Self-Adjoint Dirac Operators on Domains in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^3$$\end{document}R3

In this paper, the spectral and scattering properties of a family of self-adjoint Dirac operators in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(\Omega ; \mathbb {C}^4)$$\end{document}L2(Ω;C4), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^3$$\end{document}Ω⊂R3 is either a bounded or an unbounded domain with a compact \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document}C2-smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with boundary conditions as of Robin type. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document}Ω is an exterior domain, 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 physical point of view, they 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 by physicists in [30][31][32]34,43] to study confinement of quarks. 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. [1,25,29,58]. From the mathematical point of view, Dirac operators with special boundary conditions can be seen as the relativistic counterpart of Laplacians with boundary conditions as, e.g., of Robin type. Moreover, Dirac operators with boundary conditions are also closely related to Dirac operators with singular δ-shell interactions supported on surfaces for special choices of the interaction strengths in the so-called confinement case, i.e., when the δ-potential is impenetrable for the particle; cf. [5,12,16,37].
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 , denoted by ϑ ∈ Lip a (∂Ω), 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 (1.6) 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 existing mathematical literature on such types of Dirac operators contains different approaches. In differential geometry, there are several articles dealing with self-adjoint Dirac operators on smooth manifolds, see, for instance, [7,8,59]. The class of boundary conditions treated in [8] contains also the physically particularly interesting MIT bag boundary conditions, which will be rigorously defined below and which yield a vanishing normal flux at the boundary of Ω ⊂ R 3 . It was already shown in [61] that the 2D Dirac operator with so-called zigzag boundary conditions (in the massless case) is self-adjoint and that zero is an eigenvalue of infinite multiplicity, see also [39]. The zigzag boundary conditions arise from the termination of a lattice in a graphene quantum dot, when the direction of the boundary is perpendicular to the bonds [41]. Very recent related publications in the 2D case are [22,23], 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 [22,23,61] are based on complex analysis techniques, which are not available in the 3D situation. We also refer to [28,47,48,57] 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, A ϑ was not directly investigated for general boundary parameters in 3D, as far as we know only the particular MIT bag operator is well studied. We emphasize the recent papers [2,56] for the analysis of general properties of the MIT bag operator and [3,9,24,54,62], where it is shown that the MIT bag boundary conditions and their 2D analogues can be interpreted as infinite mass boundary conditions (i.e., Ω is surrounded by a medium with infinite mass). The strategy developed in [56] employing Calderón projections can also be used to study the self-adjointness of Dirac operators of the form (1.1). However, this approach does not allow directly a systematic spectral analysis of these operators. Finally, we mention that in recent years the self-adjointness and the spectral and scattering properties of the closely related Dirac operators with singular δ-shell interactions were studied comprehensively in [4][5][6]10,12,15,16,42,[49][50][51]55,56].
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 [22] used to describe graphene quantum dots (cf. Remark 5.2). Furthermore, Dirac operators with boundary conditions as in (1.1) can be viewed as relativistic counterparts of Schrödinger operators with Robin-type (and possibly other) boundary conditions, as it is argued for the MIT bag model in [2]. This could be made rigorous by computing the non-relativistic limit [63,Chapter 6]. Another important goal in the present paper is to allow variable parameters in the boundary conditions. To the best of our knowledge, this is a novelty in the 3D case and it requires substantial technical effort. We believe that many of our results in this regard are also of interest for studying similar problems for Dirac operators with singular δ-shell interactions with varying strength.
Our 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 [17], but proved to be useful in many other situations, see, e.g., [11,19,20]. Quasi boundary triples were also applied more recently in [10,15] to Dirac operators with singular potentials. 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 . These operators also appeared in [4,5,10,15,56] in the study of Dirac operators with δ-shell potentials, and many of their properties were derived there; see also [26,33,52] for earlier results related to this framework. In the present paper, in particular to handle non-constant boundary parameters ϑ, additional mapping properties of these integral operators are required and, in fact, this analysis covers a great part of this 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 Kreintype resolvent formula that relates the resolvent of A ϑ via a perturbation term to the resolvent of a reference operator, which in our model is the MIT bag operator T MIT . More precisely, making use of the quasi boundary triple in Theorem 4.1 and the properties of the corresponding γ-field γ and Weyl function M in Proposition 4.2 we conclude the self-adjointness of A ϑ in L 2 (Ω; C 4 ) and the relation for all λ ∈ ρ(A ϑ ) ∩ ρ(T MIT ), where ϑ ∈ Lip a (∂Ω) is any real-valued Hölder continuous function with a > 1 2 ; cf. Theorem 5.3. Our arguments here also rely on the self-adjointness of the reference operator T MIT proved in [8,56] for C 2 -boundaries; cf. Proposition 3.3. Based on (1.2), we show several spectral and scattering properties of A ϑ , which are, of course, different for bounded and for unbounded domains Ω. It turns out in Theorem 5.4 that in the case of an unbounded domain Ω with a compact C 2 -boundary 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 implied by (1.2), which states that λ ∈ σ p (A ϑ ) ∩ (−m, m) if and only if 0 ∈ σ p (ϑ − M (λ)). Furthermore, in Theorem 5.7 we provide Schatten-von Neumann estimates on the differences of resolvent powers of A ϑ and T MIT and, in particular, conclude in Corollary 5.8 that the resolvent power difference (A ϑ − λ) −3 − (T MIT − λ) −3 is a trace class operator for any λ ∈ C\R, which leads to a trace formula and also implies the existence and completeness of the wave operators for the scattering pair {A ϑ , T MIT }. If Ω is a bounded C 2 -domain, then the spectrum of A ϑ is purely discrete and all eigenvalues of A ϑ can be characterized by a modified Birman-Schwinger principle formulated in Proposition 5.5, which again can be viewed as a consequence of the abstract quasi boundary triple approach.
The above-mentioned results are proved for any real-valued ϑ ∈ Lip a (∂Ω) with a > 1 2 under the additional 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 [22,61] and also in the analysis of Dirac operators with singular interactions in [4,10,15,16,56].
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 again ω ∈ Lip a (∂Ω) is any real-valued Hölder continuous function with a > 1 2 . 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. Sect. 5.2.
In the last part of this paper, we briefly explain the connection of the Dirac operators A ϑ and A [ω] on domains with Dirac operators with δ-shell interactions. More precisely, operators and boundary conditions of the form (1.1) and (1.3) appear in the treatment of Dirac operators in R 3 with singular δ-shell potentials supported on Σ = ∂Ω ± in the confinement (or decoupling) case, where Ω + is a bounded C 2 -domain and Ω − = R 3 \Ω + is an exterior domain. Note that so far such operators B η,τ were only studied for constant interaction strengths η, τ ∈ R. It is 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 (Ω − ; C 4 ), respectively, and it turns out that these operators are exactly of the form (1.1) with a certain constant ϑ ∈ R. In Sect. 5.3, we allow variable real-valued coefficients ϑ, η, τ ∈ Lip a (∂Ω) with a > 1 2 , and we specify in Proposition 5.14 relations between the functions ϑ, η, τ such that [ω] can be used to describe a relativistic particle actually living in R 3 , but which is confined to Ω + or Ω − for all time, see [5,Section 5].

