Badreddine Benhellal

SPECTRAL PROPERTIES OF THE DIRAC OPERATOR COUPLED WITH δ-SHELL INTERACTIONS

Keywords: February 18, 2021. 2010 Mathematics Subject Classification. 81Q10, 81V05, 35P15, 58C40 Dirac operators, self-adjoint extensions, shell interactions, critical interaction strength, Quantum confinement, Semmes-Kenig-Toro domains, Uniformly rectifiable domains

Let Ω ⊂ R 3 be an open set, we study the spectral properties of the free Dirac operator H := -iα • ∇ + mβ coupled with the singular potential Vκ = (ǫI 4 + µβ + η(α • N ))δ ∂Ω , where κ = (ǫ, µ, η) ∈ R 3 . In the first instance, Ω can be either a C 2 -bounded domain or a locally deformed half-space. In both cases, the self-adjointness is proved and several spectral properties are given.

 to the case of a locally deformed half-space, by giving a complete description of the essential spectrum of H + Vκ, for the so-called critical combinations of coupling constants. In the second part of the paper, the case of bounded rough domains is investigated. Namely, in the non-critical case and under the assumption that Ω has a VMO normal, we show that H + Vκ is still self-adjoint and preserves almost all of its spectral properties. More generally, under certain assumptions about the sign or the size of the coupling constants, we are able to show the self-adjointness of the coupling H + (ǫI 4 + µβ)δ ∂Ω , when Ω is bounded uniformly rectifiable. Moreover, if ǫ 2 -µ 2 = -4, we then show that ∂Ω is impenetrable. In particular, if Ω is Lipschitz, we then recover the same spectral properties as in the VMO case. In addition, we establish a characterization of regular Semmes-Kenig-Toro domains via the compactness of the anticommutator between (α • N ) and the Cauchy operator associated to the free Dirac operator.

Finally, we study the coupling Hυ = H + iυβ(α • N )δ ∂Ω . In particular, if Ω is a bounded C 2 domain, then we show that H ±2 is essentially self-adjoint and generates confinement.

Intoduction

In this paper we investigate in R 3 the self-adjointness character and the spectral properties of the coupling H ǫ,µ + V η,υ , where H ǫ,µ is the Dirac operator with electrostatic and Lorentz scalar δ-shell interactions, formally written as: H ǫ,µ := H + V ǫ,µ = H + (ǫI 4 + µβ)δ ∂Ω , λ, µ ∈ R, (1.1) and the potential V η,υ is given by

V η,υ = (η(α • N ) + iυβ (α • N )) δ ∂Ω , η, υ ∈ R, (1.2)
here H is the free Dirac operator (see section 2 for notations), ∂Ω is the boundary of an open set Ω of R 3 , N is the unit normal vector field at ∂Ω which points outwards of Ω and the δ-potential is the Dirac distribution supported on ∂Ω. In relativistic quantum mechanics, the Dirac Hamiltonian H ǫ,µ + V η,υ describes the dynamics of the massive relativistic particles of spin-1/2 in the external potential V ǫ,µ + V η,υ . From this physical point of view, the singular interactions given by the coupling constants ǫ, µ, η and υ are called respectively electrostatic, Lorentz scalar, magnetic and anomalous magnetic potential (we refer to [START_REF] Thaller | The Dirac equation[END_REF] for more information on the classification of external fields). The surface ∂Ω supporting the interactions is called a shell.

Recently, Dirac operators with δ-shell interactions have been studied extensively. Namely, the coupling H with the electrostatic and the Lorentz scalar δ-shell interactions (i.e H ǫ,µ ); we refer to the survey [START_REF] Ourmiéres-Bonafos | Dirac operators and shell interactions: a survey[END_REF] for a review on the topic. To our knowledge, the spectral study of the Dirac operator H ǫ,µ goes back to the papers [START_REF] Dittrich | Dirac operators with a spherically δ-shell interaction[END_REF] and [START_REF] Dominguez-Adame | Exact solutions of the Dirac equation with surface delta interactions[END_REF], where the authors studied the spherical case (i.e ∂Ω is a sphere). Moreover, in [START_REF] Dittrich | Dirac operators with a spherically δ-shell interaction[END_REF] the authors point out that under the assumption ǫ 2µ 2 = -4, the shell becomes impenetrable. Physically, this means that at the time t = 0, if the particle in consideration (an electron for example) is in the region Ω (respectively in R 3 \ Ω), then during the evolution in time, it cannot cross the surface ∂Ω to join the region R 3 \ Ω (respectively Ω ) for all t > 0. Mathematically, this means that the Dirac operator in consideration decouples into a direct sum of two Dirac operators acting respectively on Ω and R 3 \ Ω with appropriate boundary conditions. In particular, when ǫ = 0, this phenomenon has been known to physicists since the 1970's (cf. [START_REF] Chodos | New extended model of hadrons[END_REF] and [START_REF] Johnson | The MIT bag model[END_REF] for example); and its mathematical model described by the Dirac operator with MIT boundary conditions has been the subject of several mathematical papers (we refer to the recent paper [START_REF] Behrndt | Self-Adjoint Dirac Operators on Domains in R 3[END_REF] as well as the references cited there). All these physical motivations made the mathematical study of Dirac operators with δ-shell interactions a very important subject. However, unlike the non-relativistic counterpart (i.e.

Schrödinger operators with δ-shell interactions) the study of relativistic δ-interactions has known a long period of silence. Indeed, apart from [START_REF] Shabani | Exactly solvable models of relativistic δ-sphere interactions in quantum mechanics[END_REF] where the authors studied the scattering theory and the non-relativistic limit of H ǫ,µ (in the spherical case), the spectral study of H ǫ,µ has been forgotten for two decades. Since then, it has been relaunched in [START_REF] Arrizabalaga | Shell interactions for Dirac operators[END_REF], where the authors developed a new technique to characterize the self-adjointness of the free Dirac operator coupled with a measure-valued potential. As a particular case, they dealt with the pure electrostatic δ-shell interactions (i.e µ = 0) supported on the boundary of a bounded regular domain, and they proved that the perturbed operator is selfadjoint for all ǫ = ±2. The same authors continue the spectral study of the electrostatic case; for instance, the existence of point spectrum and related problems; see [START_REF] Arrizabalaga | Shell interactions for Dirac operators: on the point spectrum and the confinement[END_REF] and [START_REF] Arrizabalaga | An isoperimetric-type inequality for electrostatic shell interactions for Dirac operators[END_REF]. In [START_REF] Arrizabalaga | Shell interactions for Dirac operators: on the point spectrum and the confinement[END_REF] they add the scalar Lorentz interaction and they show that under the condition ǫ 2µ 2 = -4, H ǫ,µ still generates the phenomenon of confinement.

Subsequently, the concept of quasi-boundary triples and their Weyl functions were used in [START_REF] Behrndt | On the spectral properties of Dirac operators with electrostatic δ-shell interaction[END_REF] to study the Dirac operators with electrostatic δ-shell interactions. In this paper, the authors prove the self-adjointness for all ǫ = ±2, and investigate several spectral properties, adding the scattering theory and asymptotic properties of the model. In all the above papers, the case ǫ = ±2 (known as the critical interaction strengths) has not been considered. This gap has been covered in [START_REF] Ourmères-Bonafos | A strategy for self-adjointness of Dirac operators: applications to the MIT bag model and δ-shell interactions[END_REF], then in [START_REF] Behrndt | On Dirac operators with electrostatic δ-shell interactions of critical strength[END_REF] with different approaches. Indeed, in this particular case, it turns out that the Dirac operator with electrostatic δ-shell interactions is essentially self-adjoint, and functions in the domain of the closure are less regular comparing to the non critical case. Moreover, the authors in [START_REF] Behrndt | On Dirac operators with electrostatic δ-shell interactions of critical strength[END_REF] show that if ∂Ω contains a flat part, then the point 0 belongs to the essential spectrum of H ±2,0 . Similar phenomenon appears when we study the Dirac operator H ǫ,µ . In fact, in this case, the critical combinations of coupling constants are ǫ 2µ 2 = 4; see [START_REF] Behrndt | On Dirac operators in R 3 with electrostatic and Lorentz scalar δ-shell interactions[END_REF] for example. The self-adjointness in the critical case ǫ 2µ 2 = 4 was proved for the two dimensional analogue of H ǫ,µ in [START_REF] Behrndt | Two-dimensional Dirac operators with singular interactions supported on closed curves[END_REF], where the authors considered δ-interactions supported on a smooth closed curve. Furthermore, by making use of complex analysis and periodic pseudo-differential operators techniques, they show that

Sp ess (H ǫ,µ ) = -∞, -m ∪ - mµ ǫ ∪ m, +∞ . (1.3)
Of course, such techniques are no longer available in the three dimensional case. Nevertheless, at this stage, one may ask the following question: (Q1) In the three dimensional setting, when ǫ 2µ 2 = 4, does (1.3) hold true?

Another issue that arises when we study such a coupling problems is the regularity of the surface ∂Ω. In fact, to our knowledge, all the works which deal with Dirac operators coupled with δ-shell interactions have been done for Ω at least C 2 -bounded domain (except in [START_REF] Pizzichillo | Self-Adjointness of two dimensional Dirac operators on corner domains[END_REF], where the particular case of two dimensional Dirac operator with pure Lorentz scalar δ-interactions was studied, with ∂Ω a closed curve with finitely many corners). The following question has already been asked in [START_REF] Ourmiéres-Bonafos | Dirac operators and shell interactions: a survey[END_REF]:

(Q2) Until what extent the results on self-adjointness of H ǫ,µ also hold for Lipschitz domains ?

The main objective of the current manuscript is to study questions (Q1) and (Q2) for the coupling H ǫ,µ + V η,0 (i.e υ = 0). Unlike most existing works, instead of treating the δ-interactions as a transmission problem, in this paper we made the choice to follow the strategy introduced in [START_REF] Arrizabalaga | Shell interactions for Dirac operators[END_REF].

Let us present the context we are considering and summarize the main results of our work. We shall assume that the open set Ω satisfies (for instance) one of the following hypotheses:

(1) Ω is a C 2 -bounded domain.

(2) Ω := Ω ν := {(x, t) ∈ R 2 × R : t > νφ(x)}, where ν ∈ R and φ : R 2 → R is a C 2 -smooth, compactly supported function.

We define the Dirac operators H κ := H ǫ,µ + V η,0 , on the domain

dom(H κ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ L 2 (∂Ω) 4 , u |∂Ω = -Λ + [g] , κ := (ǫ, µ, η) ∈ R 3 ,
where Φ is an appropriate fundamental solution of the unperturbed operator H, and Λ ± are bounded linear operators acting on L 2 (∂Ω) 4 (see Notation 2.1). We mention that the operator Λ ± appears in several works when the quasi-boundary triples theory is used to study the Dirac operator H ǫ,µ , see [START_REF] Behrndt | On Dirac operators with electrostatic δ-shell interactions of critical strength[END_REF]Lemma 5.4] and [START_REF] Behrndt | Two-dimensional Dirac operators with singular interactions supported on closed curves[END_REF]Proposition 4.3] for example. We point out that the consideration of the second assumption is motivated by [START_REF] Exner | Asymptotics of the bound state induced by δ-interaction supported on a weakly deformed plane[END_REF], where the Schrödinger operator with δ-shell interaction was considered.

As a first step of the current paper, we study the self-adjointness character of H κ , when Ω satisfies the assumption (1) or [START_REF] Arrizabalaga | Shell interactions for Dirac operators[END_REF]. We begin by proving that H κ is self-adjoint when ǫ 2µ 2η 2 = 4 (i.e in the non-critical case), and we show that dom(H κ ) ⊂ H 1 (R 3 \ ∂Ω) 4 , which means that functions in dom(H κ ) have a Sobolev regularity; cf. Theorem 3.1. To prove this result we develop a strategy very close to [START_REF] Ourmères-Bonafos | A strategy for self-adjointness of Dirac operators: applications to the MIT bag model and δ-shell interactions[END_REF], it is based essentially on the fact that the anticommutators of Cauchy operator C ∂Ω (see (2.17) for the definition) with β or with (α • N ) have a regularizing effect. Indeed, as it was observed in several works (see [START_REF] Arrizabalaga | Shell interactions for Dirac operators: on the point spectrum and the confinement[END_REF] for example), the operator Λ ∓ Λ ± involves the above anticommutators and it turns out that in the non-critical case, the regularization effect of these anticommutator pushes Λ + to regularize the functions in dom(H κ ) to have the H 1 -Sobolev regularity. When ǫ 2µ 2η 2 = 4, which is actually the critical case, we show that H κ is essentially self-adjoint (i.e H κ is self-adjoint).

In addition, we point out the relation between the self-adjointness of H κ and the operator Λ + , which is essentially the idea behind the concept of quasi boundary triples theory (see Subsection 3.2).

As a second step, we turn to the spectral study of H κ . We focus on the case where Ω satisfies the second assumption and we show several spectral properties of H κ . Namely, in the non-critical case, we prove that Sp ess (H κ ) = (-∞, -m] ∪ [m, +∞), moreover the discrete spectrum of H κ in the gap (-m, m) is finite. In the critical case, we give a complete characterization of the essential spectrum of H κ when Ω satisfies the second assumption. More precisely, we show that

Sp ess (H κ ) = -∞, -m ∪ - mµ ǫ ∪ m, +∞ ,
which answers positively to the question (Q1), hence generalizing the result of [START_REF] Behrndt | Two-dimensional Dirac operators with singular interactions supported on closed curves[END_REF] to this kind of surfaces. The proof is based on the use of compactness and localization arguments. We remark that even after adding the perturbation by the potential V η,0 , the point which appears in the gap remains the same (see the discussion after Theorem 4.2 for more details). All these results will be proven using an adapted Birman-Schwinger principle, a Krein-type resolvent formula and compactness arguments.

We mention that the above results are well known when η = 0, the interaction is not critical and Ω satisfies the first assumption; cf. [START_REF] Behrndt | On the spectral properties of Dirac operators with electrostatic δ-shell interaction[END_REF] for example. However, when Ω satisfies the second assumption, the situation is more delicate, in particular the use of compactness arguments.

In the cases cited above, the C 2 -regularity is essential to use our technique, especially when ∂Ω is unbounded, and the combination of the coupling constants is critical. Nevertheless, in the noncritical case, if Ω is bounded then one can do more. In fact, one of the reasons for choosing to work with the strategy of [START_REF] Arrizabalaga | Shell interactions for Dirac operators[END_REF] is that in the non-critical case, the fact that Λ + is Fredholm implies the selfadjointness of H κ (see [START_REF] Arrizabalaga | Shell interactions for Dirac operators[END_REF]Theorem 2.11]). Moreover, the compactness of the anticommutators implies that Λ + is a Fredholm operator. Fredholm's character and (or) invertibility of the boundary integral operator is one of the important tools for the analysis of strongly elliptic boundary value problems;

such techniques have been exploited since a long time to solve for example the Dirichlet or Neumann problem on Lipschitz domains; cf. [START_REF] Jerison | The Neumann problem on Lipschitz domains[END_REF], [START_REF] Verchota | Layer potentials and regularity for the Dirichlet problem for Laplace's equation[END_REF] and [START_REF] Dahlberg | Hardy spaces and the Neumann problem in L p for Laplace's equation in Lipschitz domains[END_REF]. As we will see later (see Lemma 3.1), {β, C ∂Ω } is nothing else than the trace of the matrix valued Single-layer potential, which is therefore a compact operator on L 2 (∂Ω) 4 , even if Ω is a Lipschitz domain. So we can naturally ask the following question:

(Q3) Given a bounded domain Ω, what is the necessary regularity on ∂Ω so that the anticommutator {α • N , C ∂Ω } gives rise to a compact operator on L 2 (∂Ω) 4 ?

One of the main results of this article is the answer to this question, see Theorem 5.4. Looking closely at the anticommutator {α • N , C ∂Ω }, we observe that it involves a matrix version of the principal value of the harmonic double-layer K, its adjoint K * and the commutators [N k , R j ], where R j are the Riesz transforms. Hence the situation is more clear. In fact, from the harmonic analysis and geometric measure theory point of view, it is shown that the boundedness of Riesz transforms characterizes the uniform rectifiability of ∂Ω; cf. [START_REF] Nazarov | On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1[END_REF], for example. In addition, functional analytic properties of the Riesz transforms (such as the identity 3 j=1 R 2 j = -I ) and the analogue version of the strongly singular part of (α • N )C ∂Ω in the Clifford algebra Cl 3 , i.e. the Cauchy-Clifford operator (especially its self-adjointness and compactness character) are strongly related to the regularity and geometric properties of the domain Ω, for more details we refer to [START_REF] Hofmann | Hardy spaces, singular integrals and the geometry of Euclidean domains of locally finite perimeter[END_REF] and [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]. The most important fact which allow us to establish some results for the Lipschitz class, is that the compactness of K, K * and [N j , R j ] characterizes the class of regular SKT (Semmes-Kenig-Toro) domains; see [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]. However, regular SKT domains are not necessarily Lipschitz domains and vice versa. So, to stay in the context of compact Lipschitz domains, we shall suppose that Ω satisfies the following property:

(3) Ω is a bounded Lipschitz domain with normal N ∈ VMO(∂Ω, dS) 3 .

This assumption characterizes the intersection of the Lipschitz class with the regular SKT class. Moreover, the hypothesis (3) is the answer to the question (Q3). In fact, we prove the following :

Ω satisfies the assumption (3) ⇐⇒ {α • N , C ∂Ω } is compact in L 2 (∂Ω) 4 .
(1.4) see Theorem 5.4. Once we have established that and hence proved the compactness of {α•N , C ∂Ω }, the self-adjointness of H κ will be an easy consequence of [START_REF] Arrizabalaga | Shell interactions for Dirac operators[END_REF]Theorem 2.11]. Moreover, we prove almost all the spectral properties as in C 2 -smooth case. Another geometric type result that we establish in this article, is a characterization of the class of regular SKT domains via the compactness of the 4 , see Proposition 5.2. More precisely, using the material provided in [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF], we show that if Ω is a two-sided NTA domain with a compact Ahlfors regular boundary, then it holds that

anticommutator {α • N , C ∂Ω } in L 2 (∂Ω)
Ω is a regular SKT domain ⇐⇒ {α • N , C ∂Ω } is compact in L 2 (∂Ω) 4 . (1.5)
At this stage, beyond the two classes of domains which are characterized by (1.4) and (1.5), the compactness arguments mentioned previously are no longer valid. So, in order to go further in our study we change the strategy and we turn to the invertibility arguments, which are rather valid in a more general context. Indeed, we investigate the case of bounded uniformly rectifiable domains (see Section 5 for the definitions). In one direction, making the assumption that 0

< |ǫ 2 -µ 2 | < 1/ C ∂Ω 2 L 2 (∂Ω) 4 →L 2 (∂Ω) 4 ,
we then show that H ǫ,µ is self-adjoint, cf. Theorem 5.6. In another direction, assuming that

µ 2 > ǫ 2 or 16 W 2 L 2 (Σ) 2 →L 2 (Σ) 2 < ǫ 2 -µ 2 < 1/ W 2 L 2 (Σ) 2 →L 2 (Σ)
2 (here W is the sroungly singular part of C ∂Ω defined in (5.18)), we also prove the self-adjointness of H ǫ,µ . In particular, if Ω is Lipschitz, we then recover the same spectral properties as in the case of the assumption (3). Moreover, we show that H ǫ,µ generates confinement when ǫ 2µ 2 = -4, cf. Theorem 5.7 and Proposition 5.4.

Having established the above results, and in order to enrich the knowledge on the connections between the smoothness of Ω and the Sobolev regularity of functions in dom(H κ ), we consider the class of Hölder's domains C 1,γ , with γ ∈ (0, 1), and we prove that the functions in dom(H κ ) have the 4 . Moreover, the technique developed before for the C 2 -smooth surfaces remains valid to prove such a result (cf. Remark 5.5).

H s -Sobolev regularity, with s > 1/2. In particular, we show that if γ ∈ (1/2, 1), then dom(H κ ) ⊂ H 1 (R 3 \ ∂Ω)
The last part of this paper is devoted to the spectral study of

H υ := H + iυβ(α • N )δ ∂Ω , the Dirac
operator with anomalous magnetic δ-interactions. We mention that while preparing this manuscript, it turns out that the authors of the paper [START_REF] Cassano | General δ-Shell Interactions for the two-dimensional Dirac Operator: Self-adjointness and Approximation[END_REF] (which will appear soon) worked on the two-dimensional analog of this problem at the same time, and our results intersect on this point (see Section 6 for more details). Assuming that Ω satisfies the assumption (1), one of the most important properties that we show for this operator is that, in the critical case υ 2 = 4, H υ is essentially self-adjoint and it decouples in a direct sum of two Dirac operators acting respectively on Ω and R 3 \ Ω, with boundary conditions in H -1/2 (∂Ω). Thus, H ±2 generates confinement and hence ∂Ω becomes impenetrable. Moreover, the inner part of H ±2 which acts on Ω coincide with so-called Dirac operator with zig-zag boundary condition, see Section 6.

Organisation of the paper. The structure of the paper is as follows. In the second section, we set up the necessary notations and recall the relevant material from [START_REF] Arrizabalaga | Shell interactions for Dirac operators[END_REF]. In Section 3, we study the self-adjointness of H κ , when ∂Ω satisfies the first and the second assumption. Section 4 is devoted to the spectral study of H κ . We focus namely on the case where Ω is a locally deformed half space and we give a complete description of the essential spectrum of H κ , for the critical combinations of coupling constants. Section 5 is the heart of the paper and it contains our most important contributions.

Here we consider the Dirac operator H κ , where Ω is a bounded, uniformly rectifiable domain. First, we recall some definitions related to the class of regular SKT domains. Then, in Subsection 5.1, we investigate the case of a domain Ω satisfying assumptions (3). After this, the general case of uniformly rectifiable domains is considered in Subsection 5.2, for the Dirac operator H ǫ,µ . As a last step, the class of Hölder's domains C 1,γ is considered in Subsection 5.3. Finally, in Section 6, we study the spectral properties of the Dirac operator H υ , for all possible combinations of interaction strengths.

Notations and Preliminaries

We consider a surface Σ ⊂ R 3 dividing the space into two regions Ω ± . More precisely, we assume that Σ satisfies one of the hypotheses:

(H1) Σ = ∂Ω + with Ω + a C 2 -bounded domain. (H2) Σ := Σ ν := {(x 1 , x 2 , x 3 ) ∈ R 3 : x 3 = νφ(x 1 , x 2 )}, where ν ∈ R + and φ : R 2 → R is a C 2 -
smooth, compactly supported function. We denote by L φ the Lipschitz constant of φ and by F we denote the flat part of Σ ν i.e.

