Let a ∈ ( 1 2 , 1] and let ϑ ∈ Lip a (∂Ω) be real-valued. We define A ϑ = T ↾ ker(Γ 1 − ϑΓ 0 ), which in a more explicit form is given by Remark 5.2. The boundary conditions in (5.1) are the 3D analogue of the boundary conditions used in [21]. In fact, let Ω ⊂ R 2 be a bounded C 2 -domain. In [21] the boundary conditions for all x ∈ ∂Ω are treated. Here σ = (σ 1 , σ 2 ) and σ 3 are the Pauli spin matrices in (1.5), ν = (ν 1 , ν 2 ) is the normal vector field at ∂Ω, and σ · ν = σ 1 ν 1 + σ 2 ν 2 . To see that (5.3) is equivalent to the boundary conditions in [21], one has to note that σ · t = −iσ 3 (σ · ν), where t = (−ν 2 , ν 1 ) is the tangential vector at ∂Ω. We use the splitting and remark that Q ± is the 2D-analogue of P ± from (4.1). Hence, we can rewrite (5.3) as With the help of the relations iσ 3 (σ · ν)Q ± = ±Q ± and Q − = σ 3 Q + σ 3 , we find that (5.3) is equivalent to By multiplying this equation with which exists since cos[η(x)] / ∈ {0, 1} is assumed, we see that (5.3) is equivalent to which is the 2D analogue of the boundary conditions in (5.2) for ϑ = sin(2η) 2 cos η(1−cos η) . It follows immediately from the abstract Green's identity that A ϑ is symmetric for any real-valued function ϑ. In order to prove self-adjointness we shall use Theorem 2.3, which also leads to a resolvent formula in terms of the resolvent of the MIT bag operator T MIT in (3.4) and the γ-field and Weyl function. We note that in (5.33) an explicit formula for (T MIT − λ) −1 is shown. Then the operator A ϑ in (5.2) is self-adjoint in L 2 (Ω; C 4 ) and the resolvent formula holds for all λ ∈ ρ(A ϑ ) ∩ ρ(T MIT ).
Next, we discuss the basic spectral properties of the operator A ϑ . Since these are of a very different nature whether Ω is bounded or unbounded the two cases are treated separately. Assume first that Ω is an unbounded C 2 -domain with compact boundary. The proof of (ii) is based on the same argument as the proof of [15,Proposition 3.9].
Theorem 5.4. Let Ω be the complement of a bounded C 2 -domain, let a ∈ ( 1 2 , 1], let ϑ ∈ Lip a (∂Ω) be a real-valued function such that |ϑ(x)| = 1 for all x ∈ ∂Ω, and let A ϑ be defined by (5.2). Then the following statements hold: Proof. We first deal with (i). Let γ and M be the γ-field and the Weyl function corresponding to the quasi boundary triple {G Ω , Γ 0 , Γ 1 }, respectively, from Proposition 4.2, and let γ(λ) ∈ B(G Ω , L 2 (Ω; C 4 )) and M (λ) ∈ B(G Ω ) be the closures of γ(λ) and M (λ), λ ∈ ρ(T MIT ). For λ ∈ ρ(A ϑ ) ∩ ρ(T MIT ) the resolvent formula in Theorem 5.3 can be written in the form To verify assertion (ii), consider the quadratic form Since A ϑ is a self-adjoint operator it follows that a is a closed, non-negative form and by [38, Theorem VI 2.1] the unique self-adjoint operator representing this form is A 2 ϑ . Note that the number of eigenvalues (counted with multiplicities) of A ϑ in the gap of the essential spectrum (−m, m) is equal to the number of eigenvalues of A 2 ϑ below m 2 (counted with multiplicities).
To estimate the number of eigenvalues of A 2 ϑ with the help of the quadratic form a let 0 < r < R such that ∂Ω ⊂ B(0, r) and choose real-valued functions g 1 , g 2 ∈ C ∞ (Ω; C) with the properties 0 ≤ g 1 , g 2 ≤ 1, g 1 ↾ (B(0, r) ∩ Ω) ≡ 1, g 2 ↾ B(0, R) c ≡ 1, and g 2 1 + g 2 2 ≡ 1. Note that the properties of g 1 and g 2 imply that the mapping is an isometry. Our next goal is to rewrite the form a as a sesquilinear form in L 2 (Ω ∩ B(0, R); C 4 ) ⊕ L 2 (R 3 \ B(0, r); C 4 ). For that we will often identify functions defined in Ω with their restrictions onto Ω ∩ B(0, R) or onto R 3 \ B(0, r) and we also identify functions on Ω ∩ B(0, R) or R 3 \ B(0, r) with their extensions by zero onto Ω. In both cases, we will use the same letters for the restrictions and the extended functions.