Structure of the Paper
In Sect. 2, we review the definitions of quasi boundary triples and their associated Weyl functions. Then, in Sect. 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 with the resolvent of the free Dirac operator. Next, in Sect. 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. In Sect. 5.1, we first define A ϑ with the help of the quasi boundary triple from Sect. 4 and then conclude its self-adjointness and the resolvent formula in Theorem 5.3. This allows to prove various spectral properties in Theorem 5.4, Proposition 5.5, Theorem 5.7, and Corollary 5.8. Section 5.2 is then devoted to the study of the operator A [ω] with the boundary conditions (1.3), while in Sect. 5.3 we discuss the above-mentioned connection of A ϑ and A [ω] to the operator B η,τ formally given in (1.4). Finally, in "Appendix A" 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 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 The upper/lower complex half plane is denoted by C ± . The square root √ · is fixed by Im The open ball of radius r > 0 centered at x ∈ R 3 is denoted by B(x, r). For a C 2domain Ω ⊂ R 3 , we write ∂Ω for its boundary and σ is the two-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 ktimes 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 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 satisfy s k (K) ≤ κk −1/p for all k ∈ N, see [40] or [19, 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 the 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 for all x ∈ X and y ∈ Y . Finally

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 [17,18]. 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.
The concept of quasi boundary triples is a generalization of ordinary and generalized boundary triples; cf. [14,27,35,36]. 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 with 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 [14,35]. Note that (2.1) implies, in particular, that Γ 0 ker(T − λ) is injective for λ ∈ ρ(A 0 ).