F := {x = (x 1 , x 2 , νφ(x 1 , x 2 )) ∈ Σ ν : (x 1 , x 2 ) / ∈ supp(φ)}. (2.1)
We parameterize Σ ν by the mapping (2.2) τ : R 2 -→ R 3

x -→ (x, νφ(x))

For x = (x, νφ(x)) ∈ Σ ν , we express the surface mesure on Σ ν via the formula dS(x) = J ν (x)dx, where J ν is the Jacobian given by

J ν (x) = 1 + ν 2 |∇φ(x)| 2 . (2.3)
Throughout the paper, we shall work on the Hilbert space L 2 (R d ) 4 (respectivelly, L 2 (Ω ± ) 4 ) with respect to the Lebesgue measure. D(Ω ± ) 4 denotes the usual space of indefinitely differentiable functions with compact support, and D ′ (Ω ± ) 4 is the space of distributions defined as the dual space of D(Ω ± ) 4 . We define the unitary Fourier-Plancherel operator

F : L 2 (R d ) 4 → L 2 (R d ) 4 as follows F [u](ξ) = 1 (2π) d/2 R d e -ix•ξ u(x)dx, ∀ξ ∈ R d , (2.4)
and by F -1 we denote the inverse Fourier-Plancherel operator 4 , given by

F -1 : L 2 (R d ) 4 → L 2 (R d )
F -1 [u](x) = 1 (2π) d/2 R d e iξ•x u(ξ)dξ, ∀x ∈ R d . (2.5) Given x ∈ R d-1
, by F x we abbreviate the partial Fourier-Plancherel operator on the variable x. Given s ∈ [-1, 1], we denote by H s (R d ) 4 the Sobolev space of order s, defined as

H s (R d ) 4 := {u ∈ L 2 (R d ) 4 : R d (1 + |ξ| 2 ) s |F [u](ξ)| 2 dξ < ∞}. (2.6)
The Sobolev space H 1 (Ω ± ) 4 is defined as follows: 4 we denote the usual L 2 -space over Σ. Given s ∈ [0, 1], if Σ satisfies (H2), we then define the Sobolev spaces H s (Σ) 4 in terms of the Sobolev spaces over R 2 as usual. That is given g ∈ L 2 (Σ) 4 , we define g φ (x) = g(x, νφ(x)), for x ∈ R 2 . Then

H 1 (Ω ± ) 4 = {ϕ ± ∈ L 2 (Ω ± ) 4 : there exists φ± ∈ H 1 (R 3 ) 4 such that φ± | Ω± = ϕ ± }. (2.7) By L 2 (Σ, dS) 4 := L 2 (Σ)
H s (Σ) 4 := {g ∈ L 2 (Σ) 4 : g φ ∈ H s (R 2 ) 4 }, for all s ∈ [0, 1], (2.8)
and then define H -s (Σ) 4 to be the completion of L 2 (Σ) 4 with following norm:

g H -s (Σ) 4 := g φ J ν H -s (R 2 ) 4 , for all s ∈ [0, 1]. (2.9)
Recall that H -s (Σ) 4 is a realisation of the dual space of H s (Σ) 4 ; see [START_REF] Mclean | Strongly Elliptic Systems and Boundary Integral Equations[END_REF] for example. Now, if Σ satisfies (H1), we then define the Sobolev spaces H s (Σ) 4 using local coordinates representation on the surface Σ; see [START_REF] Mclean | Strongly Elliptic Systems and Boundary Integral Equations[END_REF]. By t Σ : 4 we denote the classical trace operator. For a function u ∈ H 1 (R 3 ) 4 , with a slight abuse of terminology we will refer to t Σ u as the restriction of u on Σ.

H 1 (Ω ± ) 4 → H 1/2 (Σ)
Let x ∈ Σ and a > 0, denote the nontangential approach regions of opening a at the point x by

Γ Ω± a (x) = {y ∈ Ω ± : |x -y| < (1 + a)dist(y, Σ)}. (2.10) We fix a > 0 large enough such that x ∈ Γ Ω± a (x) for all x ∈ Σ. If x ∈ Σ, then U ± (x) := lim Γ Ω ± a (x)∋y-→ nt x U (y) (2.11) is the nontangential limit of U with respect to Ω ± at x. If a > 0 is fixed, we shall write Γ Ω± (x) instead of Γ Ω± a (x).
Let α = (α 1 , α 2 , α 3 ) and β be the 4 × 4 Hermitian and unitary matrices given by

α k = 0 σ k σ k 0 for k = 1, 2, 3 β = I 2 0 0 -I 2 , (2.12)
where σ = (σ 1 , σ 2 , σ 3 ) are the Pauli matrices defined by

σ 1 = 0 1 1 0 , σ 2 = 0 -i i 0 , σ 3 = 1 0 0 -1 . (2.13)
We denote by N and δ Σ the unit normal vector field at Σ which points outwards of Ω + and the Dirac distribution supported on Σ respectively. Given m > 0, we consider the Dirac operator [42, subsection 1.4]) and its spectrum is given by

H κ = H + V κ = -iα • ∇ + mβ + (ǫI 4 + µβ + η(α • N ))δ Σ , κ := (ǫ, µ, η) ∈ R 3 . (2.14) in the Hilbert space L 2 (R 3 ) 4 , where H = -iα • ∇ + mβ is the free Dirac operator defined on H 1 (R 3 ) 4 . It is well known that (H, H 1 (R 3 ) 4 ) is self-adjoint (see
Sp(H) = Sp ess (H) = (-∞, -m] ∪ [m, +∞).
The rest of this section will be devoted to give a first definition of the Hamiltonian H κ . For this and for the convenience of the reader, we recall the relevant material from [START_REF] Arrizabalaga | Shell interactions for Dirac operators[END_REF] (without detailed proofs), thus making our exposition self-contained.

φ z (x) = e i √ z 2 -m 2 |x| 4π|x| z + mβ + (1 -i z 2 -m 2 |x|)iα • x |x| 2 , for all x ∈ R 3 \ {0}, (2.15)
see for example [START_REF] Thaller | The Dirac equation[END_REF]Section 1.E]. Next, we define the following operators 4 , for all g ∈ L 2 (Σ) 4 . In particular, Φ z gives rise to a bounded operator from H 1/2 (Σ) 4 onto 4 , we set

Φ z : L 2 (Σ) 4 -→ L 2 (R 3 ) 4 g -→ Φ z [g](x) = Σ φ z (x -y)g(y)dS(y), for all x ∈ R 3 \ Σ, (2.16) then, Φ z : L 2 (Σ) 4 -→ L 2 (R 3 ) 4 is a bounded operator. Furthermore, (H -z)Φ z [g] = 0 holds in D ′ (Ω ± )
H 1 (R 3 \ Σ) 4 ; cf. [8, Proposition 4.2]. Given x ∈ Σ and g ∈ L 2 (Σ)
C z Σ [g](x) = lim ρց0 |x-y|>ρ φ z (x -y)g(y)dS(y) and C z ± [g](x) = lim Γ Ω ± (x)∋y-→ nt x Φ z [g](y), (2.17)
Then, we have the following lemma.

Lemma 2.1. Let C z Σ and C z ± be as above. Then C z Σ [g](x) and C z ± [g](x) exist for dS-a.e. x ∈ Σ, and 4 are linear bounded operators. Furthermore, the following hold:

C z Σ , C z ± : L 2 (Σ) 4 → L 2 (Σ)
(i) C z ± = ∓ i 2 (α • N ) + C z Σ ,(Plemelj-Sokhotski jump formula). (ii) (C z Σ (α • N )) 2 = -1 4 I 4 . In particular, C z Σ 1 2 .
Proof. If Σ satisfies (H1), then the proof is analogous to the one of [START_REF] Arrizabalaga | Shell interactions for Dirac operators: on the point spectrum and the confinement[END_REF]Lemma 2.2], where the authors use essentially the Green's theorem and the following well known result on the trace of derivatives of a single-layer potential. Indeed, given g ∈ L 2 (Σ), then for dS-a.e. x ∈ Σ, we have

lim Ω±∋y-→ nt x y -w 4π|y -w| 3 g(w)dS(w) = ∓ 1 2 g(x)N (x) + lim ρց0 |x-w|>ρ
xw 4π|x -w| 3 g(w)dS(w). (2.18) Note that this result is also true if Σ satisfies (H2), see [START_REF] Medková | The Laplace equation[END_REF]Theorem 5.4.7] for example. Thus, one can adapt the proof of [START_REF] Arrizabalaga | Shell interactions for Dirac operators[END_REF]Lemma 3.3] in this case. The detailed verification of items (i) and (ii) being left to the reader. Remark 2.1. Note that in the same setting, Lemma 2.1 still holds true if for example Σ is a compact Lipschitz surface or the graph of a Lipschitz function φ : R 2 → R; see [START_REF] Axelsson | Harmonic analysis of Dirac operators on Lipschitz domains[END_REF] and [2, Remark 3.14].

Moreover, since (Φ z ) * = (Hz) -1 ⇂ Σ , by duality arguments, it follows that the operator Φ z gives rise to a bounded operator from L 2 (Σ) 4 onto H 1/2 (R 3 \ Σ) 4 ; cf. [9, Subsection 3.3]. Hence, the non-tangential limit in Lemma 2.1(i) coincides with the trace operator for all data in H 1/2 (Σ) 4 . 4 , for all z ∈ (-m, m). 4 . This proves the first statement. The second statement is a direct consequence of the fact that φ z (yx) = φ z (xy).

Corollary 2.1. Let z ∈ C \ ((-∞, -m] ∪ [m, ∞)). Then, the operator C z Σ is bounded from H 1/2 (Σ) 4 onto itself. Moreover, it holds that (C z Σ ) * = C z Σ in L 2 (Σ) 4 . In particular, C z Σ is a self-adjoint operator in L 2 (Σ)
Proof. Given g ∈ H 1/2 (Σ) 4 . Since Φ z [g] ∈ H 1 (R 3 \ Σ) 4 , it follows that C z ± [g] ∈ H 1/2 (Σ) 4 . Thus, from Lemma 2.1 (i) we deduce that 2C z Σ [g] = (C z + + C z -)[g] ∈ H 1/2 (Σ)
Notation 2.1. Let κ = (ǫ, µ, η) ∈ R 3 such that sgn(κ) := ǫ 2µ 2η 2 = 0. We define the operators Λ z ± as follows:

Λ z ± = 1 sgn(κ) (ǫI 4 ∓ (µβ + η(α • N ))) ± C z Σ , ∀z ∈ C \ ((-∞, -m] ∪ [m, ∞)) . (2.19) Since (α • N ) is C 1 -smooth
and symmetric, it easily follows that Λ z ± are bounded (and self-adjoint for z ∈ (-m, m)) from L 2 (Σ) 4 onto itself, and bounded from H 1/2 (Σ) 4 onto itself.

In the sequel, we shall write Φ, C Σ , C ± and Λ ± instead of Φ 0 , C 0 Σ , C 0 ± and Λ 0 ± , respectively. Now we are in position to give the first definition of the Hamiltonian with δ-interactions supported on Σ, the main object of the present paper. Definition 2.1. Let κ = (ǫ, µ, η) ∈ R 3 such that sgn(κ) = 0. The Dirac operator coupled with a combination of electrostatic, Lorentz scalar and normal vector field δ-shell interactions of strength ǫ, µ and η respectively, is the operator 4 and defined on the domain

H κ = H + V κ , acting in L 2 (R 3 )
dom(H κ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ L 2 (Σ) 4 , t Σ u = -Λ + [g] , (2.20)
where

V κ (ϕ) = 1 2 (ǫI 4 + µβ + η(α • N )))(ϕ + + ϕ -)δ Σ , (2.21) with ϕ ± = t Σ u + C ± [g].
Hence, H κ acts in the sens of distributions as H κ (ϕ) = Hu, for all ϕ = u + Φ[g] ∈ dom(H κ ).

Self-adjointness of H κ

In this section, we study the self-adjointness of the Dirac operator H κ . In our setting, it turns out that the special value sgn(κ) = 4 plays a critical role in the analysis of the spectral properties of H κ . Before stating the main result of this part, some notations and auxiliary results are needed. Proposition 3.1. ( [START_REF] Ourmères-Bonafos | A strategy for self-adjointness of Dirac operators: applications to the MIT bag model and δ-shell interactions[END_REF], [START_REF] Behrndt | On Dirac operators with electrostatic δ-shell interactions of critical strength[END_REF]) Let Φ z and C z Σ be as in Lemma 2.1. Then, the following hold true: (i) The trace operator t Σ (which until now was defined on H 1 (Ω ± ) 4 ) has a unique extension to a bounded linear operator from L 2 (Ω ± ) 4 to H -1/2 (Σ) 4 .

(ii) The operator Φ z admits a continuous extension from H -1/2 (Σ) 4 to L 2 (R 3 ) 4 , which we still denote Φ z .

(iii) The operator C z Σ admits a continuous extension C z Σ : H -1/2 (Σ) 4 → H -1/2 (Σ) 4 . Moreover, we have

C z ± [h] = (∓ i 2 (α • N ) + C z Σ )[h], C z Σ [h], g H -1/2 ,H 1/2 = h, C z Σ [g] H -1/2 ,H 1/2 , (3.1)
for any g ∈ H 1/2 (Σ) 4 and h ∈ H -1/2 (Σ) 4 .

Proof. Item (i) is the classical trace theorem, see [START_REF] Mclean | Strongly Elliptic Systems and Boundary Integral Equations[END_REF]Theorem 3.38] for example. (ii) can be proved as much the same way as in [START_REF] Ourmères-Bonafos | A strategy for self-adjointness of Dirac operators: applications to the MIT bag model and δ-shell interactions[END_REF]Theorem 2.2], see also [START_REF] Behrndt | On Dirac operators with electrostatic δ-shell interactions of critical strength[END_REF]Proposition 4.4]. Since (C z Σ ) * = C z Σ , and 4 onto itself, by duality we get the first statement of (iii). Finally, (3.1) follows by density arguments, for a detailed proof we refer to [START_REF] Behrndt | Two-dimensional Dirac operators with singular interactions supported on closed curves[END_REF]Proposition 3.5] and [START_REF] Behrndt | On Dirac operators with electrostatic δ-shell interactions of critical strength[END_REF]Proposition 4.4 (ii)].

C z Σ is bounded from H 1/2 (Σ)
In the following, we denote by Λz ± the continuous extension of Λ z ± defined from H -1/2 (Σ) 4 onto itself. Now, we can state the first main theorem of the paper, the remainder of this part will be devoted to the proof of this result. Theorem 3.1. Let H κ be as in the definition 2.1. Then, the following hold true:

(i) If sgn(κ) = 4, then H κ is self-adjoint and we have dom(H κ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ H 1/2 (Σ) 4 , t Σ u = -Λ + [g] . (3.2) (ii) If sgn(κ) = 4
, then H κ is essentially self-adjoint and we have

dom(H κ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ H -1/2 (Σ) 4 , t Σ u = -Λ+ [g] . (3.3)
Proposition 3.2. Let H κ be as in the definition 2.1. Then, H κ is closable.

Proof. As any symmetric operator on a Hilbert space with dense domain of definition always admits a closure, to prove the proposition it is suffices to show the following: 4 . Thus (i) follows from this and the fact that

(i) dom(H κ ) is dense in L 2 (R 3 ) 4 . (ii) H κ is symmetric on dom(H κ ). First, observe that C ∞ 0 (R 3 \ Σ) 4 ⊂ dom(H κ ) ⊂ L 2 (R 3 )
C ∞ 0 (R 3 \ Σ) 4 is a dense subspace of L 2 (R 3 ) 4 . Now we prove (ii), let ϕ, ψ ∈ dom(H κ ) with ϕ = u + Φ[g] and ψ = v + Φ[h]. Then, we have H κ ϕ, ψ L 2 (R 3 ) 4 -ϕ, H κ ψ L 2 (R 3 ) 4 = Hu, v + Φ[h] L 2 (R 3 ) 4 -u + Φ[g], Hv L 2 (R 3 ) 4 = Hu, Φ[h] L 2 (R 3 ) 4 -Φ[g], Hv L 2 (R 3 ) 4 = t Σ u, h L 2 (Σ) 4 + g, t Σν v L 2 (Σ) 4 .

Using the conditions t

Σ u = -Λ + [g] and t Σ v = -Λ + [h],
and that Λ + is self-adjoint, we obtain

H κ ϕ, ψ L 2 (R 3 ) 4 -ϕ, H κ ψ L 2 (R 3 ) 4 = -Λ + [g], h L 2 (Σ) 4 + g, -Λ + [h] L 2 (Σ) 4 = 0. (3.4)
Thus, H κ is symmetric on dom(H κ ) and densely defined in L 2 (R 3 ) 4 . This finishes the proof.

The following proposition gives a description of the domain of the adjoint operator (H * κ .

Proposition 3.3. Let H κ be as in the definition 2.1. Then we have

dom(H * κ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ H -1/2 (Σ) 4 , t Σ u = -Λ+ [g] . (3.5)
Proof. Let D be the set on the right-hand of (3.5). First, we prove the inclusion D ⊂ dom(H * κ ).

Given ϕ := v + Φ[h] ∈ D and ψ = u + Φ[g] ∈ dom(H κ ), then ϕ, H κ ψ L 2 (R 3 ) 4 = Hv, u L 2 (R 3 ) 4 + Φ[h], Hu L 2 (R 3 ) 4 = Hv, u L 2 (R 3 ) 4 + h, t Σ u H -1/2 ,H 1/2 = Hv, u L 2 (R 3 ) 4 + h, -Λ + [g] H -1/2 ,H 1/2 = Hv, u L 2 (R 3 ) 4 + t Σ v, g H -1/2 ,H 1/2 = Hv, ψ L 2 (R 3 ) 4 .
Which yields ϕ ⊂ dom(H * κ ) and thus D ⊂ dom(H * κ ). Now we prove the inclusion dom(H * κ ) ⊂ D. For that, let ϕ 4 , we get that Hv = U in D ′ (R 3 ) 4 and then in L 2 (R 3 ) 4 . Using this, it follows that v ∈ H 1 (R 3 ) 4 . Therefore, we deduce that 4 . Note that we actually proved that if 4 , and then in H 1/2 (Σ) 4 . This completes the proof of the proposition.

:= v + Φ[h] ∈ dom(H * κ ) and let ψ ∈ C ∞ 0 (R 3 \ Σ) 4 . Then, there exists U ∈ L 2 (R 3 ) 4 such that Hϕ, ψ D ′ (R 3 ) 4 ,D(R 3 ) 4 = v + Φ[h], Hψ D ′ (R 3 ) 4 ,D(R 3 ) 4 = ϕ, Hψ L 2 (R 3 ) 4 = U, ψ L 2 (R 3 ) 4 (3.6) Because HΦ[h] = 0 in D ′ (Ω ± )
Φ[h] = ϕ -v ∈ L 2 (R 3 ) 4 . Now, Proposition 3.1(iii) yields that h = i(α • N )( C+ -C-)[h] ∈ H -1/2 (Σ)
ϕ := v + Φ[h] ∈ dom(H * κ ), then v ∈ H 1 (R 3 ) 4 and h ∈ H -1/2 (Σ) 4 . Next, let G(H * κ ) be the graph of H * κ , then G(H * κ ) : = (ϕ, H * κ ϕ) : ϕ, H * κ ψ L 2 (R 3 ) 4 = H * κ ϕ, ψ L 2 (R 3 ) 4 , ∀ψ ∈ dom(H κ ) (3.7) = (ϕ, H * κ ϕ) : Φ[h], Hu L 2 (R 3 ) 4 = Hv, Φ[g] L 2 (R 3 ) 4 , ∀ψ ∈ dom(H κ ) (3.8) = (ϕ, H * κ ϕ) : h, t Σ u H -1/2 ,H 1/2 = t Σ v, g H 1/2 , ∀ψ ∈ dom(H κ ) (3.9) = (ϕ, H * κ ϕ) : -Λ+ [h], g H -1/2 ,H 1/2 = t Σ v, g H 1/2 , ∀ψ ∈ dom(H κ ) . (3.10) Hence, t Σ v = -Λ+ [h] holds in H -1/2 (Σ)
Given z ∈ C\((-∞, -m] ∪ [m, ∞)), it is well known that the fundamental solution of (∆+m 2 -z 2 )I 4 is given by

ψ z (x) = e i √ z 2 -m 2 |x| 4π|x| , for x ∈ R 3 . (3.11)
Moreover, the trace of the single-layer associated to (∆ + m 2z 2 )I 4 , denoted by S z , has the integral representation

S z [g](x) = Σ ψ z (x -y)g(y)dS(y), ∀x ∈ Σ and g ∈ L 2 (Σ) 4 . (3.12)
If z = 0, we simply write S := S 0 .

The next result contains the main tools to prove the self-adjointness of the Dirac operator H κ .

Recall that {A, B} = AB + BA is the usual anticommutator bracket.

Lemma 3.1. Given a ∈ (-m, m), then the following hold:

(i) The anticommutator {β, C a Σ } extends to a bounded operator from H -1/2 (Σ) 4 onto H 1/2 (Σ) 4 . In particular, if Σ satisfies (H1), then {β, C a Σ } is a compact operator in L 2 (Σ) 4 . (ii) The anticommutator {α • N , C a Σ } extends to a bounded operator from H -1/2 (Σ) 4 to H 1/2 (Σ) 4 . In particular, if Σ satisfies (H1),

then {α • N , C a Σ } is a compact operator in L 2 (Σ) 4 . (iii) If Σ satisfies (H2), then {α • N , C Σ } is a compact operator in L 2 (Σ) 4 .
Proof. We are going to prove (i). For that, observe that

1 2(m 2 -a 2 ) (mI 4 -aβ){β, C a Σ }[g](x) = S a [g](x). (3.13)
Hence, the first statement of (i) follows by [START_REF] Mclean | Strongly Elliptic Systems and Boundary Integral Equations[END_REF]Theorem 6.11] (see also [START_REF] Medková | The Laplace equation[END_REF]) for example. Furthermore, if Σ satisfies (H1), then using that the embedding H 1/2 (Σ) 4 ֒→ L 2 (Σ) 4 is compact, we then get that {β, C a Σ } is a compact operator in L 2 (Σ) 4 . This finishes the proof of (i). Now we prove (ii). Let x ∈ Σ and y ∈ R 3 , a straightforward computation using the anticommutation relations of the Dirac matrices yields

(α • N (x))(α • y) = -(α • y)(α • N (x)) + 2(N (x) • y)I 4 . (3.14) Use (3.14) to obtain (α • N (x))φ a (y) = -φ a (y)(α • N (x)) - e - √ m 2 -a 2 |y| 2iπ|y| 3 (1 + m|y|)(N (x) • y)I 4 + 2a(α • N (x))ψ a (y).
Note that there are constants C 1 and C 2 such that, for all x, y ∈ Σ, it holds that

|N (x) -N (y)| C 1 |x -y| and |N (x) • (x -y)| C 2 |x -y| 2 ,
this can proved in similar way as in Proposition 5.7. Using this, for g ∈ L 2 (Σ) 4 , we have

{α • N , C a Σ }[g](x) = Σ K a (x, y)g(y)dS(y) + 2a(α • N (x))S a [g](x) := T a,1 [g](x) + T a,2 [g](x), (3.15)
where the kernel K a is given by 4 . Hence, it remains to prove that T a,1 is bounded from H -1/2 (Σ) 4 to H 1/2 (Σ) 4 . Actually, if Σ satisfies (H1), then the result follows with the same arguments as [START_REF] Ourmères-Bonafos | A strategy for self-adjointness of Dirac operators: applications to the MIT bag model and δ-shell interactions[END_REF]Proposition 2.8], where the authors prove the statement for a = 0. Now, remark that if Σ satisfies (H2), then K a (x, y) vanishes for all x, y ∈ F and it holds that

K a (x, y) = φ a (x -y)(α • (N (y) -N (x)) - e - √ m 2 -a 2 |x-y| 2iπ|x -y| 3 (1 + m 2 -a 2 |x -y|)(N (x) • (x -y))I 4 . Since Σ is C 2 -smooth, from (i) it follows immediately that T a,2 is bounded from H -1/2 (Σ) 4 to H 1/2 (Σ)
|K a (x, y)| C|x -y| -1 . Moreover, it holds that T a,1 = T K a 1 + T K a 2 + K *
, where K * is the adjoint of the matrix valued harmonic double-layer defined by (5.17), and the kernels

K a
1 and K a 2 are given by

K a 1 (x, y) = 1 4π|x -y| 3 (α • (x -y)) (iα • (N (y) -N (x)) . K a 2 (x, y) = e - √ m 2 -a 2 |x-y| 4π|x -y| a + mβ + i m 2 -a 2 α • x -y |x -y| (α • (N (y) -N (x)) + 2i m 2 -a 2 (N (x) • (x -y)) |x -y| 2 I 4 + e - √ m 2 -a 2 |x-y| -1 4π|x -y| 3 (iα • (x -y)) (α • (N (y) -N (x)) + e - √ m 2 -a 2 |x-y| -1 2iπ|x -y| 3 (N (x) • (x -y))I 4 .
Again, one can extend K * and the integral operator with kernel K a 1 to a bounded operators from H -1/2 (Σ) 4 to H 1/2 (Σ) 4 as much the same way as in [START_REF] Ourmères-Bonafos | A strategy for self-adjointness of Dirac operators: applications to the MIT bag model and δ-shell interactions[END_REF]Proposition 2.8]. Moreover, it is clear that

K a 2 is C 1 -smooth and |K a (x, y)| = O(1)
, when |x -y| tends to zero. Using this, it easily follows that the integral operator with kernel K a 2 is bounded from L 2 (Σ) 4 to H 1 (Σ) 4 , and then one can extend it continuously to a bounded operator from H -1/2 (Σ) 4 to H 1/2 (Σ) 4 by duality and interpolation arguments. The seconde statement is a direct consequence of the Sobolev injection, and this completes the proof of (ii). Now we turn to the proof of (iii). Assume that Σ satisfies (H2), let us prove that {α • N , C Σ } is compact on L 2 (Σ) 4 . From (ii), since a = 0 we have that {α • N , C Σ } coincides with T 0,1 which is given by (3.15). Let χ be a C ∞ (Σ) cutoff function vanishing out-side the deformation F . Using that K 0 (x, y) vanishes for all x, y ∈ Σ, we then obtain that

T 0,1 = χT 0,1 χ + χT 0,1 (1 -χ) + (1 -χ)T 0,1 χ. (3.16)
Hence, the claimed result follows from (ii) and the compactness of the Sobolev embedding χH 1/2 (Σ) 4 ֒→ L 2 (Σ) 4 . This finishes the proof of the lemma. Remark 3.1. Actually the above result is not surprising since the kernels associated to the anticommutators {α • N , C a Σ } and {β, C a Σ } behave locally like |x -y| -1 , when |x -y| tends to zero. Therefore, the operators in consideration are bounded from L 2 (Σ) 4

dans H 1 (Σ) 4 because Σ is C 2 -smooth.
We are now in position to prove Theorem 3.1.