Let f ∈ dom a = dom A ϑ be fixed. Then also g 1 f, g 2 f ∈ dom a. Using the relation we find that Note that (1.6) implies (α · ∇g j ) 2 = |∇g j | 2 I 4 , which gives where b 1 and b 2 are the semibounded sesquilinear forms in L 2 (Ω ∩ B(0, R); C 4 ) and L 2 (R 3 \ B(0, r); C 4 ) given by In the following let us have a closer look at b 1 and b 2 . First we note that with the aid of (3.3) and (1.6) one has for h ∈ C ∞ 0 (R 3 \ B(0, r); . By density this extends to is the closed semibounded form associated to the self-adjoint operator B 2 := −∆ D +m 2 −V , where −∆ D is the selfadjoint Dirichlet Laplacian in R 3 \ B(0, r) and V = |∇g 1 | 2 + |∇g 2 | 2 is compactly supported in B(0, R) \ B(0, r) due to the construction of g 1 and g 2 . Thus, B 2 has only finitely many eigenvalues below m 2 ; for a proof see, e.g., [15, Proof of Proposition 3.9].
Next, we claim that b 1 is closed. In fact, let (h n ) be a sequence in dom b 1 and let h ∈ L 2 (Ω ∩ B(0, R)) such that b 1 [h n − h m ] → 0 and h n − h Ω∩B(0,R) → 0, as m, n → ∞.
By the definition of b 1 this implies that a[h n − h m ] → 0 and h n − h Ω → 0, as m, n → ∞. As a is closed we have h ∈ dom a = dom A ϑ and a[h−h n ] → 0 as n → ∞. Moreover, it follows from h n − h Ω → 0 that supp h ⊂ B(0, R). Hence, h ∈ dom b 1 and b 1 [h − h n ] → 0 as n → ∞, thus, b 1 is closed. The semibounded self-adjoint operator B 1 associated to b 1 defined on dom B 1 ⊂ dom b 1 ⊂ H 1 (Ω ∩ B(0, R); C 4 ) has a compact resolvent in L 2 (Ω ∩ B(0, R); C 4 ), which implies that the spectrum of B 1 is purely discrete and accumulates only to ∞.
By combining the above considerations we are now prepared to show the claim of assertion (ii). First, we have by (5.5) where it was used that U is an isometry, and U (dom a) ⊂ dom (b 1 ⊕ b 2 ). Hence, it follows from the min-max principle that the number of eigenvalues of A 2 ϑ below m 2 is less or equal to the number of eigenvalues of the operator B 1 ⊕ B 2 associated to b 1 ⊕ b 2 below m 2 . As we have seen above, the number of eigenvalues of B 1 and B 2 below m 2 is finite. Hence, also the number of eigenvalues of B 1 ⊕ B 2 below m 2 is finite. This shows that the number of eigenvalues of A 2 ϑ below m 2 is finite, which yields the claimed result.
Finally, item (iii) is an immediate consequence of Theorem 2.3 (i).
If Ω is a bounded C 2 -domain, then dom A ϑ ⊂ H 1 (Ω; C 4 ) is compactly embedded in L 2 (Ω; C 4 ) and hence the spectrum of A ϑ is purely discrete. It is clear that the Birman Schwinger principle from Theorem 2.3 can be used to detect discrete eigenvalues of A ϑ that belong to ρ(T MIT ). The next result, which is a direct consequence of Proposition 2.4 and Proposition 3. this can be done in the same way as in [4,Theorem 3.7], see also the discussion of this result. If Ω is not connected, then there exists a bounded set Ω 1 and an unbounded connected domain Ω 2 such that Ω = Ω 1 ∪ Ω 2 . This implies that also A ϑ decomposes as A ϑ = A ϑ,1 ⊕ A ϑ,2 , where A ϑ,j is a self-adjoint operator of the form (5.2) in L 2 (Ω j ; C 4 ), j ∈ {1, 2}. By the same reasoning as above A ϑ,2 has no eigenvalues in R \ [−m, m]. Therefore the embedded eigenvalues of A ϑ are those of A ϑ,1 , which can be found with the help of Proposition 5.5.