Definition 2.2.
Let S be a densely defined, closed, symmetric operator in H, let T be a linear operator such that T = S * , and let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T ⊂ S * .
(i) The γ-field associated with {G, Γ 0 , Γ 1 } is the mapping 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, [17, Proposition 2.6] and [18, 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  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 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, which 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 [20,Corollary 3.4]. Note that there are also similar characterizations for the other types of the spectrum available in [20], 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 Proof. Define the boundary mappings Γ Θ 0 , } is a quasi boundary triple for T ⊂ S * with the additional property T ker Γ Θ 0 = A Θ . 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 For a fixed λ ∈ C\R, this is the mapping which is determined uniquely by the relation For such an f λ , we compute 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 [20,Corollary 3.4] ) 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 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 Proof. The proof is very similar as the one of Proposition 2.4, so we only sketch the main differences here. Define the mappings Γ 1 } for λ ∈ C\R, then we can apply again [20,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 2.3 (i). Hence, 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 After all these preparations, the claim of the proposition follows from [20,Corollary 3.4] applied to the quasi boundary triple {G, Γ 1 }, as S is simple.

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 Sects. 4 and 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.

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 non-trivial 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 Propositions 5.5 and 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 [14,Section 8.3]) and if Ω is unbounded this fact can be found in and we conclude This implies that (T min ) 2 is simple (see (1.10)). Therefore, H 1 = {0} and hence T min is simple.

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 [2]. 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: The proof of assertion (i) follows similar considerations in [2, Theorem 1.5] for C 3domains, 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 (1.8) that holds, where the boundary condition for f ∈ dom T MIT was used in the last step. Hence, we have with χ(r) = 1 for r < 1 2 and χ(r) = 0 for r > 1 and set x n := (R + n 2 , 0, 0) , n ∈ N. Then, we define the function ψ λ n by Then, one verifies in the same way as in [15,Theorem 5.7] that ψ λ n ∈ dom T MIT , that ψ λ n converge weakly to zero, that ψ λ n Ω = const. > 0 and (T Ω MIT − λ)ψ λ n → 0, as n → ∞. Thus, (ψ λ n ) n is a singular sequence for T MIT and λ, which shows λ ∈ σ ess (T MIT ). This finishes the proof of this proposition.
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 so that f ∈ dom T MIT if and only if γ 5 f ∈ dom T . 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.
Recall that G λ is the integral kernel of the resolvent of the free Dirac operator in R 3 , see [63, Section 1.E]. We introduce the families of integral operators (3.9) It is well known that Φ λ and C λ are bounded and everywhere defined and that  (3.11) 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. [12, equation (2.12) and the discussion below]. Hence, we can define the anti-dual of Φ * λ by Some further properties of Φ λ are summarized in the following proposition. Proposition 3.6. Let λ ∈ C\((−∞, −m] ∪ [m, ∞)) and let Φ λ be the operator in (3.8). Then, the following statements hold: 1 2 ] the operator Φ λ gives rise to a bounded and everywhere defined operator Φ λ,s : H s (∂Ω; Proof of Proposition 3.6. The claim of statement (i) for s = − 1 2 follows from the definition of Φ λ,−1/2 in (3.12), and it is contained for s = 1 2 and Ω = R 3 \Σ for a C 2 -surface Σ in [15,Proposition 4.2] (see also Remark 3.7); for that, one has to note that the map γ(λ) in [15] coincides with Φ λ,1/2 . The claim for Ω follows from this by restriction and for intermediate s ∈ (− 1 2 , 1 2 ) by an interpolation argument. Assertion (ii) follows immediately from [15, Propositions 4.4 and 2.6] by noting that Φ λ,−1/2 coincides with γ(λ) in [15].
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 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; C 4 ) → L 2 (∂Ω; C 4 ) 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 : L 2 (∂Ω; C 4 ) → H r (∂Ω; C 4 ) is bounded, and hence C λ ϑ − ϑC λ = ι r (C λ ϑ − ϑC λ ) is compact in L 2 (∂Ω; C 4 ).
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 holds for all x, y ∈ ∂Ω, x = y. Hence, the first estimate in (A.4) is satisfied.
To show the second one, we take x, y, z ∈ ∂Ω with |x − y| ≤ 1 4 (3.14) 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 Using (3.7), we find Using these two observations in (3.18), we conclude 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 important ingredients to prove the selfadjointness of Dirac operators on domains later.  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 with 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 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. We now discuss that the derivatives of the integral operators Φ λ and C λ belong to certain (weak) Schatten-von Neumann ideals. For that, we use the following result on operators with range in the Sobolev space H s (∂Ω; C). Its proof follows word-by-word the one of [11,Proposition 2.4]; hence, we omit it here. With the help of Proposition 3.13, one can show in exactly the same way as in [10, Lemma 4.5] the following result; note that the operators γ(λ) and M (λ) in [10] coincide with Φ λ and C λ , respectively.