Proof of Theorem 3.1 (i) Assume that sgn(κ) = 4. From the definition of Λa ± , a simple computation using Lemma 2.1(ii) gives

Λa ± Λa ∓ = 1 sgn(κ) -( C a Σ ) 2 + µ sng(κ) {β, C a Σ } + η sgn(κ) {α • N , C a Σ } = 1 sgn(κ) - 1 4 -C a Σ (α • N ){α • N , C a Σ } + µ sgn(κ) {β, C a Σ } + η sgn(κ) {α • N , C a Σ }.
(3.17) 4 . From (3.17), we have

Let g ∈ H -1/2 (Σ) 4 such that Λ+ [g] ∈ H 1/2 (Σ)
g = 4(sgn(κ)) 4 -sng(κ) Λ -Λ+ + C a Σ (α • N ){α • N , C a Σ } - µ sgn(κ) {β, C a Σ } - η sgn(κ) {α • N , C a Σ } [g].
Using Lemma 3.1, it follows that g ∈ H 1/2 (Σ) 4 . Hence, given any 4 , we deduce that g ∈ H 1/2 (Σ) 4 . Thus, dom(H * κ ) = dom(H κ ) and it holds that

ϕ = u + Φ[g] ∈ dom(H * κ ), since g ∈ H -1/2 (Σ) 4 and t Σ u = Λ+ [g] in H 1/2 (Σ)
dom(H κ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ H 1/2 (Σ) 4 , t Σ u = -Λ + [g] . (3.18)
This finishes the proof of (i).

(ii) Fix κ such that sgn(κ) = 4. Since H κ is closable by Proposition 3.2, it follows that H κ ⊂ H * κ . Let us prove the other inclusion, for this given ϕ = u + Φ[g] ∈ dom(H * κ ) and let (h j ) j∈N ⊂ H 1/2 (Σ) 4 be a sequence of functions that converges to g in H -1/2 (Σ) 4 . Set 4 , and it holds that

g j := g + 2 ǫ Λ-[h j -g], ∀j ∈ N. (3.19) Then (g j ) j∈N , (Λ + [g j ]) j∈N ⊂ H 1/2 (Σ)
g j ---→ j→∞ g in H -1/2 (Σ) 4 , Λ + [g j ] ---→ j→∞ Λ+ [g], in H 1/2 (Σ) 4 . (3.20)
Indeed, remark that one can write g j as follows

g j = 2 ǫ ( Λ+ [g] + Λ-[h j ]).
Using this, (3.20) follows easily since Λ± Λ∓ are bounded from H -1/2 (Σ) 4 to H 1/2 (Σ) 4 by Lemma 3.1 and (3.17). Now let

(v j ) j∈N ⊂ H 1 (R 3 ) 4 such that t Σ v j = 2 ǫ Λ+ Λ-[h j -g], for all j ∈ N. Set u j = u -v j , and define ϕ j := u j + Φ[g j ]. It is clear that u j ∈ H 1 (R 3 ) 4 and t Σ u j = -Λ + [g j ] in H 1/2 (Σ) 4 , hence (ϕ j ) j∈N ⊂ dom(H κ ). Moreover, since (h j ) j∈N (respectively (g j ) j∈N ) converges to g in H -1/2 (Σ) 4 as j -→ ∞, using the continuity of Λ± Λ∓ it follows that (ϕ j , H κ ϕ j ) ---→ j→∞ (ϕ, H * κ ϕ) in L 2 (R 3 ) 4 . Therefore H *
κ ⊂ H κ and the Theorem is proved. In the following, we explain how to define the Dirac operator H κ via transmission condition. Let

ϕ = u + Φ[g] ∈ dom(H κ ) and set ϕ ± := ϕ| Ω± . It is clear that ϕ ± , (α • ∇)ϕ ± ∈ L 2 (Ω ± ) 4 . Now, we define δ Σ ϕ as the distribution δ Σ ϕ, ψ D ′ (R 3 ) 4 ,D(R 3 ) 4 := 1 2 Σ t Σ ϕ + + t Σ ϕ -, ψ C 4 dS(x), for all ψ ∈ D(R 3 ) 4 . (3.21)
Therefore, a simple computation in the sens of distributions yields

(H + (ǫI 4 + µβ + η(α • N ))δ Σ )ϕ =(-iα • ∇ + mβ)ϕ + 1 2 (ǫI 4 + µβ + η(α • N ))(t Σ ϕ + + t Σ ϕ -)δ Σ , =(-iα • ∇ + mβ)ϕ + ⊕ (-iα • ∇ + mβ)ϕ -+ iα • N (t Σ ϕ + -t Σ ϕ -)δ Σ + 1 2 (ǫI 4 + µβ + η(α • N ))(t Σ ϕ + + t Σ ϕ -)δ Σ .
Using the Plemelj-Sokhotski formula, it easily follows that 4 and satisfying (3.22), it holds that H κ ϕ ∈ L 2 (R 3 ) 4 . In particular, this leads to the following definition: Definition 3.1. Given κ = (ǫ, µ, η) ∈ R 3 such that sgn(κ) = 0 and m > 0. The self-adjoint Dirac operator coupled with a combination of electrostatic, Lorentz scalar and normal vector field δ-shell interactions of strength ǫ, µ and η respectively, is the operator H κ defined on the domain

1 2 (ǫI 4 + µβ + η(α • N ))(t Σ ϕ + + t Σ ϕ -)δ Σ + iα • N (t Σ ϕ + -t Σ ϕ -)δ Σ = 0, (3.22) holds in H -1/2 (Σ) 4 . Since (-iα • ∇ + mβ)ϕ + (-iα • ∇ + mβ)ϕ -∈ L 2 (R 3 ) 4 , given ϕ = (ϕ + , ϕ -) ∈ L 2 (R 3 ) 4 such that (α • ∇)ϕ ± ∈ L 2 (Ω ± )
dom(H κ ) = ϕ = (ϕ + , ϕ -) ∈ L 2 (Ω + ) 4 ⊕ L 2 (Ω -) 4 : (α • ∇)ϕ ± ∈ L 2 (Ω ± ) 4 and (3.22) holds in H -1/2 (Σ) 4 , (3.23)
and it acts in the sens of distributions as 4 . Moreover, following the same arguments as above, we conclude that the transmission condition (3.22) holds actually in H 1/2 (Σ) 4 . Therefore, It follows that

H κ (ϕ) = (Hϕ + ) ⊕ (Hϕ -), for all ϕ ∈ dom(H κ ). Remark 3.2. Assume that sgn(κ) = 0, 4. Since the operator Φ is bounded from H 1/2 (Σ) 4 to H 1 (R 3 \ Σ) 4 , it holds that ϕ ± := ϕ| Ω± ∈ H 1 (Ω ± )
dom(H κ ) = ϕ = (ϕ + , ϕ -) ∈ H 1 (Ω + ) 4 ⊕ H 1 (Ω -) 4 : (3.22) holds in H 1/2 (Σ) 4 . (3.24)
Let us make some comments on the technique developed here. As we have mentioned in the introduction, the condition on Σ of being C 2 -smooth is minimal to prove the self-adjointness of H κ , when sgn(κ) = 0. Indeed, the main ingredient that we have used is the continuous extension of the operator 4 , or equivalently, the continuous extension of the anticommutators {β, C Σ } and {α • N , C Σ }. Since {β, C Σ } involves the trace of the single-layer potential, we can always extend it to a bounded operator from H -1/2 (Σ) to H 1/2 (Σ), even if Σ is Lipschitz. However, {α•N , C Σ } involves the principal value of the double-layer potential (or its adjoint operator), and it is well known that the C 2 regularity is minimal to extend it to a continuous operator from L 2 (Σ) to H 1 (Σ). However, as we will see later, if sgn(κ) = 0 and Σ is C 1,γ -smooth and compact for some γ ∈ (1/2, 1), then we can manage to prove the self-adjointness of H κ using the technique developed in this part, see Remark 5.5 for details.

Λ ± Λ ∓ from H -1/2 (Σ) 4 to H 1/2 (Σ)
3.1. On the Dirac Operator with Electrostatic and Lorentz scalar δ-Shell interactions. We discuss in this part the self-adjointness of the Dirac operator H κ in the case η = 0, and we denote it by H ǫ,µ . This operator is well known as the Dirac operator with electrostatic and Lorentz scalar δ-shell interactions, cf. [START_REF] Arrizabalaga | Shell interactions for Dirac operators: on the point spectrum and the confinement[END_REF], [START_REF] Behrndt | On Dirac operators in R 3 with electrostatic and Lorentz scalar δ-shell interactions[END_REF], [START_REF] Behrndt | Two-dimensional Dirac operators with singular interactions supported on closed curves[END_REF]. If |ǫ| = |µ|, from Theorem 3.1 we get immediately the following result.

Proposition 3.4. Given ǫ, µ ∈ R \ {0} such that |ǫ| = |µ| and define the operators Λ ± as follows

Λ ± = 1 ǫ 2 -µ 2 (ǫI 4 ∓ µβ) ± C Σ . (3.25)
Then, the following hold:

(i) If ǫ 2 -µ 2 = 4, then H ǫ,µ is self-adjoint and we have dom(H ǫ,µ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ H 1/2 (Σ) 4 , t Σ u = -Λ + [g] . (3.26) (ii) If ǫ 2 -µ 2 = 4, then H ǫ,µ is essentially self-adjoint and we have dom(H ǫ,µ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ H -1/2 (Σ) 4 , t Σ u = -Λ+ [g] .
Now we turn to the special case ǫ = ±µ. Set P ± = (I 4 ± β)/2, then H ǫ,±ǫ is given formally by

H ǫ,±ǫ = H + P ± V ǫ,±ǫ = -iα • ∇ + mβ + 2ǫP ± δ Σ . (3.27) Define Λ + = P ± (1/2ǫ + C Σ ) P ± and Λ -= P ± (1/2ǫ -C Σ ) P ± , ∀ǫ = 0. (3.28)
Clearly , Λ ± are bounded and self-adjoint from P ± L 2 (Σ) 4 onto itself (respectively from P ± H 1/2 (Σ) 4 onto itself). In order to define rigorously H ǫ,±ǫ as in Definition 2.1, that is H ǫ,±ǫ ϕ = Hu in the sense of distributions for ϕ = u + Φ[g], with u ∈ H 1 (R 3 ) 4 and g ∈ H 1/2 (Σ) 4 . We shall take g ∈ P ± H 1/2 (Σ) 4 and assume the condition

P ± t Σ u = -P ± Λ + [g]. Indeed, if we set dom(H ǫ,±ǫ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ P ± L 2 (Σ) 4 and P ± t Σ u = -P ± Λ + [g] , (3.29)
Then, in a similar way as in Proposition 3.2 and Proposition 3.3, one can check that (H ǫ,±ǫ , dom(H ǫ,±ǫ )) is closable and its adjoint is defined on the domain

dom(H * ǫ,±ǫ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ P ± H -1/2 (Σ) 4 , P ± t Σ u = -P ± Λ+ [g] . (3.30)
where Λ± denotes the bounded extension of Λ ± from P ± H -1/2 (Σ) 4 onto itself, and we get the analogous of Theorem 3.1 in this case which reads as follows:

Proposition 3.5. Assume that ǫ = 0, then (H ǫ,±ǫ , dom(H ǫ,±ǫ )) is self-adjoint and we have dom(H ǫ,±ǫ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ P ± H 1/2 (Σ) 4 , P ± t Σ u = -P ± Λ + [g] . (3.31)
Proof. Fix ǫ = 0 and let Λ± be as above. Using the relations P ± α j = P ∓ α j and P ± β = βP ± , a simple computation yields

Λ-Λ+ = 1 4ǫ 2 P ± - 1 2ǫ P ± CΣ P ± CΣ P ± = 1 4ǫ 2 P ± - m 2 2ǫ (S) 2 P ± . (3.32)
where S is given by (3.12). Recall that Λ-Λ+ and SP ± are bounded from P ± H -1/2 (Σ) 4 into P ± H 1/2 (Σ) 4 . Thus, from (3.32) it follows that, if g ∈ P ± H -1/2 (Σ) 4 such that Λ+ [g] ∈ P ± H 1/2 (Σ) 4 , then g ∈ P ± H 1/2 (Σ) 4 . Which yields that dom(H ǫ,±ǫ ) = dom(H * ǫ,±ǫ ) and the proposition is proved.

3.2.

The operators Λ a ± . Let a ∈ (-m, m) and let Λ a ± be as in the Notation 2.1. From the proof of Theorem 3.1, It is evident that the study of the self-adjointness character of H κ is related to the properties of Λ + . The goal of this part is to establish the connection between H κ and Λ + . For this, we introduce the Laplace-Beltrami operators ∆ Σ on Σ and we define the operator L := (c -∆ Σ )I 4 (here we assume that c is big enough if Σ satisfies (H2), so that c is not in the spectrum of ∆ Σ ). It is well known that L ±1/4 is a bijective operator from H ±1/2 (Σ) 4 onto L 2 (Σ) 4 . Hence, one can write the domain of H κ as follows:

dom(H κ ) = u + ΦL 1/4 [g] : u ∈ H 1 (R 3 ) 4 , g ∈ H 1/2 (Σ) 4 and L 1/4 t Σ u = -L 1/4 Λ + L 1/4 [g] , (3.33)
which leads us to define the following operators

L a ± := L 1/4 Λ ± L 1/4 with dom(L a ± ) = g ∈ H 1/2 (Σ) 4 : Λ a ± L 1/4 [g] ∈ H 1/2 (Σ) 4 . (3.34)
Lemma 3.2. Let κ ∈ R 3 such that sgn(κ) = 0 and let L a ± as above. The following hold:

(i) If sgn(κ) = 4, then L a ± is self-adjoint with dom(L a ± ) = H 1 (Σ) 4 . (ii) If sgn(κ) = 4, then L a
± is essentially self-adjoint and we have

dom(L a ± ) = g ∈ L 2 (Σ) 4 : Λa ± L 1/4 [g] ∈ H 1/2 (Σ) 4 . (3.35) Proof. Since L 1/4 and C a Σ are self-adjoint, it follows that L a ± is symmetric. Moreover, we have C ∞ 0 (Σ) 4 ⊂ dom(L a ± ) ⊂ L 2 (Σ) 4 , which yields that dom(L a ± ) is a dense subspace of L 2 (Σ) 4 , therefore L a ± is closable. Let h ∈ dom(L a * ± ) and let g ∈ dom(L a ± ), then there is f ∈ L 2 (Σ) 4 such that h, L a ± [g] L 2 = L 1/4 h, Λ a ± L 1/4 [g] H -1/2 ,H 1/2 = Λa ± L 1/4 [h], L 1/4 [g] D ′ ,D = L -1/4 [f ], g L 2 .
Thus, 4 , and therefore Λa ± L 1/4 [h] ∈ H 1/2 (Σ) 4 . Hence we get the inclusion

Λa ± L 1/4 [h] = L -1/4 [f ] in D ′ (Σ)
dom(L a * ± ) ⊂ g ∈ L 2 (Σ) 4 : Λa ± L 1/4 [g] ∈ H 1/2 (Σ) 4 .
Now, one can check easily the other inclusion and we thus get the equality. Therefore, item (i) is an immediate consequence of Lemma 3.1 and (3.17). To prove the second item it suffices to show that L a * ± ⊂ L a ± . For this, one can take the sequence of functions defined by (3.19) (just switch the roles of Λa ± and Λa ∓ ) and use the fact that Λa 4 , we omit the details. This finishes the proof of the lemma.

± Λa ∓ are continuous from H -1/2 (Σ) 4 to H 1/2 (Σ)
Taking into account the above lemma, from (3.4) and (3.7) it easily follows that:

H κ is (essentially) self-adjoint ⇐⇒ L + is (essentially) self-adjoint. (3.36)
As it was mentioned in the introduction, the operator L + appears in this form when we study the self-adjoint extension of H κ from the point of view of the boundary triples theory (see [START_REF] Behrndt | On Dirac operators with electrostatic δ-shell interactions of critical strength[END_REF] and [START_REF] Behrndt | Two-dimensional Dirac operators with singular interactions supported on closed curves[END_REF], for a more general view of the theory we refer to [START_REF] Behrndt | Boundary Value Problems, Weyl Functions, and Differential Operators[END_REF] and [START_REF] Brüning | Spectra of self-adjoint extensions and applications to solvable Schrödinger operators[END_REF] for example). Indeed, denote by

S := H ⇂ H 1 0 (R 3 \Σ) 4 and let T := H with dom(T ) = {u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ L 2 (Σ) 4 },
and define the linear mappings Γ 1 , Γ 2 : dom(T ) -→ L 2 (Σ) 4 by

Γ 1 (ϕ) = g and Γ 2 (ϕ) = t Σ u + C Σ [g], (3.37)
Then, {L 2 (Σ) 4 , Γ 1 , Γ 2 } is a quasi-boundary triples for T = S * (adapt the arguments of [START_REF] Behrndt | On Dirac operators with electrostatic δ-shell interactions of critical strength[END_REF] or [START_REF] Behrndt | Two-dimensional Dirac operators with singular interactions supported on closed curves[END_REF]). Moreover, if we denote by Γ1 the extension of Γ 1 , that is Γ1 : T -→ H -1/2 (Σ) 4 , we then get that {L 2 (Σ) 4 , L -1/4 Γ1 , L 1/4 Γ 2 } is an ordinary boundary triple for T = S * . Now it is easy to check that

H κ = T ⇂ Kr((ǫI 4 + µβ + η(α • N ))Γ 2 + Γ 1 ) = T ⇂ Kr(Γ 2 -C Σ Γ 1 + Λ + Γ 1 ). (3.38)
Thus, after transforming the quasi-boundary triples to an ordinary boundary triples (see [START_REF] Behrndt | On Dirac operators with electrostatic δ-shell interactions of critical strength[END_REF]Theorem 4.5] for example) it follows that: H κ is self-adjoint (respectively essentially self-adjoint) if and only if L + is self-adjoint (respectively essentially self-adjoint); see [START_REF] Behrndt | On Dirac operators with electrostatic δ-shell interactions of critical strength[END_REF]Corollary 2.8].

Spectral properties

In this section, we examine the spectral properties of the operator H κ . First, we give a necessary condition for the existence of the points spectrum in the gap (-m, m) and a Krein-type resolvent formula. More precisely, we have the following. Proposition 4.1. Let H κ be as in the definition 2.1. The following hold: 4 and we have

(i) Given a ∈ (-m, m), then Kr(H κ -a) = 0 ⇐⇒ Kr( Λa + ) = 0 (Birman-Schwinger principle). (ii) For all z ∈ C \ R the operator Λz + is bounded invertible from H -1/2 (Σ) 4 to H 1/2 (Σ)
(H κ -z) -1 = (H -z) -1 -Φ z ( Λz + ) -1 (Φ z ) * . (4.1)
Proof. (i) Let us prove the implication (⇒). Assume there is a ∈ (-m, m) such that H κ ϕ = aϕ for some ϕ = u + Φ a [g] ∈ dom(H κ ). Using the definition of H κ we get

Hu = aϕ = a(u + Φ[g]). (4.2)
From this we deduce that (Ha)Hu = ag holds in D ′ (R 3 ) 4 , and therefore

Hu = a Φa [g], (4.3)
it is clear that if a = 0, then u = 0 which yields Kr( Λa + ) = 0. Now assume that a = 0, then from (4.2) and (4.3)

it follows that u = (Φ a -Φ)[g]. Since ϕ = u + Φ a [g] ∈ dom(H κ ), it holds that t Σ u = -Λ+ [g]. Hence, by Lemma 3.1(iii) we get that t Σ u = ( C a Σ -CΣ )[g] = -Λ+ [g],
therefore Kr( Λa + ) = 0. We turn now to prove the implication (⇐), so fix a ∈ (-m, m) such that Λa + [g] = 0 for some g ∈ H -1/2 (Σ) 4 . Then, if a = 0, we set ϕ = Φ[g] ∈ dom(H κ ) and we have H κ ϕ = 0, which gives the result in this case.

Now suppose that a = 0, let u = aH -1 Φ a [g] ∈ H 1 (R 3 ) 4 and set ϕ = u + Φ[g]. Then Hu = aΦ a [g] and (H -a)u = aΦ[g] in D ′ (R 3 ) 4 , this amounts to saying that H κ ϕ = Hu = a(u + Φ[g]) = aϕ and u = Φ a [g] -Φ[g]. Furthermore, it can be seen easily that t Σ u = ( C a Σ -CΣ )[g] = -Λ+ [g]. Thus ϕ ∈ dom(H κ
) and H κ ϕ = aϕ, which yields Kr(H κa) = 0. This ends the proof of (i).

(ii

) Fix z ∈ C \ R. Since H κ is self-adjoint it follows that (H κ -z) -1
is well defined and bounded.

Moreover, from (i) it follows that Kr( Λz + ) = ∅, as otherwise z would be a non-real eigenvalue of H κ . Let u ∈ L 2 (R 3 ) 4 and set ϕ := (H κz) -1 u ∈ dom(H κ ). Taking into account the decomposition dom(H κ ) = dom(H) + Kr(H λ,µz) and the fact that Φz : H -1/2 (Σ) 4 -→ Kr(H κz) is a bounded bijective operator (this can be proved as much as the same way as see behrendt 2018), we deduce that there is a unique functions

v ∈ H 1 (R 3 ) 4 and g ∈ H -1/2 (Σ) 4 such that ϕ = v +Φ z [g]. Moreover one has (H κ -z)ϕ = (H -z)v, and thus v = (H -z) -1 u, which means actually that ϕ = (H -z) -1 u + Φ z [g]. Next, remark that iα • N (t Σ ϕ + -t Σ ϕ -) = g and 1 2 (t Σ ϕ + + t Σ ϕ -) = (H -z) -1 u ⇂ Σ + C z Σ [g]. (4.4)
Using that (Hz) -1 u ⇂ Σ = (Φ z ) * u and the transmission condition (3.22), we obtain that Λz 4 . Since this is true for all u ∈ L 2 (R 3 ) 4 , therefore Rn( Λz 4 is a bounded bijective operator. Summing up, we have proved that

+ [g] = (Φ z ) * u ∈ H 1/2 (Σ)
+ ) = H 1/2 (Σ) 4 . Hence Λz + : H -1/2 (Σ) 4 -→ H 1/2 (Σ)
0 ∈ ρ( Λz + ) and (H κ -z) -1 u = (H -z) -1 u + Φ z [( Λz + ) -1 (Φ z ) * u] holds for all u ∈ L 2 (R 3 ) 4 ,
(i) For all a ∈ (-m, m), one has a ∈ Sp p (H κ ) ⇐⇒ 0 ∈ Sp p (L a + ), a ∈ Sp ess (H κ ) ⇐⇒ 0 ∈ Sp ess (L a + ).
(ii) For all z ∈ ρ(H κ ) ∩ ρ(H), the operator L z + is bounded invertible from L 2 (Σ) 4 to L 2 (Σ) 4 and we have

(H κ -z) -1 = (H -z) -1 -Φ z L 1 4 (L z + ) -1 L 1 4 (Φ z ) * . (4.5)
In order to avoid repetitions, we focus namely in the remainder of this section on the spectral properties of H κ when Σ satisfies the assumption (H2). When Σ is compact we prove them in general framework, see Section 5 for more details. Notation 4.1. For all ν 0, we denote by H ν κ (respectively Φ z ν , Λz +,ν and (Φ z ν ) * ) the operator H κ (respectively Φ z , Λz + and (Φ z ) * ) whenever Σ = Σ ν , i.e Σ satisfies (H2), and we write H k (respectively Φ z , Λz + and (Φ z ) * ) instead of H 0 κ (respectively Φ z 0 , Λz +,0 and (Φ z 0 ) * ).