Next, we compare the differences of powers of the resolvents of A ϑ and T MIT and show that these operators belong to certain weak Schatten-von Neumann ideals. In the proof of this result we will make several times use of ST ∈ S r,∞ for S ∈ S p,∞ , T ∈ S q,∞ , and Moreover, for holomorphic operator functions A(·), B(·), C(·) the formula (see, e.g., [18, equation (2.7)]) will be employed several times. Furthermore, if the operator function A(·) is holomorphic and invertible with bounded everywhere defined inverses, then also A(·) −1 is holomorphic and one has cf. [18, equation (2.8)]. The proof of the following theorem is based on Lemma 3.14 and on the same strategy as in [18] or in [9,Theorem 4.6]. Hence, we have to assume some additional smoothness of ∂Ω.
Theorem 5.7. Let Ω be a C 2 -domain with compact boundary, let T MIT be the MIT bag operator in (3.4), let a ∈ ( 1 2 , 1], let ϑ ∈ Lip a (∂Ω) be a real-valued function such that |ϑ(x)| = 1 for all x ∈ ∂Ω, and let A ϑ be defined by (5.2). Moreover, let l ∈ N and, if l > 2, assume that Ω has a C l -smooth boundary. Then holds for all λ ∈ C \ R.
Proof. Let λ ∈ C \ R be fixed. By Proposition 4.2 we have Hence, using Proposition 3.6 and Proposition 3.12 we find In a similar way one gets With the resolvent formula from Theorem 5.3 and (5.7) we obtain We are going to study now all the terms on the right hand side of (5.11) and show that they belong to certain Schatten-von Neumann ideals. For this purpose we claim that d k dλ k for k ∈ {1, . . . , l − 1}. This will be shown by induction. First, for k = 1 we have by (5.8) d dλ Hence, the statement for k = 1 holds by Lemma 3.14 and Proposition 3.12. Let us assume now that the statement holds for k = 1, . . . , q with q < l − 1. With the aid of (5.7) we get d q+1 dλ q+1 Similarly, using (5.9), (5.7), Lemma 3.14, and (5.6) we obtain By taking adjoints, this implies that also Note that (5.13) yields this can be shown in the same way as (5.12). Thus, using (5.11), (5.14), (5.15), (5.16), and (5.6), we finally get that which is the claimed result.
In the following corollary we discuss the special case l = 3 in Theorem 5.7. Then the difference of the third powers of the resolvents of A ϑ and T MIT belongs to the trace class ideal. By [55,Chapter 0,Theorem 8.2] or [52,Problem 25] this implies that the wave operators for the scattering pair {A ϑ , T MIT } exist and are complete, and hence the absolutely continuous parts of A ϑ and T MIT are unitarily equivalent. Moreover, we state an explicit formula for the trace of (A ϑ − λ) −3 − (T MIT − λ) −3 in terms of the Weyl function M ; this formula can be shown in exactly the same way as in [9,Theorem 4.6].
Corollary 5.8. Assume that Ω has a C 3 -smooth boundary, let a ∈ ( 1 2 , 1], and let ϑ ∈ Lip a (∂Ω) be a real-valued function such that |ϑ(x)| = 1 for all x ∈ ∂Ω. Let A ϑ be defined by (5.2) and let T MIT be the MIT bag operator in (3.4). Then the operator (A ϑ − λ) −3 − (T MIT − λ) −3 belongs to the trace class ideal and holds for all λ ∈ C \ R. Moreover, the wave operators for the scattering system {A ϑ , T MIT } exist and are complete, and the absolutely continuous parts of A ϑ and T MIT are unitarily equivalent.

Self-adjointness and spectral properties of A [ω]
. To complement the class of boundary conditions (5.1) discussed in the previous section now boundary conditions of the form will be treated; here ω : ∂Ω → R is Hölder continuous of order a > 1 2 , as before. In particular, (5.17) for ω ≡ 0 leads to the MIT bag operator T MIT introduced in (3.4). Of course, if ω is invertible, then (5.1) and (5.17) are equivalent by setting ϑ = ω −1 , but if ω = 0 on some parts of ∂Ω, then this correspondence is only formal.
