General $\delta$-shell interactions for the two-dimensional Dirac operator: self-adjointness and approximation

In this work we consider the two-dimensional Dirac operator with general local singular interactions supported on a closed curve. A systematic study of the interaction is performed by decomposing it into a linear combination of four elementary interactions: electrostatic, Lorentz scalar, magnetic, and a fourth one which can be absorbed by using unitary transformations. We address the self-adjointness and the spectral description of the underlying Dirac operator, and moreover we describe its approximation by Dirac operators with regular potentials.


INTRODUCTION
In the present paper we study the two-dimensional Dirac operator with a singular interaction supported on a closed curve.Our main motivation is to treat the most general local interactions.Besides electrostatic δ-shell interactions and the Lorentz scalar δ-shell interactions we include into the analysis the magnetic δ-shell interactions, which correspond to the magnetic field supported on a curve.The main two questions addressed in the present paper are self-adjointness of the underlying Dirac operator and its approximation by Dirac operators with regular potentials.
Recall that the Dirac operator was firstly introduced in relativistic quantum mechanics to describe the dynamics of spin- 1  2 particles (see, e.g. the monograph [64]), and was later associated to the evolution of quasi-particles in new materials, such as the graphene (see, e.g.[27]).Dirac operators with singular interactions supported on sets of lower dimensions serve as idealized models for Dirac operators with more realistic (regular) potentials.
Hamiltonians with interactions supported on sets of zero Lebesgue measure have been studied intensively in the mathematical physics.At first the case of Schrödinger operators with singular interactions was investigated; see, e.g.[2,14,31].In the recent years the focus partially shifted to the Dirac operators with singular interactions.While for Schrödinger operators quadratic forms are a convenient tool to define the underlying Hamiltonian [21], in the Dirac setting more subtle techniques are necessary due to the lack of semi-boundedness.The case of one-dimensional Dirac operators with point-interactions is well understood [2,23,37,51,19].Three-dimensional Dirac operators with singular interactions supported on surfaces are considered in, e.g.[4,5,6,8,9,12,11].Finally, the two-dimensional case without the magnetic interaction has recently been analysed in [13,53].The interest to include the magnetic δ-shell interaction stems from applications in modern physics [52,47,35].The closely related model of magnetic links in three dimensions has been recently considered in [54,55,56].
The approximation of Dirac operators with singular interactions by Dirac operators with regular interactions provides a justification of the idealized model under consideration.In the one-dimensional setting the analysis is performed in [63,39,40,65], a generalization to three-dimensions has recently appeared in [46,45].In the present manuscript, we modify to our setting some techniques that worked efficiently in the one-dimensional case.
Recall that the action of the two-dimensional free Dirac operator D 0 is given by the differential expression where m ∈ R is the mass and σ 1 , σ 2 , and σ 3 are the Pauli matrices: It is self-adjoint on dom D 0 := H 1 (R 2 ; C 2 ) ⊂ L 2 (R 2 ; C 2 ) and essentially self-adjoint on C ∞ c (R 2 ; C 2 ).The spectrum of D 0 is purely absolutely continuous and The interaction under consideration will be supported on the boundary Σ := ∂Ω of a C ∞ -smooth bounded simply connected open set Ω ⊂ R 2 .The curve Σ splits the Euclidean space into disjoint union R 2 = Ω + ∪ Σ ∪ Ω − , where Ω + := Ω and Ω − := R 2 \ Ω + .We will call the curve Σ a shell.Let us denote the outer unit normal to Ω + and the unit vector tangent to the boundary Σ in x ∈ Σ by n ≡ (n 1 , n 2 ) = n(x) and t ≡ (t 1 , t 2 ) = t(x), respectively.For definiteness, we put t 1 = −n 2 and t 2 = n 1 .For any C 2 -valued function f defined on R 2 , we set f ± = f ↾ Ω ± .When it is defined in a suitable sense, we denote T D ± f ± the Dirichlet trace of f ± at Σ, and we define the distribution δ Σ f by where ds means integration with respect to the arc-length of Σ.