4.1. Non-critical case. This part deal with the basic spectral properties of H κ in the non-critical case. The results are mainly known for the Dirac operator coupled with a combination of electrostatic and Lorentz scalar δ-interactions (i.e η = 0) supported on a closed, bounded and sufficiently smooth surface (see [START_REF] Behrndt | On Dirac operators in R 3 with electrostatic and Lorentz scalar δ-shell interactions[END_REF] for example ), and they are still true when Σ satisfies (H2) and η = 0.

Theorem 4.1. Let κ ∈ R 3 such that sgn(κ) = 0, 4 and suppose that Σ satisfies (H2). Then the following hold:

(i) Sp ess (H ν κ ) = (-∞, -m] ∪ [m, +∞). In particular, we have Sp(H κ ) = Sp ess (H κ ). (ii) Sp disc (H ν κ ) ∩ (-m, m) is finite, for all ν 0.
When ν = 0, Theorem 4.1 gives us a simple way to construct functions of dom(H κ ). Indeed, since 0 / ∈ Sp(H κ ) it follows that Λ + is invertible and we get then

dom(H κ ) = u + Φ[-Λ -1 + [t Σ u]] : u ∈ H 1 (R 3 ) 4 .
Proof. (i) First, we show the result for ν = 0, the case ν > 0 is an immediate consequence of the Proposition 4.1. Fix a ∈ (-m, m) and set

Γ ±m,±a (ξ) = [α • (ξ 1 , ξ 2 , 0) ± mβ ± a].
Since the α j 's anticommute with β, by a simple computation we get the following properties

Γ m,a (ξ) 2 = |ξ| 2 + m 2 -a 2 + 2aΓ m,a (ξ), Γ -m,-a (ξ)Γ m,a (ξ) = |ξ| 2 + m 2 -a 2 -2mβΓ m,a (ξ), Γ m,-a (ξ)Γ m,a (ξ) = |ξ| 2 + m 2 -a 2 . (4.6)
Using the Fourier-Plancherel operator, it is not hard to prove that Λ a + is unitary equivalent to the following multiplication operator:

Π a + := 1 sgn(κ) (ǫI 4 -µβ -η(α • N )) + 1 2 |ξ| 2 + m 2 -a 2 Γ m,a (ξ). (4.7)
Moreover, take into account the properties (4.6), a simple computation shows that Π a + is invertible and it's inverse is given explicitly by

(Π a + ) -1 = C -1 1 + ǫa + µm |ξ| 2 + m 2 -a 2 - (ǫ + µβ + η(α • N )) 2 |ξ| 2 + m 2 -a 2 Γ m,a (ξ) (ǫ + µβ + η(α • N )), (4.8)
where

C = 4 -sgn(κ) 4 + ǫa + µm |ξ| 2 + m 2 -a 2 . Therefore Sp(H κ )∩(-m, m) = ∅. As H κ is self-adjoint, we have then Sp(H κ ) ⊂ (-∞, -m]∪[m, +∞).
Now we turn to prove the other inclusion, for that we construct a singular sequence for H κ and a.

Let a ∈ (-∞, -m) ∪ (m, ∞) and define ϕ :

     R 3 -→ C 4 (x, x 3 ) -→ ξ 1 -iξ 2 a -m , 0, 0, 1 t e ix•ξ , here ξ = (ξ 1 , ξ 2 ) and |ξ| 2 = a 2 -m 2 . Observe that we have (-iα • ∇ + mβ -a)ϕ = 0. Let R > 0, χ ∈ C ∞ 0 (R 2 ) and θ ∈ C ∞ 0 ([0, ∞[) such that θ = 1 for 2R > |x| > R., 0 for R |x| R/2.
For n ∈ N ⋆ , we define the sequences of functions

ϕ +,n (x, x 3 ) = ϕ(x, x 3 )χ(x/n)θ(x 3 /n) for x 3 > 0, ϕ -,n (x, x 3 ) = ϕ(x, x 3 )χ(x/n)θ(-x 3 /n) for x 3 < 0. (4.9) It's clear that ϕ ±,n ∈ H 1 (Ω ± ) and t Σ ϕ ±,n = 0, thus ϕ n := (ϕ +,n , ϕ -,n ) ∈ dom(H κ ). Moreover, we have ϕ n 2 L 2 (R 3 ) 4 = ϕ +,n 2 L 2 (Ω+) 4 + ϕ -,n 2 L 2 (Ω-) 4 = n 3 2a a -m χ 2 L 2 (R 2 ) θ 2 L 2 (R+)
and

(-iα • ∇ + mβ -a) ϕ n 2 L 2 = (-iα • ∇ + mβ -a) ϕ +,n 2
L 2 (Ω+) + (-iα • ∇ + mβ -a) ϕ -,n 2 L 2 (Ω-) n 4a a -m ∇η 2 L 2 (R 2 ) θ 2 L 2 (R+) + χ 2 L 2 (R 2 ) θ ′ 2 L 2 (R+)
Thus, we get

(-iα • ∇ + mβ -a) ϕ n L 2 (R 3 ) 4 ϕ n L 2 (R 3 ) 4 ----→ n→∞ 0.
Summing up, we have proved

(-∞, -m) ∪ (m, ∞) ⊂ Sp(H κ ) ⊂ (-∞, -m] ∪ [m, ∞).
Since the spectrum of a self-adjoint operator is closed, the end-points {-m, m} also belong to the spectrum, and hence we get

Sp(H κ ) = (-∞, -m] ∪ [m, ∞).
It is clear that the spectrum is purely essential because (non-degenerate) intervals have no isolated points, this yields Sp(H κ ) = Sp ess (H κ ), for ν = 0. Now assume that ν > 0, let's determine the essential spectrum of H ν κ . For that, recall that F is the flat part of Σ ν given by (2.1); fix z ∈ C \ R and let T : L 2 (R 3 ) 4 → L 2 (R 3 ) 4 be the bounded operator defined by

T = Φ z ν (Λ z +,ν ) -1 (Φ z ν ) * -Φ z (Λ z + ) -1 (Φ z ) * . (4.10)
Then T is a compact operator in L 2 (R 3 ) 4 . Indeed, observe that T can be written as follows: 4 to L 2 (R 3 ) 4 , we get therefore that T 2 and T 3 are compact operators on L 2 (R 3 ) 4 . Now, let χ be a C ∞ (Σ ν ) cutoff function vanishing out-side the deformation F , then we write T 1 as follows

T = Φ z ν (Λ z +,ν ) -1 -Φ z (Λ z + ) -1 (H -z) -1 ⇂ F +Φ z ν (Λ z +,ν ) -1 (H -z) -1 ⇂ Σν \F -Φ z (Λ z + ) -1 (H -z) -1 ⇂ Σ0\F := T 1 + T 2 + T 3 . Since Σ ν \ F is compact for all ν 0, it follows that the injection H 1/2 (Σ ν \ F ) 4 ֒→ L 2 (Σ ν ) 4 is compact. Using this and the fact that (H -z) -1 ⇂ Σν \F is bounded from L 2 (R 3 ) 4 to H 1/2 (Σ ν \ F ) 4 , it holds that (H -z) -1 ⇂ Σν \F is a compact operator from L 2 (R 3 ) 4 to L 2 (Σ) 4 . As Φ z ν (Λ z +,ν ) -1 is bounded from L 2 (Σ)
T 1 = Φ z ν χ(Λ z +,ν ) -1 -Φ z χ(Λ z + ) -1 (H -z) -1 ⇂ F + Φ z ν (1 -χ)(Λ z +,ν ) -1 -Φ z (1 -χ)(Λ z + ) -1 (H -z) -1 ⇂ F := T 4 + T 5 .
From the definition of Φ z ν , it is clear that T 5 = 0. Moreover, as 4 , using again the compactness of the Sobolev embedding, we obtain that T 4 is also a compact operator on L 2 (R 3 ) 4 . Therefore T is a compact operator in L 2 (R 3 ) 4 . Now, using Proposition 4.1(ii) it follows that T = (H ν κz) -1 -(H κz) -1 . Therefore, by Weyl's theorem we conclude that H ν κ has the same essential spectrum as H κ . This complete the proof of (i). In order to prove item (ii) we follow the idea of [START_REF] Holzmann | Dirac operators with Lorentz scalar shell interactions[END_REF] and [START_REF] Behrndt | On Dirac operators in R 3 with electrostatic and Lorentz scalar δ-shell interactions[END_REF]. Fix ν > 0 and let Q be the quadratic form associated to (H ν κ ) 2 with domain dom(H ν κ ), that is

(Λ z +,ν ) -1 (H -z) -1 ⇂ F is bounded from L 2 (R 3 ) 4 to H 1/2 (Σ ν )
Q[ϕ] = (α • ∇)ϕ + 2 L 2 (Ω+) 4 + (α • ∇)ϕ - 2 L 2 (Ω-) 4 + m 2 ϕ 2 L 2 (R 3 ) 4 + (-iα • N )t Σν ϕ + , mβt Σν ϕ + L 2 (Σν ) 4 -(-iα • N )t Σν ϕ -, mβt Σν ϕ -L 2 (Σν ) 4 .
Given R > 0 such that (Σ ν \ F ) ⊂ B(0, R), and define the closed and semi-bounded sesquilinear forms

Q ext [ϕ] = (α • ∇)ϕ + 2 L 2 (Ω+\B(0,R)) 4 + (α • ∇)ϕ - 2 L 2 (Ω-\B(0,R)) 4 + m 2 ϕ 2 L 2 (R 3 \B(0,R)) 4 + (-iα • N )t Σν ϕ + , mβt Σν ϕ + L 2 (F ) 4 -(-iα • N )t Σν ϕ -, mβt Σν ϕ -L 2 (F ) 4 .
with domain

dom(Q ext ) = ϕ = (ϕ + , ϕ -) ∈ H 1 (Ω + \ B(0, R)) 4 ⊕ H 1 (Ω -\ B(0, R)) 4 : t Σν ϕ = Θ λ,µ t Σν ϕ -, and
Q int [ϕ] = (α • ∇)ϕ + 2 L 2 (Ω+∩B(0,R)) 4 + (α • ∇)ϕ - 2 L 2 (Ω-∩B(0,R)) 4 + m 2 ϕ 2 L 2 (B(0,R)) 4 + (-iα • N )t Σν ϕ + , mβt Σν ϕ + L 2 (Σν ∩B(0,R)) 4 -(-iα • N )t Σν ϕ -, mβt Σν ϕ -L 2 (Σν ∩B(0,R)) 4 .
with domain 4 is compact, we conclude that the resolvent of the operator H int associated to Q int is compact. Hence, H int has a finite purely discrete spectrum in the gap (-m, m). Moreover, from (i) we deduce that inf Sp(

dom(Q int ) = ϕ = (ϕ + , ϕ -) ∈ H 1 (Ω + ∩ B(0, R)) 4 ⊕ H 1 (Ω -∩ B(0, R)) 4 : t Σν ϕ = Θ λ t Σν ϕ -, Set Q = Q ext ⊕ Q int . Then, dom(Q) ⊂ dom( Qext ) and Qext [ϕ] Q[ϕ], hold for all ϕ ∈ dom(Q). Since the injection H 1 (Ω + ∩B(0, R)) 4 ⊕H 1 (Ω -∩B(0, R)) 4 ֒→ L 2 (B(0, R))
Q ext ) = inf Sp ess (Q ext ) m 2 .
Therefore, the min-max principle gives us

inf Sp ess (Q) inf Sp ess (Q ext ) m 2 and #{Sp disc (H ν λ ) ∩ (-m, m)} < ∞. (4.11) Thus, Sp disc (H ν κ ) ∩ (-m, m
) is finite and the theorem is finally proved.

Critical case.

From now, we assume that sgn(κ) = 4. The goal of this subsection is to prove the following result.

Theorem 4.2. Given κ = (ǫ, µ, η) ∈ R 3 such that sgn(κ) = 4 and let H κ be as in Theorem 3.1. If Σ satisfies (H2), then for all ν 0 it holds that

Sp ess (H ν κ ) = -∞, -m ∪ - mµ ǫ ∪ m, +∞ . (4.12)
In particular, we have the equality Sp(H 0 κ ) = Sp ess (H 0 κ ).

A few comments are in order. Actually, one can imagine that the operator H κ is unitary equivalent to H ǫ1,µ1 , for some ǫ 1 , µ 1 ∈ R, such that ǫ 2 1µ 2 1 = 4 and ǫ 1 /ǫ = µ 1 /µ. Indeed, in [START_REF] Mas | Dirac operators, shell interactions, and discontinuous Gauge functions across the boundary[END_REF] and [START_REF] Cassano | General δ-Shell Interactions for the two-dimensional Dirac Operator: Self-adjointness and Approximation[END_REF] it has been shown that the potential ηα • N can always be absorbed as a change of gauge. So the existence of such a unitary transformation is not excluded. Another way to understand Theorem 4.2 comes from the way in which we have presented the operator H κ . In fact, in this paper we introduced the operator H κ as the perturbation of the coupling H+(ǫI 4 +µβ)δ Σ with the singular potential η(α•N )δ Σ , however, the right way is to say that H κ is the perturbation of H + η(α • N ))δ Σ with the singular potential (ǫI 4 +µβ)δ Σ . The reason is very simple. Indeed, for all η ∈ R, the operator H+η(α

•N ))δ Σ is self-adjoint even if Σ is Lipschitz, cf. Proposition 5.5. Moreover, Sp(H + η(α • N ))δ Σ ) = -∞, -m ∪ m, +∞ .
As in the non-critical case, from Theorem 4.2 we get a simple way to describe functions belonging to the domain of H κ when Σ = R 2 × {0}. Indeed, we have the following result.

Corollary 4.2. Assume that Σ := R 2 × {0} and let H κ be as above. The following hold:

(i) If µ = 0, then dom(H κ ) = u + Φ[-Λ-1 + [t Σ u]] : u ∈ H 1 (R 3 ) 4 . (4.13) (ii) If µ = 0, then dom(H κ ) = dom(H κ ) + Φ[Kr( Λ+ )]
Proof. Assertion (i) is a direct consequence of Theorem 4.2 and Proposition 4.1. Assertion (ii) follows using the same arguments as those in [2, Proposition 3.10], we omit the details.

The main properties of the operators L a ± which are relevant for us to prove Theorem 4.2 are collected in the following proposition. Proposition 4.2. Let κ = (ǫ, µ, η) ∈ R 3 such that sgn(κ) = 0 and let L a ±,κ := L a ± be as in Lemma 3.2. Then, for all a ∈ (-m, m), it holds that

0 ∈ Sp(L a +,κ ) ⇐⇒ 0 ∈ Sp(L -a +,κ ) ⇐⇒ 0 ∈ Sp(L -a -,κ ).
where κ = (-ǫ, µ, -η). In particular, a ∈ Sp(H κ ) if and only if -a ∈ Sp(H κ).

Proof.

Fix κ = (ǫ, µ, η) ∈ R 3 such that sgn(κ) = 0. Following [7, Proposition 4.2], for f ∈ L 2 (Σ) 4 we define C(f ) = iβα 2 f c , T (f ) = γ 5 βf, γ 5 := -iα 1 α 2 α 3 = 0 I 2 I 2 0 , (4.14)
where f c is the the complex conjugate of f . Remark that α 2 c = -α 2 , γ 5 β = -βγ 5 and γ 5 (α • x) = (α • x)γ 5 , for all x ∈ R 3 . Using this, it easily follows that C 2 (f ) = f and T 2 (f ) = -f . Moreover, a simple computation using the anticommutation relations of Dirac matrices yields

Λ ±a ±,κ [T (f )] = T (Λ ∓a ∓,κ [f ]), Λ a +,κ [C(f )] = -C(Λ -a +, k[f ]), Λ -a +, k[C(f )] = -C(Λ a +,κ [f ]). (4.15)
Fix a ∈ (-m, m) and assume that 0 ∈ Sp(L a + ). Then, there exists a sequence of functions

(g j ) j∈N ⊂ dom(L a + ) ⊂ L 2 (Σ) 4 , such that ||g j || L 2 (Σ) 4 = 1 and L a +,κ g j L 2 (Σ) 4 ---→ j→∞ 0. Hence, if we set f j = C(g j ) and h j = T (g j ), then it is clear that ||h j || L 2 (Σ) 4 = ||f j || L 2 (Σ) 4 = 1, f j ∈ dom(L -a +,κ
) and h j ∈ dom(L -a -,κ ), hold for all j ∈ N . Now using (4.15) it follows that

L -a +,κ [f j ] L 2 (Σ) 4 = L -a -,κ [h j ] L 2 (Σ) 4 = L a +,κ [g j ] L 2 (Σ) 4 .
Therefore 0 ∈ Sp(L -a +,κ ) and 0 ∈ Sp(L -a -,κ ). The reverse implications follow in the same way and this proves the first statement. The last statement is a direct consequence of the first one and Corollary 4.1. This completes the proof. Proof. Given a ∈ (-m, m), once the claimed statement is shown for L a + , by Proposition 4.2 we get the result for L a -. As in Theorem 4.1, on the Fourier side, one can check that L a + is unitary equivalent to the following multiplication operator:

Π a + := ξ 1 sgn(κ) (ǫI 4 -(µβ + η(α • N ))) + 1 2 |ξ| 2 + m 2 -a 2 Γ m,a (ξ) . (4.17)
Since sgn(κ) = 4, from (4.8) it follows that Π a + is invertible for all a = -mµ/λ, and we have

( Π a + ) -1 = 1 ξ 1 + |ξ| 2 + m 2 -a 2 ǫa + µm - (ǫ + (µβ + η(α • N ))) 2(ǫa + µm) Γ m,a (ξ) (ǫ + (µβ + η(α • N ))).
Furthermore it holds that

1 ξ Π a + 1 - (λ + µβ + η(α • N )) 2 |ξ| 2 + m 2 -a 2 Γ m,a (ξ) = 0, for a = - mµ ǫ .
From this, it follows that 0 is an eigenvalue of the operators L -mµ/ǫ + with infinite multiplicity, and

thereby 0 ∈ Sp ess (L -mµ/ǫ +
). Thus, we conclude that 0 ∈ Sp(L a + ) if and only if aǫ = -mµ. Now we turn to prove the last statement for the operator L -mµ/ǫ + , similar arguments give the result for

L mµ/ǫ - . A simple computation yields det( Π a + -θ) = θ θ -ξ a |ξ| 2 + m 2 -a 2 + ǫ 2 θ1 2 , (4.18)
where det( Π a +θ) is the determinant of ( Π a +θ). By studying the variations of the non-trivial root θ 1 , we obtain that

Sp(L -mµ/ǫ + ) = {0} ∪ θ 1 ([0, ∞)) = {0} ∪ ǫ 2 - µ ǫ 2 -µ 2 , ∞ if ǫ > 0, Sp(L -mµ/ǫ + ) = θ 1 ([0, ∞)) ∪ {0} = -∞, ǫ 2 + µ ǫ 2 -µ 2 ∪ {0} if ǫ < 0.
Take into account that sgn(κ) = 4, we then get that 0 is an isolated eigenvalue of L -mµ/ǫ + with infinite multiplicity, which completes the proof of the proposition. We are now in a position to complete the proof of our main result in this subsection.

Proof of Theorem 4.2. Assume that Σ satisfies (H2) and fix ν 0. The result will follow from the following statements:

(a) -∞, -m ∪ m, +∞ ⊂ Sp ess (H ν κ ). (b) {-mµ/ǫ} ∈ Sp ess (H ν κ ) and {mµ/ǫ} / ∈ Sp ess (H ν κ ). (c) Sp ess (H ν κ ) ∩ [(-m, m) \ {-mµ/ǫ, mµ/ǫ}] = ∅.
We are going to show (a). For that, given a ∈ (-∞, -m) ∪ (m, ∞) and let (ϕ n ) n∈N be the sequence of functions defined by (4.9) with R = sup{|x| : x ∈ Σ ν \ F }. By construction, it is clear that (ϕ n ) n∈N is a singular sequence for H ν κ and a. Therefore we get the inclusion (-∞, -m)∪(m, +∞) ⊂ Sp ess (H ν κ ), which yields (a). Now, we turn to the proof of (b). and we consider the operator Υ ν : L 2 (Σ ν ) 4 -→ L 2 (Σ ν ) 4 defined by:

Υ -mµ/ǫ ν := L 1 4 ν D ν B ν L 1 4 ν = L 1 4 ν (Λ mµ/ǫ -,ν Λ -mµ/ǫ +,ν )( Λ-mµ/ǫ +,ν Λmµ/ǫ -,ν )L 1 4 ν . (4.19)
Observe that

B ν =C -mµ/ǫ Σ (α • N ){α • N , C -mµ/ǫ Σ } -2 mµ ǫ C -mµ/ǫ Σ S -mµ/ǫ + mµη 2ǫ (α • N )S -mµ/ǫ + η 4 {α • N , C -mµ/ǫ Σ }, (4.20)
and

D ν =C -mµ/ǫ Σ (α • N ){α • N , C -mµ/ǫ Σ } -2 mµ ǫ S -mµ/ǫ C -mµ/ǫ Σ + mµη 2ǫ (α • N )S -mµ/ǫ + η 4 {α • N , C -mµ/ǫ Σ }. (4.21)
As L

1 4
ν is an isomorphism, using Lemma 3.1 it easily follows that Υ -mµ/ǫ ν is a bounded, self-adjoint operator on L 2 (Σ ν ) 4 . Moreover, taking into account the above assumption, by Proposition 4.2 it holds that dim Kr(Υ

-mµ/ǫ ν ) < ∞, since dim Kr( Λmµ/ǫ -,ν ) = dim Kr( Λmµ/ǫ -,ν ) < ∞, for all ν > 0. Now we introduce the unitary transformation U : L 2 (Σ ν ) 4 -→ L 2 (R 2 ) 4 , defined by U g(x) = J 1/2 ν (x)g(τ (x)). We claim that U Υ -mµ/ǫ ν U -1 -Υ -mµ/ǫ 0 is a compact operator on L 2 (Σ ν ) 4 . Indeed, let χ be a C ∞ (Σ)
cutoff function vanishing out-side the deformation F , we then get

U Υ -mµ/ǫ ν U -1 -Υ -mµ/ǫ 0 := (U Υ -mµ/ǫ ν U -1 -Υ -mµ/ǫ 0 )χ + (U Υ -mµ/ǫ ν U -1 -Υ -mµ/ǫ 0 )(1 + χ).
Since the embedding χL 2 (R 2 ) 4 ֒→ H -1/2 (R 2 ) 4 is compact, and U Υ -mµ/ǫ ν U -1 is bounded from 4 , for all ν 0, it follows that (U Υ

H -1/2 (R 2 ) 4 to L 2 (R 2 )
-mµ/ǫ ν U -1 -Υ -mµ/ǫ 0 )χ is a compact operator on L 2 (R 2 ) 4 . Next, observe that (U Υ -mµ/ǫ ν U -1 -Υ -mµ/ǫ 0 )(1 + χ) := T 1 + T 2 ,
where 4 , for all v 0, and the embedding 4 , we then get that T 1 is a compact operator on L 2 (R 2 ) 4 . We now apply this argument again to the operator T 2 , and we get

T 1 = U L 1 4 ν D ν B ν χL 1 4 ν U -1 (1 + χ) -L 1 4 0 D 0 B 0 χL 1 4 0 (1 + χ), T 2 = U L 1 4 ν D ν B ν -L 1 4 0 D 0 B 0 (1 -χ)L 1 4 0 (1 + χ). Recall that L 1 4 ν U -1 is bounded from L 2 (R 2 ) 4 to H -1/2 (Σ ν )
χH -1/2 (Σ ν ) 4 ֒→ H -1 (Σ ν ) 4 is compact. Since B ν is bounded from H -1 (Σ ν ) 4 to L 2 (Σ ν ) 4 (see Remark 3.1) it follows that B ν χL 1 4 ν U -1 is a compact operator from L 2 (R 2 ) 4 to L 2 (Σ ν ) 4 . Now, as U L 1 4 ν D ν is bounded from L 2 (Σ ν ) 4 to L 2 (R 2 )
T 2 = U L 1 4 ν D ν χB ν -L 1 4 0 D 0 χB 0 (1 -χ)L 1 4 0 (1 -χ) + U L 1 4 ν D ν (1 -χ)B ν -L 1 4 0 D 0 (1 -χ)B 0 (1 -χ)L 1 4 0 (1 -χ) := T 3 + T 4 .
Then in the same manner, it is easy to check that T 3 is a compact operator on L 2 (R 2 ) 4 . Now set

Bν := (1 -χ)B ν (1 -χ) = (1 -χ)( Λ-mµ/ǫ +,ν Λmµ/ǫ -,ν )(1 -χ),
Observe that

Bν = C -mµ/ǫ Σ χ(α • N ){α • N , C -mµ/ǫ Σ } -2 mµ ǫ C -mµ/ǫ Σ χS -mµ/ǫ + mµη 2ǫ χ(α • N )S -mµ/ǫ + η 4 χ{α • N , C -mµ/ǫ Σ } -2 mµ ǫ χC -mµ/ǫ Σ (1 + χ)S -mµ/ǫ -2χ mµ ǫ C -mµ/ǫ Σ (1 + χ)S -mµ/ǫ (1 -χ) + (1 + χ) -2 mµ ǫ C -mµ/ǫ Σ (1 + χ)S -mµ/ǫ + mµη 2ǫ (α • N )S -mµ/ǫ (1 -χ) := B ν,1 + B 2 .
Here we used the fact that

(1 + χ)(α • N ){α • N , C -mµ/ǫ Σ }(1 + χ)
vanishes identically. Therefore we obtain that

T 4 = U L 1 4 ν D ν B ν,1 -L 1 4 0 D 0 B 0,1 L 1 4 0 (1 -χ) + U L 1 4 ν D ν -L 1 4 0 D 0 B 2 L 1 4
0 (1χ) := T 5 + T 6 .