More precisely, let {G Ω , Γ 0 , Γ 1 } be the quasi boundary triple from Theorem 4.1 and assume that ω : ∂Ω → R is Hölder continuous of order a > 1 2 . Then the Dirac operator A [ω] acting in L 2 (Ω; C 4 ) with boundary conditions (5.17) is defined by Similarly to Section 5.1, one can prove now several results about the spectral properties of A [ω] . The following assertions follow from Theorem 2.5 and the Krein type resolvent formula from Theorem 5.9 in the same way as in Theorem 5.4. Theorem 5.10. Let Ω be the complement of a bounded C 2 -domain, let a ∈ ( 1 2 , 1], let ω ∈ Lip a (∂Ω) be a real-valued function such that |ω(x)| = 1 for all x ∈ ∂Ω, and let A [ω] be defined by (5.18). Then the following statements hold: Finally, also the proof of Theorem 5.7 can be adapted in a straightforward way to obtain a similar result for A [ω] . A summary of the counterpart of Theorem 5.7 and Corollary 5.8 reads as follows: Theorem 5.12. Let Ω be a C 2 -domain with compact boundary, let T MIT be the MIT bag operator in (3.4), let a ∈ ( 1 2 , 1], let ω ∈ Lip a (∂Ω) be a real-valued function such that |ω(x)| = 1 for all x ∈ ∂Ω, and let A [ω] be defined by (5.18). Moreover, let l ∈ N and, if l > 2, assume that Ω has a C l -smooth boundary. Then holds for all λ ∈ C \ R. In particular, for l = 3 the operator in (5.19) belongs to the trace class ideal and holds for all λ ∈ C \ R. Moreover, the wave operators for the scattering system {A [ω] , T MIT } exist and are complete, and the absolutely continuous parts of A [ω] and T MIT are unitarily equivalent.
In particular, in both situations (i) and (ii) the operator B η,τ in (5.20) is self-adjoint in L 2 (R 3 ; C 4 ).

Ω±
[ω] the self-adjoint operators defined in (5.18) acting in L 2 (Ω ± ; C 4 ). Now we verify for suitable ω, η, τ ∈ Lip a (Σ), which is the counterpart of the identity (5.21). As above one may use (5.29) to translate results for Dirac operators with δ-interactions to the operators A

Ω±
[ω] , and vice versa. The next proposition provides the necessary relations between the functions ω, η, τ . The proof follows the same strategy as the proof of Proposition 5.14.
Moreover, for λ ∈ C \ ((−∞, −m] ∪ [m, ∞)) we consider the integral operator R λ : L 2 (Ω; C 4 ) → L 2 (Ω; C 4 ), ∈ Ω, f ∈ L 2 (Ω; C 4 ), (5.32) with G λ defined by (3.7). Note that R λ is the compression of the resolvent of the free Dirac operator A in R 3 , that is, cf. [54, Section 1.E]. The resolvent formulas in the next preparatory lemma are now an immediate consequence of [11,Theorem 3.4], Proposition 5.14 (ii), and Proposition 5.15 (ii). We emphasize that in contrast to the previous discussion the coefficients are first assumed to be real constants since Dirac operators with electrostatic and Lorentz scalar δ-shell interactions have been studied in this case only. To avoid confusion we use a subindex here.
Next, we show that the multiplication operator with a Hölder continuous function ϑ ∈ Lip a (∂Ω) is bounded in H s (∂Ω; C) for any 0 ≤ s < a.
Lemma A.2. Let 0 ≤ s < a ≤ 1 and ϑ ∈ Lip a (∂Ω). Then the operator given by the multiplication with ϑ is bounded in H s (∂Ω; C).
Proof. Throughout the proof let C be a generic constant, which changes its value several times, and let ϕ ∈ H s (∂Ω; C) be fixed. In order to get the desired result we estimate in both terms on the right hand side separately. First, since ϑ ∈ Lip a (∂Ω) and ∂Ω is bounded, we have ϑ L ∞ (∂Ω) < ∞. Therefore Given now 0 < a ≤ 1 and an integral kernel k : ∂Ω × ∂Ω → C such that |k(x, y)| ≤ C |x − y| 2−a for all x = y, |k(x, z) − k(y, z)| ≤ C |x − y| a |x − z| 2 for all |x − y| < It is well known that if the integral kernel satisfies (A.4), then T is bounded in L 2 (∂Ω; C) and, in particular, the integral on the right hand side of (A.5) exists a.e. on ∂Ω, see, e.g., [33,Proposition 3.10]. In the following theorem, which is the main result of this appendix, we show that T has even better mapping properties between Sobolev spaces on ∂Ω: Theorem A.3. Let 0 < s < a ≤ 1. Then T defined by (A.5) gives rise to a bounded operator T : L 2 (∂Ω; C) → H s (∂Ω; C).