We are interested in the Dirac operator in L 2 (R 2 ; C 2 ) given by the formal expression where η, τ, λ, ω are smooth real-valued functions, and where we used the notation (1.4) For any given x ∈ Σ the matrices I 2 , σ 3 , σ • t(x), and σ • n(x) constitute a basis of the Hermitian 2× 2 matrices, so at every point there is the most general Hermitian matrix as a coefficient of δ Σ .The electrostatic δ-shell interaction ηI 2 δ Σ and the Lorentz scalar δshell interaction τ σ 3 δ Σ describe a distribution of charges and masses on the curve Σ, respectively.The novelty in our treatment is the magnetic δ-shell interaction λ(σ • t)δ Σ , which describes a magnetic field supported on Σ.We remark that the vector potential associated with the latter interaction is given by A Σ = λ(t 1 δ Σ , t 2 δ Σ ) and we will show in Appendix B that the underlying magnetic field is given by B Σ = λ∂ n δ Σ , where ∂ n δ Σ stands for the double layer distribution.Finally, we prove that, under some restrictions on parameters η, τ, λ, ω, the interaction term ω(σ • n)δ Σ can be gauged away in the spirit of [44,50]; see Theorem 2.1 for details.In particular, this term can be always gauged away when all parameters are constant.Due to this observation we may focus on the case when ω = 0 and other parameters are smooth real-valued functions.
To this aim we build an ordinary boundary triple for S * , which is a modification of the boundary triple constructed in [13].This modification is necessary to treat the magnetic δ-shell interaction.In this construction the operator D η,τ,λ acts as D 0 on Ω ± and is subject to the local boundary conditions on Σ, which involves the parameters η, τ, λ, the tangential vector t, and the normal vector n.The construction boils the question of self-adjointness of D η,τ,λ down to self-adjointness of a certain first-order pseudo-differential operator on Σ.The latter is shown by conventional techniques under the condition − λ 2 = 0 everywhere on Σ. (1.5) The case when the above expression on the left-hand side vanishes is called critical and needs a special treatment.In the present paper we cover in Section 7 the special sub-case of purely magnetic critical shell interaction (η = τ = 0 and λ = ±2), in which case D 0,0,±2 is defined as a self-adjoint operator by a different and more direct method.It is also remarkable that the condition η 2 − τ 2 − λ 2 = −4 is necessary and sufficient for the confinement to take place, where by confinement we understand that the operator D η,τ,λ can be decomposed into the orthogonal sum with respect to the decomposition L 2 (R 2 ; C 2 ) = L 2 (Ω + ; C 2 )⊕L 2 (Ω − ; C 2 ); cf.Section 2.3 for details.In particular, the choice η = τ = 0 and λ = ±2 gives rise to zig-zag boundary conditions.
We finish this introduction pointing out that when concluding the preparation of this manuscript we learnt that the three dimensional analogue of the magnetic δ-shell interaction introduced here was being considered in the non-published work [17]; the reader may see Section 9 for more details.
Organization of the paper.In Section 2 we formulate and discuss all the main results of the present paper.Section 3 contains preliminary material that is used throughout the paper.In Section 4 we obtain spectral relations for the point spectrum of the Dirac operator with δ-shell interactions under special transforms of the interaction strengths and, moreover, we show how the fourth interaction ω(σ • n)δ Σ can be eliminated by a properly constructed unitary transform.Further, in Section 5 we provide a condition on the interaction strengths, which gives the confinement.In Section 6 we analyse the non-critical case, prove self-adjointness of the underlying Dirac operator and obtain its basic spectral properties.Self-adjointness and spectral properties of the Dirac operator with purely magnetic critical interaction are investigated in Section 7. In Section 8 we construct strong resolvent approximations of Dirac operators with δ-shell interactions by sequences of Dirac operators with suitably scaled regular potentials.Possible generalization for higher dimensions is briefly discussed in Section 9.
The paper is complemented by two appendices.In Appendix A we focus on exponentials of 2 × 2 matrices of a special structure.Finally, in Appendix B we compute the magnetic field associated with the magnetic δ-shell interaction.

MAIN RESULTS
We briefly discuss here the main results of this paper, referring to the various sections below for more detailed results.For an open set Ω ⊂ R 2 , we define (2.1) and . By means of an integration by parts it can be seen (see e.g.[49]) that D η,τ,λ,ω is the operator representing the formal differential expression (1.3).