Again, using the compactness of the Sobolev injection, one can show that T 5 is a compact operator on L 2 (R 2 ) 4 . Next, remark that

D ν (1 -χ) = C -mµ/ǫ Σ χ(α • N ){α • N , C -mµ/ǫ Σ } -2 mµ ǫ χS -mµ/ǫ C -mµ/ǫ Σ + mµη 2ǫ (α • N )S -mµ/ǫ + η 4 χ{α • N , C -mµ/ǫ Σ } (1 + χ) + (1 + χ) -2 mµ ǫ S -mµ/ǫ (1 + χ)C -mµ/ǫ Σ + mµη 2ǫ (α • N )S -mµ/ǫ (1 + χ) := D ν,1 (1 + χ) + D 2 (1 + χ). Note that (U (1 -χ)L 1 4 ν -(1 -χ)L 1 4 0 )D 2 B 2 L 1 4
0 (1χ) = 0. Therefore, we obtain that

T 6 = U L 1 4 ν D ν,1 -L 1 4 0 D 0,1 B 2 L 1 4 0 (1 -χ) + U χL 1 4 ν -χL 1 4 0 D 2 B 2 L 1 4 0 (1 -χ).
Then, we conclude as above that T 6 is a compact operator on L 2 (R 2 ) 4 . Therefore, U Υ ). This contradicts the fact that dim Kr(Υ -mµ/ǫ ν ) < ∞, which proves (b).

-mµ/ǫ ν U -1 - Υ -mµ/ǫ 0 is compact in L 2 (R 2
We now show (c), so assume that a ∈ (-m, m) \ {-mµ/ǫ, mµ/ǫ}. We introduce the operator G a ν : L 2 (Σ ν ) 4 -→ L 2 (Σ ν ) 4 defined by:

G a ν := L 1 4 ν (Λ a -,ν Λ a +,ν )( Λa -,ν Λa +,ν )L 1 4 ν . Clearly, G a ν is bounded self-adjoint in L 2 (Σ ν ) 4 , since Λa -,ν Λa +,ν = Λa +,ν
Λa -,ν . Moreover, by definition we have

0 ∈ Sp ess (L a ±,ν ) =⇒ 0 ∈ Sp ess (G a ν ). (4.22)
As Λa +,0 and Λa -,0 are bounded, invertible operators for all a ∈ (-m, m) \ {-mµ/ǫ, mµ/ǫ}, from Proposition 4.3 it follows that 0 ∈ Sp ess (G a 0 ) if and only if a = ∓mµ/ǫ. Next, we claim that: if a = ∓mµ/ǫ, then 0 / ∈ Sp ess (G a ν ). To prove this, let U be the unitary transformation defined in the proof of (b), and set T = U G a ν U -1 -G a 0 . Then, T is a compact operator in L 2 (R 2 ) 4 . Indeed, this may be handled in much the same way as in the proof of the previous statement, we omit the details. Therefore Sp ess (G a ν ) = Sp ess (G a 0 ) holds by Weyl's theorem. This proves the claim because 0 ∈ Sp ess (G a 0 ) if and only if a = ∓mµ/ǫ. Using this, from (4.22) it follows that, if a = ∓mµ/ǫ then 0 / ∈ Sp ess (L a ±,ν ). Therefore, Corollary 4.1 yields that Sp ess (H ν κ ) ∩ [(-m, m) \ {-mµ/ǫ, mµ/ǫ}] = ∅, which yields (c).

Summing up, from (a) and (b) we obtain that -∞, -m ∪ {-mµ/ǫ} ∪ m, +∞ ⊂ Sp ess (H ν κ ). From (b) and (c) we get the inclusion Sp ess (H ν κ ) ⊂ -∞, -m ∪ {-mµ/ǫ} ∪ m, +∞ . Since the essential spectrum of a self-adjoint operator is closed, we get then the equality (4.12). This completes the proof of the theorem.

Actually in the case Σ = R 2 × {0}, one can check directly using the separation of variables that a = -mµ/ǫ is an eigenvalue of H κ with infinite multiplicity. Indeed, let a = -mµ/ǫ and ϕ ∈ dom(H κ ) such that:

(H κ -a)ϕ = 0, in L 2 (R 3 ) 4 . (4.23)
A simple computation yields the following relations

1 2 (ǫI 4 -µβ + ηα 3 ) + iα 3 1 2 (ǫI 4 + µβ -ηα 3 ) -iα 3 = (2 -iη)I 4 , 1 2 (ǫI 4 -µβ + ηα 3 ) + iα 3 1 2 (ǫI 4 + µβ -ηα 3 ) -iα 3 = iα 3 (λ + µβ). (4.24)
Hence, using this relation and the Definition 3.1, another way of stating (4.23) is to say:

(H -a)ϕ = 0 for all x 3 = 0, (2 -iη)t Σ ϕ + = -iα 3 (ǫ + µβ)]t Σ ϕ -for x 3 = 0. (4.25)
Since (H + a)(Ha) = (-∆ + m 2a 2 )I 4 , one get that ϕ is also solution of the following equation

(-∆ + m 2 -a 2 )I 4 ϕ = 0,
for all x 3 = 0 Thus, applying Fourier-Plancherel operator on x = (x 1 , x 2 ), we get that (4.26)

F x [ϕ] (ξ, x 3 ) =    e -x3 √ |ξ| 2 +m 2 -a 2 F x [ψ + ] (ξ) for x 3 >0, e x3 √ |ξ| 2 +m 2 -a 2 F x [ψ -] (ξ) for x 3 <0,
for some ψ ± ∈ H 1 2 (R 2 ). Since (H + a)ϕ = 2aϕ, by applying the inverse Fourier-Plancherel operator, we obtain that

ϕ(x, x 3 ) =        1 2π R 2 e ix•ξ e -x3 √ |ξ| 2 +m 2 -a 2 Γ i (ξ)F x [ψ + ] (ξ)dξ for x 3 > 0, 1 2π R 2 e ix•ξ e x3 √ |ξ| 2 +m 2 -a 2 Γ -i F x [ψ -] (ξ)dξ for x 3 < 0, (4.27)
where

Γ ±i (ξ) = α • (ξ 1 , ξ 2 , ±i |ξ| 2 + m 2 -a 2 ) + mβ + a . From this, it is clear that ϕ ± , (α•∇)ϕ ± ∈ L 2 (Ω ± ). Now, if we set ψ -= - η -i2 ǫ 2 -µ 2 (ǫ + µβ)α 3 ψ + , then we get (2 -iη)Γ +i (ξ)F x [ψ + ] (ξ) = -iα 3 (ǫI 4 -µβ)Γ -i (ξ)F x [ψ -] (ξ). (4.28)
Thus (ϕ + , ϕ -) satisfies the transmission condition. Therefore, for all ψ ∈ H

1 2 (R 2 ) the function ϕ(x, x 3 ) =        1 2π R 2 e ix•ξ e -x3 √ |ξ| 2 +m 2 -a 2 Γ i (ξ)F x [ψ + ] (ξ) for x 3 > 0, -i 4π R 2 e ix•ξ e x3 √ |ξ| 2 +m 2 -a 2 Γ -i (λ -µβ)α 3 F x [ψ + ] (ξ) for x 3 < 0, (4.29)
is an eigenvector associated to the eigenvalue a = -mµ/ǫ.

δ-interactions supported on compact Ahlfors regular surfaces

As the title of this section indicates, here we study the spectral properties of the Dirac operator H κ in the non-critical case, when the δ-interactions are supported on the boundary of a bounded rough domain Ω + . More precisely, we assume that Ω + is uniformly rectifiable (see the definitions below), as usual ∂Ω + = Σ ⊂ R 3 divide the space into two regions Ω ± . Before stating the main results, we need to give a few preliminary definitions. Namely, we recall the necessary definitions related to the notion of bounded regular Semmes-Kenig-Toro domains developed by S. Hofmann, M. Mitrea, and M. Taylor in [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]. Let Ω ⊂ R 3 be open with locally finite perimeter, that is the characteristic function of Ω, denoted by ½ Ω , satisfies the following:

ω := ∇½ Ω , (5.1)
with ω a locally finite R 3 -valued measure and the equality (5.1) can be understood in the sense of distributions as follows:

Ω divf dx = ∂Ω N , f dι, ∀f ∈ C ∞ 0 (R 3 , R 3 ), (5.2)
here ω = -N ι, ι is a locally finite positive measure, supported on ∂Ω, and N ∈ L ∞ (∂Ω) is an R 3valued function, satisfying |N (x)| = 1, ι-a.e. x. Moreover, if we denote by B(x, r) the ball of radius r centred on x, then for ι-a.e. x, it holds that

lim rց0 1 ι(B(x, r)) B(x,r)
N dι = N (x). (5.3) Definition 5.1 (Ahlfors regularity). We say that a closed set E ⊂ R 3 is 2-dimensional Ahlfors regular if there exist 0 < a b < ∞ such that ar 2 H 2 (B(x, r) ∩ E) br 2 , ∀x ∈ E, r ∈ (0, diam(E)), (5.4) where H 2 is the two dimensional Hausdorff measure and diam(E) denotes the diameter of E, that is diam(E) := sup x,y∈E |x -y|. If Ω ⊂ R 3 is an open set such that ∂Ω is Ahlfors regular, then we say that Ω is an Ahlfors regular domain. Definition 5.2 (uniformly rectifiable domains). We say that a compact set E ⊂ R 3 is uniformly rectifiable provided that it is Ahlfors regular and the following holds. There exist ρ, M ∈ (0, ∞) (called the uniform rectifiability constants of E) such that for each x ∈ E, r ∈ (0, 1], there is a Lipschitz map φ : B r → R 3 (where B r is a ball of radius r in R 2 ) with Lipschitz constant L φ M , such that

H 2 (E ∩ B(x, r) ∩ φ(B r )) ρr 2 .
(5.5)

A nonempty, proper and bounded open subset Ω ⊂ R 3 is called uniformly rectifiable, provided that ∂Ω is uniformly rectifiable and also H 2 (∂Ω \ ∂ * Ω ′ ) = 0, here ∂ * Ω is defined as follows:

∂ * Ω := {x ∈ ∂Ω : (5.3) holds, with |N (x)| = 1} (5.6)
Definition 5.3 (Corkscrew condition). We say that an open set Ω ⊂ R 3 satisfies the Corkscrew condition if there are constants C > 1 and r 0 > 0 such that for all x ∈ ∂Ω and r ∈ (0, r 0 ) there exists y ∈ Ω (which depends on x and r), such that |x -y| < r and dist(y, ∂Ω) > r/C. Also Ω satisfies the exterior corkscrew condition if R 3 \ Ω satisfy the interior corkscrew condition. Finally, Ω satisfies the two-sided corkscrew condition if it satisfies both the interior and exterior corkscrew conditions. Definition 5.4 (Harnack Chain condition). We say that an open set Ω ⊂ R 3 satisfies the Harnack Chain condition if for every ǫ > 0 and every pair of points x 1 , x 2 ∈ Ω ∩ B(y, r/4) for some y ∈ ∂Ω, r ∈ (0, r 0 ) (with reference to C and r as in Definition 5.3), and if dist(x j , ∂Ω) > ǫ and |x 1 -

x 2 | < 2 k ǫ,
there is a chain of open balls B 1 , . . . , B n ⊂ Ω, n Ck such that each B j has a radius r j , with x 1 ∈ B 1 , x 2 ∈ B n and B j ∩ B j+1 = ∅ for 1 j n and having the following property

r j C < dist(B, ∂Ω) < Cr j and diam(B j ) 1 C min (dist(x 1 , ∂Ω), dist(x 2 , ∂Ω)) .
One remark is in order here. Generally speaking, the Corkscrew condition is a quantitative, scale invariant version of openness, and the Harnack Chain condition is a scale invariant version of path connectedness.

Definition 5.5 (two-sided NTA domains). We say that a nonempty, proper open set Ω of R 3 is an NTA (non-tangentially accessible) domain if Ω satisfies both the two-sided Corkscrew and Harnack Chain conditions. Furthermore, we say that Ω is a two-sided NTA domain if both Ω and R 3 \ Ω are non-tangentially accessible domains.

Definition 5.6 (Separation property). A nonempty, bounded proper subset Ω ⊂ R 3 is said to satisfy the separation property if there exists r 0 > 0 such that for each x ∈ ∂Ω and r ∈ (0, r 0 ] there exists a 2-dimensional plane P (x, r) in R 3 passing through x and a choice of unit normal vector N x,r to P (x, r) such that y + tN x,r ∈ B(x, r) : y ∈ P (x, r), t < -r/4 ⊂ Ω, and

y + tN x,r ∈ B(x, r) : y ∈ P (x, r), t > r/4 ⊂ R 3 \ Ω.
Moreover, if Ω is unbounded, we also require that ∂Ω divides R 3 into two distinct connected components and that R 3 \ Ω has a non-empty interior.

Definition 5.7 (Reifenberg flatness). Let E ⊂ R 3 be a compact set and let δ ∈ (0, 1/4 √ 2). We say that E is δ-Reifenberg flat if there exists r 0 > 0 such that for every x ∈ E and every r ∈ (0, r 0 ] there exists a 2-dimensional plane P (x, r) which contains x such that where dS = H 2 ⇂ ∂Ω and BMO(∂Ω, dS) stands for the space of functions with bounded mean oscillation, relative to the surface measure dS = H 2 ⇂ ∂Ω. Definition 5.10 (Regular SKT Domains). We say that a bounded open set Ω ⊂ R 3 is a regular SKT domain provided Ω is δ-Reifenberg flat domain, for some δ ∈ (0, δ 0 ), and whose geometric measure theoretic outward unit normal ν ∈ VMO(∂Ω, dS) 3 . Here VMO(Σ, dS) stands for the Sarason space of functions with vanishing mean oscillation on Σ, relative to the surface measure dS = H 2 ⇂ ∂Ω.

In what follows, we always suppose that Ω + is a bounded, uniformly rectifiable domain, and we set ∂Ω + := Σ. Hence Σ is compact, 2-dimensional Ahlfors regular. Moreover, if we let Ω -= R 3 \ Ω + , then Ω -is uniformly rectifiable (see [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]Proposition 3.10]) and we have R 3 = Ω + ∪ Σ ∪ Ω -. As we mentioned in the introduction, in this section, the self-adjointness of the Dirac operator H κ will be derived using the main result of [START_REF] Arrizabalaga | Shell interactions for Dirac operators[END_REF]. However, the way the result [2, Theorem 2.11] is stated doesn't take into account the case when Σ is Ahlfors regular, therefore, a few comments on how to extend it should be in order. In fact, the only thing left to do, is to give the "good" trace theorem for the functions of the Sobolev space H 1 (R 3 ) in the case of Ahlfors regular surfaces, and this is the purpose of the next proposition. Of course [START_REF] Arrizabalaga | Shell interactions for Dirac operators[END_REF]Theorem 2.11] remains valid in a more general context, but to go further in our analysis we restrict ourselves to the class of bounded uniformly rectifiable domains. Let dS = H 2 ⇂ Σ, recall that the Besov space B 2 1/2 (Σ) (see [START_REF] Jonsson | Function spaces on subsets of R n[END_REF]Chapter V] for example), consists of all functions g ∈ L 2 (Σ) for which

|x-y|<1 |g(x) -g(y)| 2 |x -y| 3 dS(y)dS(x) < ∞. (5.8)
The Besov space is equipped with the norm

g 2 B 2 1/2 (Σ) := Σ |g(x)| 2 dS(x) + |x-y|<1 |g(x) -g(y)| 2
|x -y| 3 dS(y)dS(x). (5.9)

In the following, we use the notation B(x,r) U (y)dy for the mean of U in B(x, r). Given U ∈ H 1 (R 3 ), we set Proposition 5.1. Let Σ be as above. Then the trace operator t Σ (which until now was defined on D(R 3 )) extend to a bounded linear operator T Σ from H 1 (R 3 ) to B 2 1/2 (Σ) (where T Σ is given by (5.10)) with a bounded linear inverse operator E from B 2 1/2 (Σ) to H 1 (R 3 ). In other words, B 2 1/2 (Σ) is the trace to Σ of H 1 (R 3 ) and T Σ E is the identity operator.

T Σ u(x) := lim
Throughout this section, we often use the fact that the trace operator T Σ coincide with t Σ , when Ω + is a Lipschitz domain (i.e H 1/2 (Σ) is the trace to Σ of H 1 (R 3 )). Now, using Proposition 5.1 instead of [2, Proposition 2.6], and taking into account [2, Lemma 2.8 and Lemma 2.10] and [2, Remark 2.12], the main result of [START_REF] Arrizabalaga | Shell interactions for Dirac operators[END_REF] (more precisely [2, Theorem 2.11 (iii)]) reads as follows:

Theorem 5.1. Let Ω + and Σ be as above. Let Λ :

L 2 (Σ) 4 -→ L 2 (Σ) 4 be a bounded linear self-adjoint operator. Define T = H + V with dom(T ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ L 2 (Σ) 4 and T Σ u = Λ[g] (5.11) and T (u + Φ[g]) = Hu, i.e V (u + Φ[g]) = -g for all u + Φ[g] ∈ dom(T ). If Λ is Fredholm, then (T, dom(T )) is self-adjoint.
For the convenience of the reader, we begin our study with the subclass of bounded Lipschitz domains with VMO normals, where we can discuss the spectral properties of the Dirac operator H κ , for any κ ∈ R 3 with sgn(κ) = 0, 4. This is the purpose of the next subsection.

δ-interactions supported on the boundary of a Lipschitz domain with VMO normal.

In what follows, unless otherwise specified, we always suppose that Σ satisfies the following property:

where the kernels K j , j = 1, 2, are given by

K 1 (x, y) = e i √ z 2 -m 2 |x-y| 4π|x -y| (α • N (x)) z + mβ + z 2 -m 2 α • x -y |x -y| + e i √ z 2 -m 2 |x-y| 4π|x -y| z + mβ + z 2 -m 2 α • x -y |x -y| (α • N (y)) + e i √ z 2 -m 2 |x-y| -1 4π|x -y| 3 [(α • N (x))(iα • (x -y)) + (iα • (x -y)) (α • N (y))] . K 2 (x, y) = i 4π|x -y| 3 ((N (x))(α • (x -y)) + α • (x -y))(N (y))) .
Using the estimate

e i √ z 2 -m 2 |x| -1 z 2 -m 2 |x|, (5.14) it easily follows that sup 1 k,j 4 |K 1 (x -y)| = O(|x -y| -1 ) when |x -y| -→ 0.
(5.15) Once (5.15) has been established, working component by component and using [START_REF] Folland | Introduction to partial differential equations[END_REF]Lemma 3.11], one can show that T K1 is a compact operator in L 2 (Σ) 4 . Now it is straightforward to check that

T K2 [g](x) = K[g](x) + K * [g](x) + 3 j=1 3 k=1 k =j α j α k [N j , R k ] [g](x).
(5. [START_REF] Coifman | Extensions of Hardy spaces and their use in analysis[END_REF] where K denotes the matrix valued harmonic double layer and K * is the associated adjoint operator, and R k are the matrix versions of the Riesz transforms. That is, for x ∈ Σ and g ∈ L 2 (Σ) 4 , we have

K[g](x) = lim ρց0 |x-y|>ρ N (y) • (x -y) 4π|x -y| 3 I 4 g(y)dS(y), K * [g](x) = lim ρց0 |x-y|>ρ N (x) • (x -y) 4π|x -y| 3 I 4 g(y)dS(y), R k [g](x) = lim ρց0 |x-y|>ρ
x ky k 4π|x -y| 3 I 4 g(y)dS(y).

(5.17)

Since the adjoint of a compact operator is a compact operator and α j 's are constants matrices, using Theorem 5.3 and working component by component, we get that T K2 is a compact operators in L 2 (Σ) 4 . 4 and this finishes the proof of the lemma. Note that Lemma 5.1 is not valid for general Lipschitz surfaces. In fact, it turns out that assuming (H3) means that we are excluding the special class of corner domains. Indeed, from Theorem 5.3 we know that any bounded Lipschitz domain Ω is an NTA domain and ∂Ω is Ahlfors regular. However, in our context (i.e R 3 = Ω + ∪ Σ ∪ Ω -), the presence of any angle θ = 0 implies that dist(N , VMO(∂Ω, dS) 3 ) > 0, where the distance is taken in BMO(∂Ω, dS) 3 , cf. [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]Proposition 4.38] and the discussion that precedes it. Hence, Ω + is not a regular SKT domain and then by [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]Theorem 4.47], the principale value of the harmonic double layer K and the commutators [N j , R k ], 1 j, k 3, are not compact on L 2 (Σ). So, {α • N , C Σ } is not a compact operator on L 2 (Σ), and thus the assumption (H3) is sharp. To make this clearer, we have the following result:

Therefore {α • N , C z Σ } is a compact operator in L 2 (Σ)
Theorem 5.4. Let Ω + be a bounded Lipschitz domain, such that the decomposition

R 3 = Ω + ∪ Σ ∪ Ω - holds, where ∂Ω + = Σ. Then, Σ satisfies (H3) if and only if {α • N , C Σ } is compact in L 2 (Σ) 4 .
Proof. The first implication follows from Lemma 5.1 by taking z = 0. Let us prove the reverse implication, so assume that {α • N , C Σ } is compact in L2 (Σ) 4 . Given g ∈ L 2 (Σ) 2 , we define the bounded, linear operator W : L 2 (Σ) 2 -→ L 2 (Σ) 2 as follows:

W [g](x) = lim ρց0 |x-y|>ρ iσ • (x -y)
4π|x -y|3 g(y)dS(y), (5.18) where σ = (σ 1 , σ 2 , σ 3 ) are the Pauli matrices defined by (2.13). That W is bounded in L 2 (Σ) 2 is a consequence of Lemma 2.1 and Remark 2.1. Now, from (5.13) it holds that

{α • N , C Σ } = T K1 + {σ • N , W } 0 0 {σ • N , W } , (5.19)
where T K1 is a compact operator in L 2 (Σ) 4 . From this, it follows that

{α • N , C Σ } is compact in L 2 (Σ) 4 ⇐⇒ {σ • N , W } is compact in L 2 (Σ) 2 . (5.20)
Hence, it remains to show that

{σ • N , W } is compact in L 2 (Σ) 2 =⇒ Σ satisfies (H3). (5.21)
For that, from Theorem 5.3, we know that Ω + is a two-sided NTA domain and Σ is Ahlfors regular.