A Quasi Boundary Triple for Dirac Operators on Domains
In this section, we introduce a quasi boundary triple which is useful to define self-adjoint Dirac operators on domains via suitable boundary conditions on ∂Ω. Throughout this section, let Ω be a bounded or unbounded domain in R 3 with a compact C 2 -smooth boundary. As before, we denote the normal vector field at ∂Ω pointing outwards of Ω by ν. In the following, the operators will play an important role. The relation P − = I − P + is clear. Furthermore, using the anti-commutation relation (1.8) it is easy to see that P 2 ± = P ± and P + P − = P − P + = 0, that is, P ± are orthogonal projections in L 2 (∂Ω; C 4 ). We also note that (1.8) implies We shall make use of the spaces For convenience of notation, we simply write G Ω := G 0 Ω . As P + is an orthogonal projection in L 2 (∂Ω; C 4 ) the space G Ω is a Hilbert space. Moreover, since ∂Ω is C 2 -smooth, the normal vector field ν is Lipschitz continuous and hence it follows from Lemma A.2 that G s Ω ⊂ H s (∂Ω; C 4 ) for s ∈ [0, 1 2 ]. Furthermore, it is not difficult to check that G s Ω is a closed subspace of H s (∂Ω; C 4 ) for s ∈ [0, 1 2 ]. In the sequel the spaces G s Ω are always equipped with the norm of H s (∂Ω; C 4 ). Next, we define the operator T in L 2 (Ω; C 4 ) by and the mappings Γ 0 , Γ 1 : dom T → G Ω acting as In the following theorem, we show that {G Ω , Γ 0 , Γ 1 } is a quasi boundary triple and that T coincides with the maximal Dirac operator T max from (3.1). Moreover, it turns out that the reference operator T ker Γ 0 is the MIT bag operator studied in Sect. 3.2. Moreover, and hence, in particular, ran(Γ 0 , Proof. First, we have T * min = T max by Lemma 3.1. Furthermore, Lemma 3.1 also implies that the closure of T coincides with T max , as C ∞ (Ω; C 4 ) ⊂ dom T is dense in dom T max equipped with the graph norm. Now we verify that the abstract Green's identity is valid. For this, consider f, g ∈ dom T = H 1 (Ω; C 4 ). Then, (3.3) and the self-adjointness of α · ν yield Using that β is unitary and self-adjoint, we see that the last expression is equal to Since P + is an orthogonal projection in L 2 (∂Ω; C 4 ), we conclude which is the abstract Green's identity. Next, we check the range property (4.7). Clearly, by the definition of Γ 0 and Γ 1 , dom Γ 0 = dom Γ 1 = H 1 (Ω; C 4 ), and by standard properties of the trace map and Lemma A.2 one has ran Γ 0 ⊂ G Ω and choose f ∈ H 1 (Ω; C 4 ) such that f | ∂Ω = ϕ. Since ϕ ∈ G Ω , we have P + ϕ = ϕ and P − ϕ = 0. Hence, and using (4.2), we obtain that is, ϕ ∈ ran(Γ 0 ker Γ 1 ). To prove G 1/2 Ω ⊂ ran(Γ 1 ker Γ 0 ), let ψ ∈ G 1/2 Ω and choose g ∈ H 1 (Ω; C 4 ) such that g| ∂Ω = βψ. Since ψ ∈ G Ω , we have P + ψ = ψ and P − ψ = 0. Hence, using (4.2) we obtain Γ 0 g = P + g| ∂Ω = P + βψ = βP − ψ = 0 and Γ 1 g = P + βg| ∂Ω = P + β 2 ψ = ψ, that is, ψ ∈ ran(Γ 1 ker Γ 0 ). Together with (4.8), we conclude (4.7).
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 In the next proposition, we derive a useful formula for the inverse of M (λ). Let T −MIT be given by (3.5) and let T Ω c −MIT be the Dirac operator acting in L 2 (R 3 \Ω; C 4 ) with the same boundary conditions as T −MIT . Recall that by Proposition 3.4, we have that the latter set is contained in ρ(T MIT ) ∩ ρ(T Ω c MIT ), and that by Proposition 3.12 (ii) the operator − 1 2 β +C λ,1/2 is boundedly invertible in H 1/2 (∂Ω; C 4 ) for any number λ ∈ ρ(T −MIT ) ∩ ρ(T Ω c −MIT ).
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 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).
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
This section contains the main results of this paper. First, in Sect. 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 Sect. 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 Sect. 5.3 we relate the operators A ϑ to Dirac operators B η,τ with singular δ-shell potentials of the form (1.4). 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.