Since most of this article focuses on the case ω = 0, due to the results presented in Section 4, for the sake of brevity we also set D η,τ,λ := D η,τ,λ,0 , i.e. and Finally, we denote 2.1.Reduction to ω = 0.In Section 4 we will prove the following result.
Theorem 2.1.Given ω ∈ R and η, τ, λ ∈ C ∞ (Σ; R) such that d := η 2 − τ 2 − λ 2 is a constant function on Σ, let X be a solution to and where Roughly speaking, Theorem 2.1 implies that a spectral study for D η,τ,λ = D η,τ,λ,0 suffices to treat the general case D η,τ,λ,ω , hence the formal term ω(σ•n)δ Σ in the δ-shell interaction is indeed superfluous.In a classical (absolutely continuous) framework one would say that this term can be gauged away: this is reminiscent of a similar effect for magnetic potentials in the Coulomb gauge, see [25,Remark 1.5], [26,20,34,33].In Section 4 we show that the unitary transform U z can always be taken different from the identity except for the case (d, ω) = (−4, 0), which corresponds to confining δ-shell interactions, see Section 5.In particular, D η,τ,λ,ω never yields confinement if At the end of Section 4 we find some isospectral transformations as a byproduct of our result, and we describe the charge conjugation properties of the operator D η,τ,λ .

The non-critical case.
We say that we are in the non-critical case when (1.5) holds true, i.e. everywhere on Σ (2.9) In the following theorem we gather the properties of D η,τ,λ in the case when (2.9) holds true.We point out that the non-magnetic case (λ = 0) has been already treated in [13,Theorem 1.1] for constant η, τ ∈ R.
The proof of Theorem 2.2 mimics the strategy of the proof of [13, Theorem 1.1], taking into account the necessary modifications to treat the additional interaction λ(σ • t)δ Σ .It is provided in Section 6, where we also show a Krein-type resolvent formula, an abstract version of the Birman-Schwinger principle, and obtain the spectral properties of D η,τ,λ .

2.3.
Confining δ-shell interactions.If d = −4 everywhere on Σ, the phenomenon of confinement arises.Physically, this means that a particle initially located in Ω ± can not escape this region during the quantum evolution associated with D η,τ,λ .We describe the corresponding Hamiltonian in the following theorem, which will be proved in Section 5.
2.4.The critical case.In [13, Theorem 1.2], the critical cases when C(η, τ, λ) = λ = 0 and η, τ ∈ R are described, namely, the self-adjointness is proved and the spectral properties are analysed.We complement this result by analysing the case C(η, τ, λ) = η = τ = 0, i.e. we consider the purely magnetic critical interactions ±2(σ • t)δ Σ .In this case we prove the self-adjointness and give a detailed description of the spectrum: we take the advantage of the phenomenon of confinement and the decomposition in Theorem 2.3 and we adapt the analysis of the massless Dirac operator on a domain with the zig-zag boundary conditions given in [61] to the case with a mass.
The proof of Theorem 2.4 is provided in Section 7. Note that the presence of embedded eigenvalues was already observed in the non-critical confining case in the three dimensional setting in [5,Th. 3.7], [11,Prop. 3.3].
In the three-dimensional setting [46], the proof of the strong resolvent convergence is an adaptation to the relativistic scenario of the approach used in [7] for the case of Schrödinger operators with δ-shell interactions.In [7], the co-dimension of the shell is strictly smaller than the order of the differential operator (the Laplacian).As a consequence, the singularities of the kernels of the boundary integral operators used in [7] are weak enough to be controlled uniformly along the approximation procedure.This, in particular, leads to the convergence in the norm resolvent sense in the case of the Schrödinger operator.However, in the case of the Dirac operator, the co-dimension of the shell is exactly the same as the order of the differential operator.This has an important effect on the nature of the corresponding boundary integral operators, which now are singular integral operators instead of compact.Due to this new obstruction with respect to the Schrödinger case, the approach used in [7] was adapted in [46] to the Dirac case to show the convergence in the strong resolvent sense assuming uniform smallness of the approximating potentials.Nevertheless, the question of strong resolvent convergence can also be addressed by more direct methods, which do not require any smallness assumption on the approximating potentials, such as by proving the convergence in the strong graph limit sense and then applying Theorem VIII.26 of [59], which says that, in the self-adjoint setting, the strong graph convergence and the strong resolvent convergence are equivalent.