So Ω + satisfies the two-sided corkscrew condition and Σ is Ahlfors regular. Hence, Ω + is a uniformly rectifiable by [START_REF] David | Morceaux de graphes lipschitziens et intégrales singulières sur une surface[END_REF] (see also [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]Corollary 3.9]). Next, we claim that there exists C > 0, depending only on the dimension, the uniform rectifiability and Ahlfors regularity constants of Σ, such that dist N , VMO(∂Ω, dS) 3 Cdist {σ • N , W }, L c (L 2 (Σ) 2 ) , (5.22) where the distance in the right-hand side is measured in L(L 2 (Σ) 2 ). Here L(L 2 (Σ) 2 ) (respectively L c (L 2 (Σ) 2 )) denotes the set of bounded operators (respectively bounded and compact operators) from L 2 (Σ) 2 into itself. Now, assume for instance that (5.22) holds true. Since {σ • N , W } is compact in L 2 (Σ) 2 , by (5.22), it holds that N ∈ VMO(∂Ω, dS) 3 . Therefore, Σ satisfies (H3), which proves the theorem. Now, let us come back to the proof of (5.22). Given x, y ∈ R 3 , we define the following multiplication operator

x ⊙ y := (σ • x)(-σ • y). (5.23)
Using the anticommutation properties of the Pauli matrices, it is easy to check that:

x ⊙ x := -|x| 2 , x ⊙ y + y ⊙ x = -2(x • y)I 2 , ∀x, y ∈ R 3 .
Now, we make the observation that the multiplication operator defined by (5.23) has the same properties as the multiplication operator in the Clifford algebra Cl 3 (see [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]Section 4.6] for the precise definition). Moreover, W (σ • N ) plays the same role as the Cauchy-Clifford operator defined on L 2 (Σ) ⊗ Cl 31 (i.e. it acts on Cl 3 -valued functions), cf. [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]Section 4.6]. Thus, one can adapt the same arguments of [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]Theorem 4.46] and show that the claim (5.22) holds true, we omit the details. This completes the proof of the theorem.

As it was done in [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]Theorem 4.47], one can also characterize the class of bounded regular SKT domains via the compactness of the anticommutators {σ • N , W } in L 2 (Σ) 2 , or equivalently via the compactness of the anticommutator {α • N , C Σ } in L 2 (Σ) 4 . This is the purpose of the following proposition.

Corollary 5.1, we conclude that Λ z + : L 2 (Σ) 4 -→ L 2 (Σ) 4 is bijective and thus (5.26) makes sense. Now given v ∈ L 2 (R 3 ) 4 , we set

ϕ = (H -z) -1 v -Φ z (Λ z + ) -1 (Φ z ) * [v].
To prove item (ii), it remains to show that ϕ ∈ dom(H κ ). For this, remark that ϕ 4 . Consequently, we get that g ∈ L 2 (Σ) 4 and u ∈ H 1 (R 3 ) 4 . Moreover, using Lemma 2.1(ii), we obtain

= u + Φ[g] where u = (H -z) -1 v -(Φ z -Φ)(Λ z + ) -1 (Φ z ) * [v] and g = -(Λ z + ) -1 (Φ z ) * [v]. Note that (Λ z + ) -1 (Φ z ) * is a bounded operator from L 2 (R 3 ) 4 to L 2 (Σ) 4 and (H -z)u = v + zΦ[g] ∈ L 2 (R 3 )
t Σ u = (Φ z ) * -(C z Σ -C Σ )(Λ z + ) -1 (Φ z ) * [v] = (Φ z ) * -(Λ z + -Λ + )(Λ z + ) -1 (Φ z ) * [v] = -Λ + [g].
Thus ϕ ∈ dom(H κ ), which yields (ii). Item (iii) is a consequence of item (ii). Indeed, since 4 , using the Sobolev injection it follows that

(Φ z ) * is a bounded operator from L 2 (R 3 ) 4 to H 1/2 (Σ)
Φ z (Λ z + ) -1 (Φ z ) * is a compact operator on L 2 (R 3 ) 4 . Thus (H κ -z) -1 -(H-z) -1 is a compact operator on L 2 (R 3 ) 4 , therefore Weyl's theorem yields Sp ess (H κ ) = Sp ess (H) = (-∞, -m] ∪ [m, +∞). Finally, item (
iv) follows by Proposition des equivalences.

The reader interested on confinement may wonder if the Dirac operator H κ generates this phenomenon under the assumption that sgn(κ) = -4. Our aim in what follows is to clarify and provide an answer to this question. Let 4 and x ∈ Ω ± . Let H κ be as in Theorem 5.2 and assume that sgn(κ) = -4. Given

Φ ± : L 2 (Σ) 4 -→ L 2 (Ω ± ) 4 be the operators defined by Φ Ω± [g](x) = Φ[g](x), for g ∈ L 2 (Σ)
ϕ = u + Φ[g] ∈ dom(H κ ), we set ϕ ± := ϕ| Ω± = u| Ω± + Φ Ω± [g]. (5.27)
For simplicity, we denote by lim nt ϕ ± the nontangential limit of ϕ ± . By definition it holds that

lim nt ϕ ± = t Σ u + lim nt Φ Ω± [g] = t Σ u + (C Σ ∓ i 2 (α • N ))[g] = 1 4 (ǫ -µβ -η(α • N )) ∓ i 2 (α • N ) g, (5.28)
where in the last equality we used that t Σ u = -Λ + [g]. Now, multiplying the identity (5.28) by

1 2 (ǫ + µβ + η(α • N )) ± i(α • N ) ,
we get

1 2 (ǫ + µβ + η(α • N )) ± i(α • N ) lim nt ϕ ± = ∓iηg. (5.29)
As consequence, if sgn(κ) = -4 and η = 0, then H κ cannot generate confinement. Hence Σ is penetrable. Clearly, if we set η = 0 in (5.29), then H κ generates confinement, but we postpone this case to the next subsection, where we establish that in a more general context.

δ-interactions supported on the boundary of a bounded uniformly rectifiable domain.

Here we discuss special cases where we can show the self-adjointness of H κ , when Ω + is bounded uniformly rectifiable and η = 0. The idea is to identify some situations where the operator Λ + gives rise to a Fredholm operator, and thereby use Theorem 5.1 to get the self-adjointness of H κ . So in this subsection the domain Ω + is bounded uniformly rectifiable unless stated otherwise. Before going any further, we remind the reader that, since Ω + is bounded uniformly rectifiable, all the singular integral operators defined in this article are bounded in their corresponding L 2 -spaces; cf [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]. Given g ∈ L 2 (Σ), we define the Hardy-Littlewood maximal operator by

M Σ g(x) = sup r>0 B(x,r)∩Σ |g(y)|dS, x ∈ Σ. (5.30)
Then, by [16, p. 624], there is C > 0 such that

M Σ g L 2 (Σ) C g L 2 (Σ) (5.31)
For the convenience of the reader, in what follows we give the main ideas to establish Lemma 2.1(i) in the case of uniformly rectifiable domains. Let U be a function defined in Ω ± , we denote the nontangential maximal function of U on Σ by

N Ω± a (x) = sup{|U (y)| : y ∈ Γ Ω± (x)}. (5.32)
Recall the operator W from (5.18). Then we have the following result.

Proposition 5.3. Assume that Ω + is bounded uniformly rectifiable, then Lemma 2.1(i) is still holds true. Moreover, it holds that

((σ • N )W ) 2 = (W (σ • N )) 2 = - 1 4 I 2 . (5.33) Proof. Given z ∈ C \ ((-∞, -m] ∪ [m, ∞)). Set k(x) := φ z (x) -i(α • x |x| 2 ), for all x ∈ R 3 \ {0}, (5.34)
Then there is a constant C such that |k(ω, y)| C/|ω -y| 3/2 := k(ω, y), for ω, y ∈ Ω + . Define

T [g](x) = Σ k(x, y)g(y)dS(y), (5.35)
Clearly, T is bounded in L 2 (Σ) 4 . Now, recall the definition of Γ Ω± (x) from (2.10). Let x ∈ Σ and

ω ∈ Γ Ω± (x), then B(ω,2|x-ω|)∩Σ k(ω, y)g(y)dS(y) B(ω,2|x-ω|)∩Σ C 1 + a |x -ω| 3/2
|g(y)|dS(y). (5.36) Using the Ahlfors regularity, it follows that there is C 1 depending only on the Ahlfors regularity constant of Ω + such that

B(ω,2|x-ω|)∩Σ k(ω, y)g(y)dS(y) C 1 |x -ω| 1/2 M Σ g(x). (5.37) Let y ∈ Σ \ B(x, 2|x -ω|), then |ω -y| 2|x -y|, and thus |k(ω, y)| k(ω, y) 2 3 k(x, y). Therefore Σ\B(ω,2|x-ω|) k(ω, y)g(y)dS(y) 2 3 T [|g|](x). (5.38)
Thus, (5.37), (5.38) and the dominate convergence theorem yield that

lim Γ Ω + (x)∋ω-→ nt x Σ k(ω, y)g(y)dS(y) = Σ k(x, y)g(y)dS(y), (5.39)
holds for all g ∈ L 2 (Σ) 4 and dS-a.e. x ∈ Σ. Similarly, one can show that

lim Γ Ω -(x)∋ω-→ nt x Σ k(ω, y)g(y)dS(y) = Σ k(x, y)g(y)dS(y), (5.40)
holds for all g ∈ L 2 (Σ) 4 and dS-a.e. x ∈ Σ. Now, let φ be the fundamental solution of the massless Dirac operator -iσ • ∇, that is

φ(x) = iσ • x |x| 3 , for all x ∈ R 3 \ {0}. (5.41)
Now, we define the bounded operator Φ :

L 2 (Σ) 2 -→ L 2 (R 3 ) 2 as follows Φ[g](x) = Σ φ(x -y)g(y)dS(y), for all x ∈ R 3 \ Σ, (5.42) Observe that (-iσ • ∇) Φ[g] = 0 holds in D ′ (Ω ± ) 2 , for all g ∈ L 2 (Σ) 2 . We set W ± [g](x) := lim Γ Ω ± (x)∋y-→ nt x Φ[g](y). (5.43)
Then, by [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]Proposition 3.30] it follows that W ± [g](x) exist for dS-a.e. x ∈ Σ, and 2 are linear bounded operators. Moreover, the following holds true:

W ± : L 2 (Σ) 2 → L 2 (Σ)
W ± = ∓ i 2 (σ • N ) + W. (5.44)
Therefore, Lemma 2.1(i) follows from the above considerations and this completes the proof the first statement. The proof of the second statement is a relatively straightforward modification of the technique used in the proof of [2, Lemma 3.3](ii). Indeed, by [28, p. 2659] it follows that

N Ω± a ( Φ[g]) L 2 (Σ) 2 C g L 2 (Σ) 2 , (5.45)
for some C > 0 depending only on a as well as the Ahlfors regularity and the uniform rectifiability [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]Theorem 4.49] it holds that

constants of Σ. Using that (-iσ • ∇) Φ[g] = 0 in Ω ± , by
Φ[g] = Σ φ(x, y)(±iσ • N (y))g(y)dS(y), x ∈ Ω ± . (5.46)
Although [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]Theorem 4.49] was stated in the case of tow-sided NTA domains it also holds for uniformly rectifiable domains by the discussion on [28, p. 2758]. Now, given x ∈ Ω + and g ∈ L 2 (Σ) 2 .

Then, (5.46) yields that

Φ[(iσ • N )g](x) = Φ[(iσ • N )W + (iσ • N )g](x) (5.47)
Now, by taking the nontangential limit in (5.47) and using (5.44), we then obtain (5.33). This completes the proof of the proposition.

Given z ∈ C \ ((-∞, -m] ∪ [m, ∞)), recall that Λ z ± are defined in Notation 2.1. Let κ ∈ R 3 such that sgn(κ) = 0, 4. Then, the Dirac operator H κ = H + V κ acting in L 2 (R 3 ) 4 , is defined on the domain dom(H κ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ L 2 (Σ) 4 , T Σ u = -Λ + [g] , (5.48)
where T Σ is the trace operator defined in Proposition 5.1, and

V κ (ϕ) = 1 2 (ǫI 4 + µβ + η(α • N )))(ϕ + + ϕ -)δ Σ , (5.49) with ϕ ± = T Σ u + C ± [g],
and H κ acts in the sens of distributions as H κ (ϕ) = Hu, for all ϕ = u + Φ[g] ∈ dom(H κ ). Furthermore, as we did in the proof of Theorem 5.2, one can check easily that

V κ (ϕ) = -gδ Σ . Now, set (T, dom(T )) = (H κ , dom(H κ ), V = V κ , Λ = -Λ + .
(5.50) If Λ + is Fredholm, we then fall under the conditions of Theorem 5.1.

From now we suppose that η = 0. Thus, H κ coincide with H ǫ,µ , the Dirac operator with electrostatic and Lorentz scalar δ-shell interactions supported on Σ. The first main result on the spectral properties of the Dirac operator H ǫ,µ reads as follows: 4 . If we assume in addition that there is

Theorem 5.6. Let λ, µ ∈ R such that 0 < |ǫ 2 -µ 2 | < 1/ C Σ 2 L 2 (Σ) 4 →L 2 (Σ) 4 , then H ǫ,µ is self-adjoint. In particular, if Ω + is Lipschitz, then dom(H ǫ,µ ) ⊂ H 1/2 (R 3 \ Σ)
z 0 ∈ C \ R, such that |ǫ 2 -µ 2 | < 1/ C z0 Σ 2 L 2 (Σ) 4 →L 2 (Σ) 4 , then it holds that Sp ess (H ǫ,µ ) = (-∞, -m] ∪ [m, +∞). (5.51) Proof. Fix ǫ, µ ∈ R such that 0 < |ǫ 2 -µ 2 | < 1/ C z Σ 2 L 2 (Σ) 4 →L 2 (Σ) 4 , holds for some z ∈ C \ ((-∞, -m] ∪ [m, ∞)).
Then, from the proof of Corollary 5.1 we have

Λ z ∓ Λ z ± = 1 ǫ 2 -µ 2 -(C z Σ ) 2 + 2µ ǫ 2 -µ 2 (mI 4 + zβ)S z , (5.52) Recall that C z Σ is bounded in L 2 (Σ) 4 . Using Neumann's lemma, it follows that M z := (I -(ǫ 2 - µ 2 )(C z Σ ) 2
) is a bounded invertible operator in L 2 (Σ) 4 . Now, since (mI 4 + zβ) is bounded and S is a compact on L 2 (Σ) 4 , we therefore get that 4 . Combining this, from (5.52) it holds that

K z := 2µ ǫ 2 -µ 2 (mI 4 +zβ)S z is compact on L 2 (Σ)
I -(ǫ 2 -µ 2 )M -1 z Λ z -Λ z + = -(ǫ 2 -µ 2 )M -1 z K z , I -(ǫ 2 -µ 2 )Λ z + Λ z -M -1 z = -(ǫ 2 -µ 2 )K z M -1 z .
(5.53) 4 , then [1, Theorem 1.50 and Theorem 1.51] yields that Λ z + is Fredholm. Hence, the first statement is a direct consequence of Theorem 5.1 and the fact that Λ + is a self-adjoint, Fredholm operator on L 2 (Σ) 4 . Now, assume that Ω + is Lipschitz, then dom(H ǫ,µ ) ⊂ H 1/2 (R 3 \ Σ) 4 is a consequence of Remark 2.1. The last statement follows by the same method as in Theorem 5.5 using the Fredholm property of Λ z0 + , we omit the details.

As M -1 z Λ z -and Λ z -M -1 z are bounded operators on L 2 (Σ) 4 , M -1 z K z and K z M -1 z are compact on L 2 (Σ)
Remark 5.1. From Lemma 2.1(ii) it easily follows that C z Σ 1/2 (cf. [3, Remark 3.5]), which implies that |ǫ 2µ 2 | < 4. Hence, the combination of coupling constants ǫ and µ is not critical. Of course, we already know that the above result is false in the case ǫ 2µ 2 = 4. Note that Theorem 5.6 remains valid if one control the norm of the Cauchy operator instead of controlling the combination of interactions. However, this can influence the geometrical characterization of Σ which can imply an increase in terms of regularity.

As it was mentioned in the introduction, under the assumption that m = 0 and µ ∈ (-2, 2), it was proved in [START_REF] Pizzichillo | Self-Adjointness of two dimensional Dirac operators on corner domains[END_REF] the existence of a unique self-adjoint realization of the two dimensional Dirac operator with pure Lorentz scalar δ-shell interactions, where Σ is a closed curve with finitely many corners. it seems that their assumption is related to the assumption that we imposed in the Theorem 5.6.

Although Theorem 5.6 gives an upper bound for |ǫ 2µ 2 | so that H ǫ,µ is self-adjoint, this is not satisfactory in the sense that this bound involves C Σ 2 L 2 (Σ) 4 →L 2 (Σ) 4 , which is not easy to quantify. In what follow, we are going to remove this restriction by imposing a sign hypothesis on the coupling constants, and give a better quantitative assumption than the one of Theorem 5.6. The next theorem makes this more precise.

Theorem 5.7. Let λ, µ ∈ R such that |ǫ| = |µ|, and let (H ǫ,µ , dom(H ǫ,µ , )) be as above. Assume that ǫ and µ satisfy one of the the following assumptions: 4 . Moreover, the following statements hold true: ), for all a ∈ (-m, m).

(a) µ 2 > ǫ 2 . (b) ǫ 2 > µ 2 and 16 W 2 L 2 (Σ) 2 →L 2 (Σ) 2 < ǫ 2 -µ 2 < 1/ W 2 L 2 (Σ) 2 →L 2 (Σ) 2 . Then H ǫ,µ is self-adjoint. In particular, if Ω + is Lipschitz, then dom(H ǫ,µ ) ⊂ H 1/2 (R 3 \ Σ)
(i) Given a ∈ (-m, m), then Kr(H ǫ,µ -a) = 0 ⇐⇒ Kr(Λ a + ) = 0. (ii) For all z ∈ C \ R, it holds that (H ǫ,µ -z) -1 = (H -z) -1 -Φ z (Λ z + ) -1 (Φ z ) * . (iii) Sp ess (H ǫ,µ ) = (-∞, -m] ∪ [m, +∞). (
(vi) C 0 := sup a∈[-m,m] C a Σ < ∞. Moreover, Sp disc (H ǫ,µ ) ∩ (-m, m) = ∅ either if |ǫ -µ| < 1/C 0 and |ǫ + µ| < 1/C 0 , or if |ǫ -µ| > 4C 0 and |ǫ + µ| > 4C 0 .
Proof. To prove the theorem, in both situations, we prove that

Λ z + is Fredholm for all z ∈ C \ ((-∞, -m] ∪ [m, ∞)).
Once this is shown, we use the fact that Λ + is a bounded self-adjoint operator, and we conclude by using Theorem 5. (5.55) Hence, T z K is compact in L 2 (Σ) 4 . Therefore, in the same way as in (5.52) we get that

Λ z ∓ Λ z ± = 1 ǫ 2 -µ 2 -W 2 + (T z K ) 2 + {T z K , W } + 2µ ǫ 2 -µ 2 (mI 4 + zβ)S z := 1 ǫ 2 -µ 2 -W 2 + K, (5.56)
where K is compact in L 2 (Σ) 4 . Now observe that

W 2 = W 2 0 W 2 .
Now, if ǫ 2 < µ 2 holds. Then, using that W is a bounded self-adjoint operator in L 2 (Σ) 2 , it follows that W 2 is a nonnegative, self-adjoint operator on L 2 (Σ) 4 . From this, it follows that 1/(ǫ 2µ 2 ) belongs to the resolvent set of W 2 and hence I 4 -(ǫ 2µ 2 ) W 2 in invertible on L 2 (Σ) 4 . In another hand, assume that ǫ 2 > µ 2 and ǫ 2 - 4 . In both cases, similar arguments to those of the proof of Theorem 5.6 yield that Λ z + is Fredholm, which proves the first statement of the theorem for this two cases. Now we deal with the case 16 W 2 L 2 (Σ) 2 →L 2 (Σ) 2 < ǫ 2µ 2 . From Proposition 5.33, we have that W is invertible on L 2 (Σ) 2 and W -1 = -4(σ • N )W (σ • N ). Thus, from (5.56) it follows that

µ 2 < 1/ W 2 L 2 (Σ) 2 →L 2 (Σ) 2 hold, then Neumann's lemma yields that I 4 -(ǫ 2 -µ 2 ) W 2 is invertible on L 2 (Σ)
(ǫ 2 -µ 2 )( W -1 ) 2 Λ z -Λ z + = ( W -1 ) 2 -(ǫ 2 -µ 2 )I 4 + (ǫ 2 -µ 2 )( W -1 ) 2 K, (ǫ 2 -µ 2 )Λ z + Λ z -( W -1 ) 2 = ( W -1 ) 2 -(ǫ 2 -µ 2 )I 4 + (ǫ 2 -µ 2 )K( W -1 ) 2 .
(5.57) 2 , from this and Proposition ?? (ii) we get that 4 . Now, from (5.57) it follows that

As W L 2 (Σ) 4 →L 2 (Σ) 4 = W L 2 (Σ) 2 →L 2 (Σ)
W -1 L 2 (Σ) 4 →L 2 (Σ) 4 4 W L 2 (Σ) 4 →L 2 (Σ) 4 = 4 W L 2 (Σ) 2 →L 2 (Σ) 2 (5.58) Hence, if ǫ 2 -µ 2 > 4 W L 2 (Σ) 2 →L 2 (Σ) 2 , then ǫ 2 -µ 2 > W -1 2 L 2 (Σ) 4 →L 2 (Σ) 4 . Thus ǫ 2 -µ 2 is not in the spectrum of ( W -1 ) 2 . Thereby W -1 -(ǫ 2 -µ 2 )I 4 is invertible on L 2 (Σ)
I 4 -(ǫ 2 -µ 2 ) ( W -1 ) 2 -(ǫ 2 -µ 2 )I 4 -1 ( W -1 ) 2 Λ z -Λ z + = K 1 , I 4 -(ǫ 2 -µ 2 )Λ z + Λ z -( W -1 ) 2 ( W -1 ) 2 -(ǫ 2 -µ 2 )I 4 -1 = K 2 .
(5.59)

where K 1 and K 2 are compact operators on L 2 (Σ) 4 . Thereby, [1, Theorem 1.50 and Theorem 1.51] yields that Λ z + is Fredholm, and this finishes the proof of the first and the second statements. The proof of items (i),(ii), (iii) and (iv) runs as in the proof of Theorem 5.5. Item (v) is a direct consequence of (i). Indeed, denote by Λ a +,ǫ ′ ,µ ′ the Λ a + operator associated to the Dirac operator H -4ǫ

ǫ 2 -µ 2 , -4µ ǫ 2 -µ 2
.