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 [22]. In fact, let Ω ⊂ R 2 be a bounded C 2 -domain.
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.
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 2domain with compact boundary. The proof of (ii) is based on the same argument as the proof of [16, Proposition 3.8].

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 By Proposition 4.2 (ii), the operator γ(λ) * is compact from L 2 (Ω; C 4 ) to G Ω . Furthermore, by Lemma 4.4 the inverse (ϑ − M (λ)) −1 is bounded in G Ω . Since γ(λ) : G Ω → L 2 (Ω; C 4 ) is bounded, we deduce that the right-hand side in (5.4) is compact in L 2 (Ω; C 4 ) and hence the same holds for the left-hand side. Together with Proposition 3.3 (iii) this implies To verify assertion (ii), consider the quadratic form Since A ϑ is a self-adjoint operator, it follows that a is a closed, nonnegative form and by [44, 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 set V = |∇g 1 | 2 + |∇g 2 | 2 and conclude 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.
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  (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 with 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 Propositions 2.4 and 3.2, goes beyond the standard Birman-Schwinger principle in two ways: First, it allows to detect eigenvalues of A ϑ that may be eigenvalues of T MIT at the same time, and second, it enables to use the explicit expression for the values M (λ) of the Weyl function in Proposition 4.2 in terms of integral operators (which we have available only for λ ∈ C\((−∞, −m] ∪ [m, ∞))).

Proposition 5.5.
Let Ω be a bounded C 2 -smooth 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, σ(A ϑ ) = σ disc (A ϑ ) and λ is an eigenvalue of A ϑ if and only if there exists ϕ ∈ G 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., [19, 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. [19, equation (2.8)]. The proof of the following theorem is based on the result of Lemma 3.14 and on the same strategy as in [19] or in [10,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 Propositions 3.6 and 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 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.
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 [64,Chapter 0,Theorem 8.2] or [60,Problem 25], this implies that the wave operators for the scattering pair 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 let ω : ∂Ω → R be 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 Sect. 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.
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.
) 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] , 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. Proof. Assertion (5.29) in (i) and (ii) follows from Lemma 5.13; the remaining assertions on η, τ, ω in (i) and (ii) are easy to check. In fact, to verify (5.29) in item (i) we shall multiply the identity (5.23) by where η 2 − τ 2 = −4 was used. It follows that (5.23) is equivalent to Using the quasi boundary triples {G Ω± , Γ Ω± 0 , Γ Ω± 1 } from Theorem 4.1, we find that (5.23) is equivalent to With ω in (5.30), we now conclude from Lemma 5.13 that f = f + ⊕ f − ∈ dom B η,τ if and only if f ± ∈ dom A

Ω±
[ω] , that is, the identity (5.29) is valid. To show (5.29) in (ii), one argues in the same way as in the proof of Proposition 5.14. The details are left to the reader. Now we return to identities (5.21) and (5.29), and illustrate how one can translate known results for Dirac operators with δ-potentials to self-adjoint Dirac operators on domains studied in this paper. Some preparation is necessary to formulate Theorem 5.17. For a C 2 -domain Ω ⊂ R 3 with compact boundary, we introduce the orthogonal projection and the corresponding embedding ι Ω : L 2 (Ω; C 4 ) → L 2 (R 3 ; C 4 ), ι Ω g = g in Ω, 0 in Ω c .
Moreover, for λ ∈ C\((−∞, −m] ∪ [m, ∞)) we consider the integral operator R λ : L 2 (Ω; C 4 ) → L 2 (Ω; C 4 ), x ∈ 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. [63, Section 1.E]. The resolvent formulas in the next preparatory lemma are now an immediate consequence of [12, 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.
Proof. Throughout the proof, let C be a generic constant, which changes its value several times, and let ϕ ∈ L 2 (∂Ω; C) be fixed. Let us estimate the two terms in Let us now focus on the second term in (A.6). We set The three terms in the sum in the right-hand side of (A.8) are also estimated separately. In order to deal with the first one, let ∈ (0, a − s) be fixed. By