Originally, this approach was used in the one-dimensional setting [39,40].In this way, one can find approximating potentials for any type of δ-potential.The norm resolvent convergence of approximations is harder to tackle.However, since in the one-dimensional case one can perform very explicit calculations with the resolvents of the approximations and the resolvents of their limit operators, it can be proved as well [63,65].In the present work, we will modify the ideas of [39] to get the sequence of approximating potentials for general linear combinations of δ-shell interactions, not only for the purely electrostatic or purely Lorentz scalar δ-shell interactions as in [46].It will converge in the strong resolvent sense, without any smallness assumption on the approximating potentials.We expect that a similar approach can be applied in the three-dimensional case as well.
We can now state the main result of this subsection.
The proof of Theorem 2.5 is in Section 8.
Remark 2.6.In the case that d = 0 the phenomenon of renormalization of the coupling constants does not occur.This was already observed in the one-dimensional setting in [65].
Remark 2.8.From (2.15)-(2.17), Since in all the cases d > −4, Theorem 2.5 does not provide strong convergence to a Dirac operator with a δ-shell causing confinement (see Theorem 2.3).However, we recover the case d = −4 in the limit d → −∞.This suggests that it should be possible to get the confining cases by means of an approximation procedure in which we choose the coefficients η = η ǫ , τ = τ ǫ and λ = λ ǫ dependent on the parameter ǫ so that the associated parameter d = dǫ satisfies dǫ > −4 and dǫ → −4 uniformly in the limit ǫ → 0. Finally, the fact that As a consequence of Theorem 2.5, we get immediately the second result of this section.
Remark 2.10.In the case that d ≥ 0 the correspondence (η, τ , λ) → (η, τ, λ) is not one-to-one.One can choose the coupling constants in the approximating potentials arbitrarily large and still ends up with the same limit operator.From the physical perspective, we suppose that this surprising behaviour is possible due to the Klein effect (also called the Klein paradox).Usually, the Klein effect is related to the scattering on the electrostatic barrier when, speaking vaguely, the transmission coefficient does not depend on the height of the barrier monotonously, see [29] for an overview.
Clearly, this effect occurs for the pure electrostatic interaction, for which d > 0. On the other hand, one can push d below zero, and thus eliminate the Klein effect, by switching sufficiently strong Lorentz scalar/magnetic fields on.
We conclude the presentation of our results underlining that, in the case that η = τ = 0 and λ ∈ R \ {±2}, it is possible to give a simple direct proof of Corollary 2.9, constructing an alternative sequence of approximations without making use of "parallel coordinates", see Section 8.1.

PRELIMINARIES
In order to prove our results we need to introduce a number of mathematical objects and related results.First, we discuss in Subsections 3.1 and 3.2 planar curves and their tubular neighborhoods.Then in Subsection 3.3 we provide some identities related to Pauli matrices.Further, we give in Subsection 3.4 basic ideas on the Sobolev spaces and pseudo-differential operators on Σ.Then we recall in Subsection 3.5 the concept of the trace operator.After that we outline in Subsection 3.6 the approach of boundary triples to the extensions theory of symmetric operators.Finally, we recall some properties of the free Dirac operator in Subsection 3.7 and define several associated auxiliary integral operators on Σ in Subsection 3.8.In this preliminary section we partially follow the presentation in [13], that gives the theoretical background and the technical instruments for our analysis.We refer to it and to the references therein for the proofs of the results in this section and for further details.
3.1.Tangent, normal and curvature of Σ.We gather here some elementary facts on curves, in order to fix the notations.Details can be found, e.g. in [1].