Observe that Λ a +,ǫ ′ ,µ ′ is given by

Λ a +,ǫ ′ ,µ ′ = - 1 4 (ǫI 4 -µβ) + C a Σ . (5.60)
Using item (i) and Lemma 2.1 it follows that

a ∈ Sp disc (H ǫ,µ ) ∩ (-m, m) ⇐⇒ there is 0 = g ∈ L 2 (Σ) 4 : - 1 ǫ 2 -µ 2 (ǫI 4 -µβ)g = C a Σ [g] ⇐⇒ 4 ǫ 2 -µ 2 (ǫI 4 -µβ)((α • N )C a Σ ) 2 [g] = C a Σ [g] ⇐⇒ C a Σ ((α • N )C a Σ )[g] = 1 4 (ǫI 4 -µβ)((α • N )C a Σ )[g] ⇐⇒ there is 0 = f = ((α • N )C a Σ )[g] ∈ L 2 (Σ) 4 : Λ a +,ǫ ′ ,µ ′ [f ] = 0 ⇐⇒ a ∈ Sp disc (H -4ǫ ǫ 2 -µ 2 , -4µ ǫ 2 -µ 2 ) ∩ (-m, m).
Which proves (v). Now we turn to the proof of item (vi), the first claim of statement is contained in [START_REF] Arrizabalaga | Shell interactions for Dirac operators: on the point spectrum and the confinement[END_REF]Lemma 3.2] and [6, Proposition 3.5], for a C 2 -compact surface Σ, and the same arguments hold true in the Lipschitz case. To see the last claim, note that for all a ∈ (-m, m), we have

0 ∈ Sp disc (Λ a + ) ⇐⇒ -1 ∈ Sp disc ((ǫI 4 + µβ)C a Σ ). (5.61)
Using the first claim, it follows that if |ǫ -µ| < 1/C 0 and |ǫ + µ| < 1/C 0 , then

(ǫI 4 + µβ)C a Σ L 2 (Σ) 4 →L 2 (Σ) 4 < 1.
Therefore, -1 / ∈ Sp disc ((ǫI 4 + µβ)C a Σ ). Hence, (5.61) and (i) yield that Sp disc (H ǫ,µ ) ∩ (-m, m) = ∅. Using (v), the case |ǫ -µ| > 4C 0 and |ǫ + µ| > 4C 0 , follows by iterating the same arguments, which gives (vi). This completes the proof of theorem.

Remark 5.2. Assume that Ω + is Lipschitz. Then, following essentially the same arguments as in Theorem 5.7, one can show that , if for all a ∈ (-m, m) the following holds: 4 . However, if µ = 0, then from Theorem 5.7

16 C a Σ 2 L 2 (Σ) 4 →L 2 (Σ) 4 < ǫ 2 -µ 2 < 1/ C a Σ 2 L 2 (Σ) 4 →L 2 (Σ) 4 , then H ǫ,µ is self-adjoint with dom(H ǫ,µ ) ⊂ H 1/2 (R 3 \ Σ)
(vi) it follows that Sp disc (H ǫ,0 ) ∩ (-m, m) = ∅, see also [START_REF] Arrizabalaga | Shell interactions for Dirac operators: on the point spectrum and the confinement[END_REF]Theorem 3.3] and [START_REF] Behrndt | On the spectral properties of Dirac operators with electrostatic δ-shell interaction[END_REF]Theorem 4.4].

Remark 5.3. Note that in 5.7 (b), the combination of coupling constants ǫ and µ is not critical. Moreover, there is an interval J ⊂ R + , such that we have no information on the self-adjointness

character of H ǫ,µ , if ǫ 2 -µ 2 ∈ J.
Next, we discuss the particular case ǫ 2µ 2 = -4. Assume that Ω + is Lipschitz, given ϕ = u + Φ[g] ∈ dom(H ǫ,µ ), recall the definition of ϕ ± from (5.27). Define P ± = 1 2 (ǫ + µβ) ± i(α • N ) , then from (5.29) we know that P ± are projectors. As consequence, we have the following result. Proposition 5.4. Let ǫ, µ ∈ R such that ǫ 2µ 2 = -4, and let H ǫ,µ be as in Theorem 5.7. Then Σ is impenetrable. Moreover, if Ω + is Lipschitz then it holds that

H ǫ,µ ϕ = H Ω+ ǫ,µ ϕ + ⊕ H Ω- ǫ,µ ϕ -= (-iα • ∇ + mβ) ϕ + ⊕ (-iα • ∇ + mβ) ϕ - (5.62)
where H Ω± ǫ,µ are the self-adjoint Dirac operators defined on

dom(H Ω± ǫ,µ ) = ϕ ± := u Ω± + Φ Ω± [g], u Ω± ∈ H 1 (Ω ± ) 4 , g ∈ L 2 (Σ) 4 : P ± lim nt ϕ ± = 0 .
Proof. That Σ is impenetrable is consequence of [START_REF] Arrizabalaga | Shell interactions for Dirac operators: on the point spectrum and the confinement[END_REF]Theorem 5.4. and Theorem 5.5.] and Theorem 5.7 (i). Let us prove the second statement, so assume that Ω + is Lipschitz.

Given ϕ = u + Φ[g] = (ϕ + , ϕ -) ∈ dom(H ǫ,µ ), observe that t Σ u = -Λ + [g] ⇐⇒ t Σ u + C Σ [g] = - 1 4 (ǫI 4 -µβ)g ⇐⇒ 1 2 (lim nt ϕ + + lim nt ϕ -) = - i 4 (ǫI 4 -µβ)(α • N ))(lim nt ϕ + + lim nt ϕ -).
(5.63)

where Lemma 2.1 was used in the last step. Now from (5.63) it follows that

t Σ u = -Λ + [g] ⇐⇒ 1 2 (ǫI 4 -µβ) + i(α • N )) lim nt ϕ + = - 1 2 (ǫI 4 -µβ) -i(α • N )) lim nt ϕ - ⇐⇒ P + lim nt ϕ + = -P -lim nt ϕ -.
Thus, the second statement follows from this and the fact that P ± are projectors.

Remark 5.4. By Taking ǫ = 0 in Theorem 5.7 (i), we conclude that the free Dirac operator coupled with Lorentz scalar δ-shell interactions is always self-adjoint and Σ becomes impenetrable for µ = ±2.

In particular, if Ω + is Lipschitz, then dom(H ǫ,µ ) is included in H 1/2 (Σ) 4 , for any compact Lipschitz surface Σ.

The reason we assumed that η = 0 is due to the technique that we have use above. However, as we have already commented in Subsection 4.2, the coupling H + η(α • N )δ Σ is always self-adjoint. In fact, we have the following proposition.

Proposition 5.5. Assume that Ω + is Lipschitz. Let η ∈ R \ {0}, set κ = (0, 0, η) and let H κ be as above. Then H κ is self-adjoint and we have

dom(H κ ) = u + Φ[-4η 2 (η + 4) -1 (α • N )Λ -(α • N )[t Σ u]] : u ∈ H 1 (R 3 ) 4 .
Moreover, dom(H κ ) ⊂ H 1 (R 3 \ Σ) 4 and the spectrum of H κ is given by

Sp(H κ ) = Sp ess (H κ ) = (-∞, -m] ∪ [m, +∞). (5.64) Proof. Assume that η ∈ R \ {0} and fix z ∈ C \ ((-∞, -m] ∪ [m, ∞)). Recall that Λ z
± are given by

Λ z ± = 1 η (α • N ) ± C z Σ .
Now, using Lemma 2.1, a simple computation yields 4 , using Theorem 5.1 we then get the first statement. Now, by Remark 2.1 it follows that (Φ z ) * is bounded from L 2 (R 3 ) 4 onto H 1/2 (Σ) 4 . That Sp(H κ ) is characterized by (5.64) is a consequence of the Birman-Schwinger principle and the resolvent formula. This completes the proof of the proposition.

(η(α • N ))Λ z -(η(α • N ))Λ z + = Λ z + (η(α • N ))Λ z -(η(α • N )) = 1 + η 2 4 . Therefore, Λ z + is invertible with (Λ z + ) -1 = 4η 2 (η+4) -1 (α•N )Λ z -(α•N ). In particular, Λ z + is Fredholm, for all z ∈ C \ ((-∞, -m] ∪ [m, ∞)). As Λ + is invertible, self-adjoint in L 2 (Σ)
To finish this part, we discuss briefly the particular case µ = ±ǫ. Given ǫ ∈ R \ {0} and let H ǫ,±ǫ be as in Proposition 3.5. Note that from (3.32) we have

Λ -Λ + = 1 4ǫ 2 P ± - m 2 2ǫ (S) 2 P ± . (5.65)
Since S is bounded from H -1/2 (Σ) 4 to H 1/2 (Σ) 4 and hence compact on L 2 (Σ) 4 (see [START_REF] Mclean | Strongly Elliptic Systems and Boundary Integral Equations[END_REF]Theorem 6.11] for example), we get the following result.

Proposition 5.6. Let ǫ ∈ R \ {0} and assume that Σ is a compact Lipschitz surface. Then H ǫ,±ǫ is self-adjoint and

dom(H ǫ,±ǫ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ P ± H 1/2 (Σ) 4 , P ± t Σ u = -P ± Λ + [g] .
Proof. This readily follows from the compactness and regularization property of the operator S and Theorem 5.2. 5.3. δ-interactions supported on the boundary of a C 1,γ -domain. In this part, we discuss how the smoothness of the surface supporting the singular perturbation influence the regularity of the domain of the Dirac operator H κ in the non-critical case. As we have already seen in section 3, when Σ is a C 2 -smooth compact surface, from Theorem 3.1 and Remark 3.2 we know that the functions in dom(H κ ) are indeed in H 1 (R 3 \ Σ) 4 . However, such a result can fails if Σ is less regular. Indeed, there are two obstacles which prevent us from obtaining such a result. The first one is that (α • N )Λ + [g] should belongs to H 1/2 (Σ) 4 , which clearly fails if for example Σ is C 1 -smooth. The second reason is that we need also to extend the anticommutator {α • N , C Σ } to a bounded operator from L 2 (Σ) 4 to H 1/2 (Σ) 4 . Although again, we know that behind this operator there are components of the Riesz transforms as well as the principale value of the harmonic double layer operator and its adjoint, which do not have this property, even if Σ is C 1,1/2 -smooth.

In the following, we assume that Ω + is a bounded C 1,γ -smooth domain with γ ∈ (0, 1), and we set Σ = ∂Ω + . For all s ∈ (0, γ), we denote by H s (Σ) 4 the Sobolev space endowed with norm

g 2 H s (Σ) 4 := Σ |g(x)| 2 dS(x) + Σ Σ |g(x) -g(y)| 2 |x -y| 2(1+s) dS(y)dS(x). (5.66)
Note that this definition is equivalent to the one given in Section 2; cf. [START_REF] Mclean | Strongly Elliptic Systems and Boundary Integral Equations[END_REF]. The main result of this subsection reads as follows:

Theorem 5.8. Let κ ∈ R 3 such that sgn(κ) = 0, 4 and let H κ be as in Definition 2.1. Then H κ is self adjoint and the following hold:

(i) If γ 1/2, then for all s < γ we have

dom(H κ ) ⊂ u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ H s (Σ) 4 , t Σ u = -Λ + [g] ⊂ H 1/2+s (R 3 \ Σ) 4 . (ii) If γ > 1/2, then dom(H κ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ H 1/2 (Σ) 4 , t Σ u = -Λ + [g] ⊂ H 1 (R 3 \ Σ) 4 .
Proposition 5.7. There is a constant C > 0 such that for all x, y ∈ Σ, it hold that

|N (x) • (x -y)| C|x -y| 1+γ . (5.67)
Proof. Since Σ is C 1,γ -smooth, it suffies to prove the statement for |x -y| < 1. Without loss of generality (after translation and rotation if necessary), we may assume that x = 0 and N (x) = (0, 0, 1).

There is a C 1,γ -smooth function φ : B(0, 1) ⊂ R 2 -→ R such that φ(0) = 0, |∇φ(0)| = 0 and

B(0, 1) ∩ Σ = {x = (x 1 , x 2 , x 3 ) : x 3 = φ(x 1 , x 2 )}.
Then we get

|N (x) • (x -y)| = |y 3 | = |φ(y 1 , y 2 )| C|y| 1+γ . (5.68)
Therefore the statement is proven since Σ is compact.

In the following proposition, we prove that the anticommutator of the Cauchy operator C Σ and the multiplication operator by (α • N ) is bounded from L 2 (Σ) 4 to H s (Σ) 4 , for all s ∈ (0, γ). This result should be compared to [9, Proposition 3.10], where the authors showed that for Σ a C 2 -smooth compact surface, the commutator of the Cauchy operator C Σ with a Hölder continuous function of order a ∈ (0, 1) is bounded from L 2 (Σ) 4 to H s (Σ) 4 , for all s ∈ (0, a). In fact, both results are identical modulo a slight change of the assumptions.

Lemma 5.2. Suppose that Σ is C 1,γ . Then, for all s ∈ (0, γ), the anticommutator {α

• N , C Σ } is a bounded operator from L 2 (Σ) 4 to H s (Σ) 4 .
Proof. Let g ∈ L 2 (Σ) 4 , in the same manner as in the proof of Lemma 3.1, we can see that

{α • N , C Σ }[g](x) = y∈Σ K ′ (x, y)g(y)dS(y) + K * [g](x) := T K ′ [g](x) + K * [g](x), (5.69)
where K * is defined by (5.17) and the kernel K ′ is given by

K ′ [g](x) =φ(x -y)(α • (N (y) -N (x)) -m e -m|x-y| 2iπ|x -y| 2 (N (x) • (x -y))I 4 - e -m|x-y| -1 2iπ|x -y| 3 (N (x) • (x -y))I 4 . As Σ is C 1,γ -smooth, there is a constant C > 0 such that |N (x) -N (y)| C|x -y|.
Using this, the estimate (5.14) and Proposition 5.7, we obtain that |K ′ (x, y)| C|x -y| -1 . Hence the integral operator T K ′ is not singular, and since N is in the Hölder class C 0,γ (Σ) 4 , we then can adapt the proof of [43, Proposition 2.8] (see also [START_REF] Behrndt | Self-Adjoint Dirac Operators on Domains in R 3[END_REF]Proposition 3.10]) and show that T K ′ is bounded from L 2 (Σ) 4 to H s (Σ) 4 for all s ∈ (0, γ). Finally, the fact that K * is bounded from L 2 (Σ) 4 onto H s (Σ) 4 , for all s ∈ (0, γ), follows by [40, p. 165].

We are now in a position to complete the proof of Theorem 5.8:

Proof of Theorem 5.8. The first statement is a direct consequence of Theorem 5.2. The second statement follows by the same method as in Theorem 3.1 . Indeed, fix γ ∈ (0, 1) and assume that Σ is C 1,γ , and let g ∈ L 2 (Σ) 4 such that Λ + [g] ∈ H 1/2 (Σ) 4 . Note that multiplication by N is bounded in 4 , for all s ∈ [0, γ). Now, note that

H s (Σ) 4 for all s ∈ [0, γ) (cf. [9, Lemma A.2]) and C Σ is bounded from H 1/2 (Σ) 4 into itself. Therefore, we obtain that Λ -Λ + [g] ∈ H s (Σ)
Λ -Λ + = 1 sgn(κ) - 1 4 -{α • N , C Σ }(α • N )C Σ + 2mµ sgn(κ) S + η sgn(κ) {α • N , C Σ },
here we used the fact that {α 4 into itself and S is bounded from L 2 (Σ) 4 to H 1/2 (Σ) 4 , using Lemma 5.2, from (5.70) it follows that g ∈ H s (Σ) 4 , for all s ∈ [0, γ). Since for all s ∈ [0, 1/2] the operator Φ gives rise to a bounded operator Φ : H s (Σ) 4 -→ H 1/2+s (R 3 \ Σ) 4 (this follows by adapting the same arguments as [9, Proposition 3.6]), we get then the inclusions in (i). In particular, if γ > 1/2, we then obtain that g ∈ H 1/2 (Σ) 4 and therefore Φ[g] ∈ H 1 (R 3 \ Σ) 4 . This gives the equality in (ii) and the theorem is shown.

• N , C Σ }C Σ (α • N ) = {α • N , C Σ }(α • N )C Σ . Thus we get g = 4(sgn(κ)) 4 -sgn(κ) Λ -Λ + + {α • N , C Σ }(α • N )C Σ - η sgn(κ) {α • N , C Σ } - 2mµ sgn(κ) S [g], (5.70) As C Σ (α • N ) is bounded from L 2 (Σ)
Remark 5.5. Note that if Σ is C 1,γ with γ ∈ (1/2, 1) and sgn(κ) / ∈ {0, 4}, then using the same technique as in section 3, we can show that H κ is self-adjoint. In fact, as {α • N , C Σ } is self-adjoint, and bounded from L 2 (Σ) 4 to H 1/2 (Σ) 4 , by duality, we can extend it to a bounded operator from 4 . Hence, by iterating twice the same argument to those of the proof of Theorem 3.1, we then get that

H -1/2 (Σ) 4 to L 2 (Σ)
dom(H * κ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ H 1/2 (Σ) 4 , t Σ u = -Λ + [g] ,
which proves the self-adjointness of H κ in this case.

Quantum Confinement induced by Dirac operators with anomalous magnetic δ-shell interactions

The main goal of this section is to derive a new model of Dirac operators with δ-shell interactions which generate confinement. Let us explain how to derive this model. Using the unit c = = 1, where c is the speed of light and is the Planck's constant, the Dirac operator with an electromagnetic field is given by (see [START_REF] Thaller | The Dirac equation[END_REF]):

H = α • (-i∇ -eA(x)) + mβ + eφ el (x)I 4 , (6.1)
where e is the charge of the particle, φ el (x) is the electric field, and A(x) is the magnetic vector potential. Here the electric and magnetic field strengths are

E(x) = -∇φ el (x) - ∂A(x) ∂t , B(x) = ∇ × A(x).
In this setting, the anomalous magnetic potential is given by:

V (x) = υ iβ(α • E(x)) - 1 4 β((α × α) • B(x)) . (6.2)
here the coupling constant υ is the magnitude of the anomalous potential. Now, we put φ el (x) = |x| and A(x) = 0, we then obtain

V (x) = iυβ(α • x |x| ). Now, given R > 0, if x ∈ S 2 R = {x ∈ R 3 : |x| = R}, then
x/|x| coincide with the normal vector field N (x). Thus we get

V υ (x) := V (x) = iυβ(α • N (x)). (6.3)
Now, given a surface Σ ⊂ R 3 satisfying the assumption (H1), we can consider the following Dirac operator

H + V υ = H + V υ δ Σ , υ ∈ R. (6.4)
and called it Dirac operator with anomalous magnetic δ-shell interactions of strength υ. As we already mentioned in the introduction, when finalizing the current manuscript, it turns out that the authors of the article [START_REF] Cassano | General δ-Shell Interactions for the two-dimensional Dirac Operator: Self-adjointness and Approximation[END_REF] considered this problem in dimension two, and their work was also in the final phase of preparation. However, instead of deriving the potential V υ as we had done here, they rigorously proved that the two-dimensional analog of V υ can be approximated by regular shrinking potentials of magnetic type, and hence they justified the fact that V υ is a "magnetic" δ-shell interactions.

Recall the matrix γ 5 defined in (4.14), we set V ζ = ζγ 5 , for all ζ ∈ R. To our knowledge, the potential V ζ does not seem to have a physical interpretation, but mathematically, it has the same characteristics as the electrostatic potential when ζ = ±2; cf. Remark 6.2.

Unless otherwise specified, throughout this section we assume that Σ satisfies the assumption (H1), and we consider the Dirac operator H ζ,υ defined formally by

H ζ,υ = H + V ζ,υ = H + (ζγ 5 + iυβ(α • N )) δ Σ , ζ, υ ∈ R. (6.5)
Comparing with the operators studied before, the operator H ζ,υ is very different. Indeed, because of the presence of anomalous magnetic potential, several commutativity properties are no longer true in this case. In addition, H 0,υ (i.e ζ = 0) has the particularity of combining two important phenomena that we have seen before. In fact, as it was indicated in the introduction, in the critical case, H 0,±2 is essentially self-adjoint and Σ becomes impenetrable; see Theorem 6.2 below. Now, given z ∈ C \ ((-∞, -m] ∪ [m, ∞)), we define the operators Λ z ± as follows: 4 , for all z ∈ (-m, m). Now, using the same notations as before, the Dirac operator H ζ,υ (acting in L 2 (R 3 ) 4 ) is defined on the domain

Λ z ± = 1 ζ 2 + υ 2 (ζγ 5 + iυβ(α • N )) ± C z Σ . (6.6) Since iυβ(α • N ) is C 1 -smooth and symmetric, it follows that Λ z ± are bounded from L 2 (Σ) 4 onto itself (respectively from H 1/2 (Σ) 4 onto itself). Moreover, Λ z ± are self-adjoint on L 2 (Σ)
dom(H ζ,υ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ L 2 (Σ) 4 , t Σ u = -Λ + [g] , (6.7)
and the potential V ζ,υ is defined by:

V ζ,υ (ϕ) = 1 2 (ζγ 5 + iυβ(α • N ))(ϕ + + ϕ -)δ Σ , (6.8) with ϕ ± = t Σ u + C ± [g]. Here H ζ,υ acts in the sens of distributions as H ζ,υ (ϕ) = Hu, for all ϕ = u + Φ[g] ∈ dom(H ζ,υ ).
We remind the reader that Λz ± denotes the continuous extension of Λ z ± defined from H -1/2 (Σ) 4 onto itself.. Using the same method as in Section 3, one can show that H ζ,υ is closable and the domain of the adjoint is given by

dom(H * ζ,υ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ H -1/2 (Σ) 4 , t Σ u = -Λ+ [g] . (6.9)
In the following, we briefly discuss the basic spectral properties of H ζ,υ in the non-critical case, i.e

dom(H ζ,υ ) = u + Φ(g) : u ∈ H 1 (R 3 ) 4 , g ∈ H 1/2 (Σ) 4 , t Σ u = -Λ + [g] ⊂ H 1 (R 3 \ Σ) 4 . (6.10)
Moreover, the following statements hold true:

(i) Given a ∈ (-m, m), then Kr(H ζ,υ -a) = 0 ⇐⇒ Kr(Λ a + ) = 0. (ii) For all z ∈ C \ R, it holds that (H ζ,υ -z) -1 = (H -z) -1 -Φ z (Λ z + ) -1 (Φ z ) * . (6.11) (iii) Sp ess (H ζ,υ ) = (-∞, -m] ∪ [m, +∞). (iv) Sp disc (H ζ,υ ) ∩ (-m, m) is finite.
Recall that [A, B] = AB -BA is the usual commutator bracket. Before giving the proof of the above theorem, we need the following proposition.

Proposition 6.1. Let z ∈ C \ ((-∞, -m] ∪ [m, ∞)). Then, the commutator [β(α • N ), C z Σ ] gives rise to a bounded operator [β(α • N ), C z Σ ] : H -1/2 (Σ) 4 → H 1/2 (Σ) 4 . (6.12) In particular, [β(α • N ), C z Σ ] is compact in L 2 (Σ) 4 .
Proof. Let x, y ∈ Σ, using (3. As N is C 1 -smooth and S z is bounded from H -1/2 (Σ) 4 to H 1/2 (Σ) 4 , it follows that β[(α • N ), S z ] and {(α • N ), S z } are bounded from H -1/2 (Σ) 4 to H 1/2 (Σ) 4 . Now, that T z is bounded from H -1/2 (Σ) 4

to H 1/2 (Σ) 4 is a direct consequence of Lemma 3.1. This completes the proof of the first statement, the second statement is a consequence of the Sobolev embedding. Now we are in position to prove Theorem 6.1.

Proof of Theorem 6.1. Fix z ∈ C \ ((-∞, -m] ∪ [m, ∞)), then a simple computation yields

Λ z ∓ Λ z ± = 1 ζ 2 + υ 2 - 1 4 -C z Σ (α • N ){α • N , C z Σ } ± ζ ζ 2 + υ 2 [γ 5 , C z Σ ] ± iυ ζ 2 + υ 2 [β(α • N ), C z Σ ].
A straightforward computation yields [γ 5 , C z Σ ] = 2mγ 5 βS. Using this, it follows that ) is given by (6.10) follows using the same arguments as in the proof of Theorem 3.1. The assertions (i), (ii) and (iii) can be proved as in Theorem 5.5. Assertion (iv) is a consequence of the Sobolev injection. Indeed, one can adapt easily the proof of [7, Theorem 4.1 (ii)] to this case. We omit the details. Remark 6.1. Note that if Σ satisifies the assumption (H3), then using the same arguments as in Subsection 5.1 yield that H ζ,υ is self-adjoint with dom(H ζ,υ ) ⊂ H 1/2 (R 3 \ Σ) 4 . If Σ satisifies the assumption (H2), then Theorem 6.1 is still holds true.

Λ z ∓ Λ z ± = 1 ζ 2 + υ 2 - 1 4 -C z Σ (α • N ){α • N , C z Σ } ± 2mζ ζ 2 + υ 2 γ 5 βS z ± iυ ζ 2 + υ 2 [β(α • N ), C z Σ ].
In the following theorem, we discuss the self-adjointness of H µ,υ in the critical case, i.e ζ 2 + υ 2 = 4. We mention that assertions (a) and (c) have already been proven in [START_REF] Holzmann | A note on the three dimensional Dirac operator with zigzag type boundary conditions[END_REF], where the author studied the inner part of H 0,±2 which acts on Ω + , known as the Dirac operator with zig-zag boundary conditions (see also [START_REF] Cassano | General δ-Shell Interactions for the two-dimensional Dirac Operator: Self-adjointness and Approximation[END_REF] for the two-dimensional case). (ii) for all z ∈ C \ R the operator Λz + is bounded invertible from H -1/2 (Σ) 4 to H 1/2 (Σ) 4 and we have (H ζ,υz) -1 = (Hz) -1 -Φ z ( Λz + ) -1 (Φ z ) * . (6.17 (c) There is a sequence (λ j (m)) j∈N ⊂ Sp(H 0,υ ) such that each λ j (m) is an eigenvalue with finite multiplicity, with λ j (m) 2 > m 2 for all j ∈ N, and λ j (m) 2 -→ ∞ as j -→ ∞.

Proof. Let us show the first statement. The proof is a relatively straightforward modification of the technique used in the proof of Theorem 3.1. Indeed, as H ζ,υ is closable the only thing left to prove is the inclusion H * ζ,υ ⊂ H ζ,υ . For this, let ϕ = u + Φ[g] ∈ dom(H * ζ,υ ) and let (h j ) j∈N ⊂ H 1/2 (Σ) 4 be a sequence of functions that converges to g in H -1/2 (Σ) 4 . Set Clearly, (g j ) j∈N ⊂ H 1/2 (Σ) 4 and g j ---→ j→∞ g in H -1/2 (Σ) 4 . Since Λ + is bounded from H 1/2 (Σ) 4 onto itself, we then get that (Λ + [g j ]) j∈N ⊂ H 1/2 (Σ) 4 . Now, remark that From Lemma 3.1, Proposition 6.1 and (6.16) it follows that Λ± Λ∓ are bounded from H -1/2 (Σ) 4 to H 1/2 (Σ) 4 . Therefore, (6.23) yields that Λ + [g j ] ---→ j→∞ Λ+ [g], in H 1/2 (Σ) 4 . (6.24) Now, let (v j ) j∈N ⊂ H 1 (R 3 ) 4 such that 2t Σ v j = Λ+ (ζγ 5 + iυβ(α • N )) Λ-[h jg], for all j ∈ N (this can be easily done by taking the average of linear extensions of t Σ v j on Ω ± , since t Σ v j ∈ H 1/2 (Σ) 4 by definition). Set u j = uv j and define ϕ j := u j + Φ[g j ]. It is clear that u j ∈ H 1 (R 3 ) 4 and t Σ u j = -Λ + [g j ] in H 1/2 (Σ) 4 , hence (ϕ j ) j∈N ⊂ dom(H ζ,υ ). Moreover, since (h j ) j∈N (resp (g j ) j∈N ) converges to g in H -1/2 (Σ) 4 as j -→ ∞, using the continuity of Λ± Λ∓ it follows that (ϕ j , H ζ,υ ϕ j ) ---→ From this, it follows that a belongs to Sp(H ζ,υ ) if and only if -a belongs to Sp(H ζ,υ ), which yields (i). Item (ii) can be proved in the same way as Proposition 4.1. To prove item (iii), observe that dom(H 0,υ ) = ϕ = (ϕ + , ϕ -) ∈ L 2 (Ω + ) 4 ⊕L 2 (Ω -) 4 : (α • ∇)ϕ ± ∈ L 2 (Ω ± ) 4 and i(α • N )P -,υ t Σ ϕ + = i(α • N )P +,υ t Σ ϕ -, Since P ±,υ are projectors, it easily follows that a function ϕ = (ϕ + , ϕ -) ∈ L 2 (Ω + ) 4 ⊕ L 2 (Ω -) 4 with (α • ∇)ϕ ± ∈ L 2 (Ω ± ) 4 belongs to dom(H 0,υ ) if and only if P ∓,υ t Σ ϕ ± = 0 (this should be understood as an equality in H -1/2 (Σ) 4 ). Therefore, Σ becomes impenetrable and the decomposition (6.18) holds true. Next, we turn to the proof of the assertion (a). Following the arguments of the proof of Proposition 4.1, it is clear that the Birman-Schwinger principle applies also for a = ±m (if g ∈ Kr(Λ m ± ), then ϕ = mΦ m [g]+ Φ[g] is an eigenfunction of H 0,υ associated to the eigenvalue m). Now observe that dimRn(Λ m ± ) = ∞, otherwise Λ m ∓ Λ m ± would not be a compact operator on L 2 (Σ) 4 . Let g ∈ L 2 (Σ) 2 , using [2, Lemma 4.1] it easily follows that

Λ m + (iυβ(α • N ))Λ m -(iυβ(α • N )) = (1 - υ 2 4
) g 0 = 0. (6.28) Therefore m is an eigenvalue of H 0,υ with infinite multiplicity. By (i) it follows that -m is also an eigenvalue of H 0,υ with infinite multiplicity, which proves assertion (a). Now we are going to prove (b) and (c), for that we consider the following Dirac operators Now, let (-∆ Ω± ) be the Dirichlet realization of (-∆) in Ω ± , with domain H 2 (Ω ± ) ∩ H 1 0 (Ω ± ). Using the Weyl's theorem and the fact that H 1 0 (Ω + ) is compactly embedded in L 2 (Ω + ), it is not hard to show that Sp(-∆ Ω-+ m 2 ) = [m 2 , +∞), Sp(-∆ Ω+ + m 2 ) = Sp disc (-∆ Ω+ + m 2 ) = {m 2 + λ j , j ∈ N}, (6.33) with λ j > 0 for all j ∈ N, and λ j -→ ∞ as j -→ ∞. Now assume that υ = 2 (the case υ = -2 can be recovered with the same arguments), using the boundary condition it follows that Remark 6.2. Let ζ = ±2 and let H ±2,0 be as in Theorem 6.2. Given (ϕ + , ϕ -) ∈ dom(H ζ,0 ), we write ϕ ± = (ϕ ±,1 , ϕ ±,2 ). Then, one can write the transmission condition as follows: However, if ζ = 0, then there is no embedded eigenvalues in the essential spectrum of H 0,±2 , and we have Sp(H 0,±2 ) = Sp ess (H 0,±2 ) = -∞, -m ∪ m, +∞ . (6.39)

t Σ ϕ +,1 = iζ 2 (σ • N )t Σ ϕ -,2 , t Σ ϕ +,2 = iζ 2 (σ • N )t Σ ϕ -,

2. 1 .

 1 Integral operators associated to the Dirac operator. Here we list some well known results about integral operators associated to the fundamental solution of the Dirac operator. Given z ∈ C \ ((-∞, -m] ∪ [m, ∞)) with the convention that Im √ z 2m 2 > 0, we recall that the fundamental solution of Hz is given by

Proposition 4 . 3 .

 43 Let a ∈ (-m, m) and let L a ± be as in Lemma 3.2. Assume that ν = 0, then it holds that0 ∈ Sp(L a + ) ⇐⇒ a = -mµ ǫ and 0 ∈ Sp(L a -) ⇐⇒ a = mµ ǫ . (4.16) Moreover, 0 is an isolated eigenvalue of L -mµ/ǫ + and L mµ/ǫ with infinite multiplicity.

Remark 4 . 2 .

 42 The reader should not confuse the unbounded operator L -mµ/ǫ + with the original operator Λ -mµ/ǫ + which is indeed a bounded operator on L 2 (Σ) 4 with closed range.

) 4 .-mµ/ǫ 0 ) because 0 ∈

 400 As 0 ∈ Sp ess (Υ Sp ess (L mµ/ǫ -) by Proposition 4.3, then by Weyl's theorem we get that 0 ∈ Sp ess (Υ -mµ/ǫ ν

1 r

 1 max sup dist(y, E ∩ P (x, r)) : y ∈ E ∩ B(x, r) , sup dist(y, E ∩ B(x, r)) : y ∈ E ∩ P (x, r) < δ. Definition 5.8 (δ-Reifenberg flat domains). Given a bounded open set Ω ⊂ R 3 and δ ∈ (0, δ 0 ). We say that Ω is δ-Reifenberg flat domain provided Ω satisfies the separation property and ∂Ω is δ-Reifenberg flat.Definition 5.9 (δ-SKT Domains). Let δ ∈ (0, δ 0 ), where δ 0 is a fixed, sufficiently small number. Call a bounded set Ω ⊂ R 3 of finite perimeter a δ-SKT ( Semmes-Kenig-Toro) domain if Ω is a δ-Reifenberg flat domain, ∂Ω is Ahlfors regular and whose geometric measure theoretic outward unit normal ν is such that ||ν|| BMO(∂Ω,dS) 3 < δ,(5.7)

  x ∈ Σ where the limit exists. Then, we have the following trace theorem, see [33, Theorem 1 and Example 1] and [34, Theorem 1, p.182 ].

4 |K

 4 1 and Remark 2.1 to obtain the first statement of the theorem. So, fix λ, µ ∈ R such that |ǫ| = |µ|, and let z ∈ C \ ((-∞, -m] ∪ [m, ∞)). Then, from the definition of C z Σ it follows that (xy)| = O(|x -y| -1 ) when |x -y| -→ 0.

ζ 2 + υ 2 = 4 .Theorem 6 . 1 .

 22461 The following theorem gathers the most important properties of H µ,υ . Let ζ, υ ∈ R such that ζ 2 + υ 2 = 0, 4. Then H ζ,υ is self adjoint and we have

  [START_REF] Cassano | General δ-Shell Interactions for the two-dimensional Dirac Operator: Self-adjointness and Approximation[END_REF] a trivial verification shows thatβ(α • N (x))(α • (xy)) -(α • (xy))β(α • N (y)) =(α • (xy))β(α • (N (y) -N (x)) + (N (x) • (xy))β. (6.13) Now, let g ∈ L 2 (Σ)4 , then using the identity (6.13), similar arguments to those of Lemma 3.1 yield[β(α • N ), C z Σ ][g](x) = zβ[(α • N ), S z ][g](x) -m{(α • N ), S z }[g](x) + T z [g](x),(6.14)where the integral representation of T z is given byT z [g](x) = Σ K z (x, y)g(y)dS(y),(6.15)withK z (x, y) = β e i √ z 2 -m 2 |x-y| 4π|x -y| 3 (1i z 2m 2 |x -y|) (α • (xy))(α • (N (x) -N (y))-2 (N (x) • (xy)) I 4 .

(6. 16 )

 16 As ζ 2 + υ 2 = 0, 4, by Lemma 3.1 and Proposition 6.1 it follows that Λ z ∓ Λ z ± are Fredholm operators. Therefore Λ z + is Fredholm by [1, Theorem 1.46 (iii)]. Hence, the self-adjointness of H ζ,υ is a direct consequence of Theorem 5.1. That dom(H ζ,υ

Theorem 6 . 2 .

 62 Let ζ, υ ∈ R such that ζ 2 + υ 2 = 4 , then H ζ,υ is essentially self -adjoint and we have dom(H ζ,υ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ H -1/2 (Σ) 4 , t Σ u = -Λ+ [g] ,Moreover, the following hold true:(i) a ∈ Sp(H ζ,υ ) ⇐⇒ -a ∈ Sp(H ζ,υ ).

  )(iii) if ζ = 0, then Σ becomes impenetrable and it holds thatH 0,υ = H Ω+ υ ⊕ H Ω- υ = (-iα • ∇ + mβ) ⊕ (-iα • ∇ + mβ) ,(6.18)where H Ω± υ are the self-adjoint Dirac operators defined ondom(H Ω± υ ) = ϕ ± ∈ L 2 (Ω ± ) 4 : (α • ∇)ϕ ± ∈ L 2 (Ω ± ) 4 and P ∓,υ t Σ ϕ ± = 0 ,where the boundary condition has to be understood as an equality in H -1/2 (Σ)4 , and P ±,υ are the projectors defined by

Furthermore, we

  have (a) -m and m are eigenvalues of H 0,υ with infinite multiplicities. (b) Sp(H 0,υ ) = (-∞, -m] ∪ [m, +∞).

g j := 1 2 (ζγ 5 +

 25 iυβ(α • N )) Λ+ [g] + Λ -[h j ] , ∀j ∈ N. (6.20)

- 1 2 (ζγ 5 + 2 Λ+ (ζγ 5 +

 2525 iυβ(α • N )) Λ-[g] = -g + 1 2 (ζγ 5 + iυβ(α • N )) Λ+ [g] (6.21)Using this, it follows thatg j := g -1 2 (ζγ 5 + iυβ(α • N )) Λ-[h jg], ∀j ∈ N. (6.22)Therefore we obtainΛ+ [g jg] = -1 iυβ(α • N )) Λ-[gh j ] = Λ+ Λ-Λ-+ Λ + Λ+ Λ-[h jg].(6.23)

2 (ζγ 5 + 2 (ζγ 5 +

 2525 j→∞ (ϕ, H * ζ,υ ϕ) in L 2 (R 3 ) 4 . Therefore H * ζ,υ ⊂ H ζ,µ and hence H ζ,υ is self-adjoint with dom(H ζ,υ ) = u + Φ[g] : u ∈ H 1 (R 3 ) 4 , g ∈ H -1/2 (Σ) 4 , t Σ u = -Λ+ [g] ,(6.25) this finishes the proof of the first statement. In order to continue the proof of the theorem we use the definition of dom(H ζ,υ ) with transmission condition. As in Definition 3.1, using the Plemelj-Sokhotski formula, one can show that H µ,υ acts in the sense of distributions as (-i∇• α + mβ)⊕(-i∇ • α + mβ) on the domain dom(H ζ,υ ) = ϕ = (ϕ + , ϕ -) ∈ L 2 (Ω + ) 4 ⊕ L 2 (Ω -) 4 : (α • ∇)ϕ ± ∈ L 2 (Ω ± ) 4 and 1 iυβ(α • N )) + i(α • N ) t Σ ϕ + = -1 iυβ(α • N ))i(α • N ) t Σ ϕ -,where the transmission condition holds in H -1/2 (Σ)4 . Let us show item (ii), for that recall that the operator C is defined in (4.14). Then, a trivial computation yields that ϕ ∈ dom(H ζ,υ ) ⇐⇒ C[ϕ] ∈ dom(H ζ,υ ), (6.26) C[(-iα • ∇ + mβ)u] = -(-iα • ∇ + mβ)C[u], ∀u ∈ L 2 (R 3 ) 4 . (6.27)

D

  Ω± υ ψ = (-iα • ∇ + mβ) ψ, ψ ∈ dom(D Ω± υ ) = ϕ ± ∈ H 1 (Ω ± ) 4 : P ∓,υ t Σ ϕ ± = 0 . (6.29)Then, one can check easily that D Ω± υ are symmetric, closable operators. Moreover, it holds thatD Ω± υ = H Ω± υ . Indeed, denote by Q Ω± υ the quadratic form associated to (D Ω± υ ) 2 . Given ϕ ∈ dom(D Ω± υ ),using the Green's formula and the boundary conditions, it easily follows that:Q Ω± υ [ϕ] = (α • ∇)ϕ 2 L 2 (Ω±) 4 + m 2 ϕ 2 L 2 (Ω±) 4 . (6.30)Hence, we getQ Ω± υ [ϕ] m 2 ϕ 2 L 2 (Ω±) 4 . (6.31) Thus (D Ω± υ ) 2 is lower semi-bounded. Therefore, by [21, Theorem 6.3.2] it follows that (H Ω± υ ) 2 is the Friedrichs extension of (D Ω± υ ) 2 and it holds that Sp(H Ω± υ ) ⊂ (-∞, -m] ∪ [m, +∞). (6.32)

2 ) 2 L 2 ( 2 L 2 ( 2 L 2 2 L 2

 222222222 =⇒ ϕ 1 ∈ H 1 0 (Ω -) 2 . (6.34)Denote by QΩ± the quadratic form associated to (-∆ Ω± + m 2 )I 2 . Using (6.34) and the Green's formula, from (6.30) it follows thatQ Ω+ 2 [ϕ] = (σ • ∇)ϕ 1 Ω+) 2 + m 2 ϕ 1 Ω+) 2 + QΩ+ [ϕ 2 ], ∀ϕ ∈ dom(D Ω+ 2 ), Q Ω-2 [ϕ] = (σ • ∇)ϕ 2 (Ω-) 2 + m 2 ϕ 2 (Ω-) 2 + QΩ-[ϕ 1 ], ∀ϕ ∈ dom(D

( 6 . 35 )

 635 Using this and assertion (i), it follows that λ j (m) = ± m 2 + λ j is an eigenvalue of H 0,υ with finite multiplicity, and we haveSp(H Ω± υ ) = (-∞, -m] ∪ [m, +∞),(6.36)which yields (b) and (c). This achieves the proof of theorem.

Remark 6 . 3 .

 63 1 . (6.37) Thus, we deduce that H ζ,0 coincide with the Dirac operator coupled with the electrostatic δ-interactions of strength -ζ. Thus, in this sense, one can consider the potential V ζ as an electrostatic potential for ζ = ±2. If one assume that Σ satisfies the assumption (H2), then H ζ,υ is essentially self-adjoint, when ζ 2 + υ 2 = 4. In particular, if υ = 0, then Remark 6.2 and Theorem 4.2 yield that Sp ess (H ±2,0 ) = -∞, -m ∪ {0} ∪ m, +∞ .(6.38)

  which proves the identity (4.1). 4.1. A careful inspection of the argument used above reveals that a ∈ (-m, m) is an isolated point of Sp(H κ ) if and only if 0 is an isolated point of Sp( Λa Furthermore, item (ii) holds true for all z ∈ ρ(H κ ) ∩ ρ(H) . an immediate consequence of Lemma 3.2, Proposition 4.1 and Remark 4.1, we have the following. Corollary 4.1. Let H κ be as in the definition 2.1. The following hold:

	Remark As

+ ), and dimKr(H κ ) = dimKr( Λa + ).

  Sp ess (H ν κ ). So fix ν > 0 and assume that -mµ

	Corollary 4.1 and Proposition 4.2 it follows that 0 / ∈ Sp ess (L -mµ/ǫ +,ν B ν := Λ-mµ/ǫ +,ν Λmµ/ǫ -,ν , D ν := Λ mµ/ǫ -,ν Λ -mµ/ǫ +,ν	ǫ ) and 0 / ∈ Sp ess (L / ∈ Sp ess (H ν κ ). Then, by mµ/ǫ -,ν ). Now we set

Actually from Proposition 4.3 and Corollary 4.1, we know that item (b) holds true for ν = 0. Since Kr( Λ-mµ/ǫ +,ν ) ∩ Kr( Λ-mµ/ǫ -,ν ) = {0} holds, it sufficient to prove that -mµ ǫ ∈

  iv) a ∈ Sp p (H ǫ,µ ) if and only if -a ∈ Sp p (H -ǫ,µ ), for all a ∈ (-m, m). (v) a ∈ Sp p (H ǫ,µ ) if and only if a ∈ Sp p (H -4ǫ

	ǫ 2 -µ 2 , -4µ ǫ 2 -µ 2

Note that the algebra generated by the Pauli spin matrices σ = (σ 1 , σ

, σ

) (as an algebra on the real field) is isomorphic to Cl 3 . Thus, W (σ • N ) can be viewed as the restriction of the Cauchy-Clifford operator on L 2 (Σ) ⊗ C 2 .

Acknowledgement

I would like to thank my PhD supervisors Vincent Bruneau and Luis Vega for their encouragement, and for their precious discussions and advices during the preparation of this paper. I would also like to thank Mihalis Mourgoglou for various fruitful discussions on topics related to the analysis of PDEs in rough domains. I am particularly grateful to him for suggesting several references related to Section 5 his careful reading of a preliminary version of that section. This project is based upon work supported by the Government of the Basque Country under Grant PIFG18/06 "ERC Grant: Harmonic Analysis and Differential Equations", researcher in charge: Luis Vega.

(H3) Σ = ∂Ω + with Ω + a bounded Lipschitz domain with normal N ∈ VMO(∂Ω, dS) 3 .

Roughly speaking, the above assumption implies smallness of the Lipschitz constant of Ω + . Another way to reformulate the assumption (H3) is to say that Ω + belongs to the intersection of the class of bounded Lipschitz domains and the class of regular SKT domains, see the proof of Theorem 5.3. Now we can state the main result of this subsection.

Theorem 5.2. Let κ ∈ R 3 such that sgn(κ) = 0, 4, and assume that Σ satisfies (H3), and let H κ be as in Definition 2.1. Then H κ is self-adjoint and dom(H

Let g ∈ L 2 (Σ), then the harmonic double layer K and the Riesz transforms (R k ) 1 k 3 on Σ are defined by

x ky k 4π|x -y| 3 g(y)dS(y).

(5.12)

The following theorem is implicitly contained in [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF], but we state and prove it here for the sake of completeness. In the proof, we use the notation R 3 ± = {x ∈ R 3 : ±x 3 > 0} for the upper (respectively the lower) half space.

Theorem 5.3. Assume that Σ satisfies (H3). Then, the harmonic double layer K and the commutators [N j , R k ], 1 j, k 3, are compact operators on L 2 (Σ).

Proof. The result follows from the fact that Ω + is a bounded regular SKT domain. To see this indeed, note that bi-Lipschitz mappings preserve the class of two-sided NTA domains with Ahlfors regular boundaries, and the class of regular SKT domains is invariant under continuously differentiable diffeomorphisms, see [START_REF] Hofmann | Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains[END_REF]. Now, by definition Ω + is locally the region above the graph of a Lipschitz function φ : R 2 -→ R. Therefore, one may assume (via a partition of unity and a local flattening of the boundary) that

Let F : R 3 -→ R 3 be defined for all (x, x 3 ) ∈ R 2 × R as F (x, x 3 ) := (x, x 3 + φ(x)). Then it easily follows that F is a bijective function with inverse F -1 : R 3 -→ R 3 given by F -1 (y, y 3 ) := (y, y 3 -φ(y))

under the bi-Lipschitz homeomorphism F , which also maps R 2 × {0} onto Σ. From this, it follows that Ω + is a two-sided NTA domain and Σ is Ahlfors regular (because R 3 + is a two sided NTA domain and R 2 × {0} is Ahlfors regular). Since N ∈ VMO(Σ) by assumption, thanks to [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]Theorem 4.21], we know that Ω + is a regular SKT domain. Therefore the claimed result follows by [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]Theorem 4.47].

Proof. Given g ∈ L 2 (Σ) 4 , similar computation as in the proof of Lemma 3.1 give (iv) {σ • N , W } is a compact operator on L 2 (∂Ω) 4 .

Proof. (i) ⇒ (ii) is a consequence of [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]Theorem 4.21] and [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]Theorem 4.47]. (ii) ⇒ (iii) readily follows from (5.13) and (5.16). (iii) ⇒ (iv) is an immediate consequence of (5.20). Finally, (iv) ⇒ (i) follows from (5.22) and [START_REF] Hofmann | Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains[END_REF]Theorem 4.21].

Taking into account (3.13), from the proof of Theorem 3.1 we have that

recall that S z is defined by (3.12). As the injection H 1/2 (Σ) 4 into L 2 (Σ) 4 is compact, cf. [START_REF] Mclean | Strongly Elliptic Systems and Boundary Integral Equations[END_REF]Theorem 6.11] for example, it follows that S z is a compact operator in L 2 (Σ) 4 . Using that Then, for all ϕ = u + Φ[g] ∈ dom(T ), it holds that

Therefore, V (ϕ) = -g and hence T (ϕ) = Hu holds in the sense of distribution for all ϕ = u + Φ[g] ∈ dom(T ). Now, since |N (x)| = 1, it is clear that Λ (defined in Notation 2.1) is a bounded and selfadjoint operator on L 2 (Σ) 4 . Moreover, from Corollary 5.1 we know that Λ is Fredholm operator. Therefore, the claimed result follows directly by Theorem 5.1 and Remark 2.1.

In the next theorem, we infer some spectral properties of the Dirac operator H κ .

Theorem 5.5. Let H κ be as in Theorem 5.2. Then the following is true: Proof. The proof of item (i) follows in the same way as in Proposition 4.1. Let us show (ii). Given z ∈ C \ R, from (i) and the fact H κ is self-adjoint it is clear that Kr(Λ z + ) = 0. Hence, using this and