We recall that Ω ⊂ R 2 is a bounded open simply connected set with C ∞ boundary Σ := ∂Ω.Set ℓ := |Σ| and let γ : R ℓZ → Σ ⊂ R 2 be a smooth arc-length parametrization of Σ with positive orientation.Let where the dot stands for the derivative with respect to the arc-length s.Clearly, {n γ (s), t γ (s)} is a positively oriented basis of R 2 for any s ∈ R/ℓZ.Moreover, by the Frenet-Serret formulas, there exists function κ γ , called the signed curvature, such that ṫγ = −κ γ n γ , ṅγ = κ γ t γ .
The functions t, n, κ are independent of the particular choice of the positively oriented arc-length parametrization γ: n is the unit normal vector field along Σ which points outwards of Ω + , t is the unit tangent vector field along Σ, counter-clockwise oriented, and κ is the signed curvature of Σ.We remark that we choose the definition of the curvature κ so that it is non-negative for convex domains Ω.

Tubular neighborhoods of Σ.
Below we recall some elementary properties of tubular neighborhoods of planar curves.Details can be found in [32, Chapter 1], [43,7], see also [1, Sections 1.6 and 2.2].For β > 0, is the tubular neighborhood of Σ of width β.Let us introduce the following mapping The following theorem shows that, for all β small enough, the map L γ is a smooth parametrization of Σ β .

Sobolev spaces and pseudo-differential calculus on Σ.
We denote by T the torus T := R/Z; the space of the periodic smooth functions on the torus T and the space of periodic distributions on the torus T will be denoted by D(T) = C ∞ (T) and D(T) ′ , respectively.For f ∈ D(T) ′ , we define its Fourier coefficients using the duality pairing •, • D(T) ′ ,D(T) as follows: For s ∈ R, the Sobolev space of order s on T is defined as A linear operator H on C ∞ (T) is a periodic pseudo-differential operator on T if there exists h : T × Z → C such that: ) there exists α ∈ R such that for all p, q ∈ N 0 there exists c p,q > 0 such that there holds where the operator ω is defined for all (t, n) ∈ T × Z by (ωh The number α is called the order of the pseudo-differential operator H.The set of all pseudo-differential operators of order α on T is denoted Ψ α , and we define Recall that ℓ = |Σ| and that γ : R ℓZ → Σ is a smooth arc-length parametrization of Σ.We define the map U * : D(T) → D(Σ) as where we have set D(Σ) := C ∞ (Σ), and the map U : (3.16) The Sobolev space of order s ∈ R on Σ is defined as For all s ≥ 0, H −s (Σ) = (H s (Σ)) ′ and the duality pairing φ, ψ A linear operator H on C ∞ (Σ) is a periodic pseudo-differential operator on Σ of order α ∈ R if the operator H 0 := U HU −1 ∈ Ψ α .The set of pseudo-differential operators on Σ of order α is denoted Ψ α Σ and we set In the next proposition we gather some useful properties of the pseudo-differential operators on Σ (for proofs see [60, Sections 5.8 & 5.9]).
(i) For all s ∈ R, A extends uniquely to a bounded linear operator, denoted by the same letter, from H s (Σ) to H s−α (Σ).
(ii) There holds The operator Λ α .We describe an example of pseudo-differential operator that is useful for our purposes.For α ∈ R, consider the operator Thanks to (3.15) and (3.17), one can show that Σ .Due to Proposition 3.2 (i), Λ α extends uniquely to a bounded linear operator from H s (Σ) to H s− α 2 (Σ), for any s ∈ R; and such extension is in fact an isomorphism, by the definition of H s (Σ).Of course, Λ α can also be seen as an unbounded operator on H s (Σ), for all s ∈ R.
In particular, the operator Λ := Λ 1 is used repeatedly in the paper, and its action on vector valued functions is understood component-wise.It is useful to remark that for all φ, ψ ∈ L 2 (Σ) we have  [3,Lemma 15,Lemma 18]), the Dirichlet trace operators (Σ; C 2 ) extend uniquely to the bounded linear operators . and the following holds: 3.6.Theory of the boundary triples.In this section we review the theory of the boundary triples, referring to [13,22,28,57] and to the monographs [10,62] for details.
We start with the definition of a boundary triple for a symmetric operator.
Definition 3.4.Let A be a closed densely defined symmetric operator in a Hilbert space H.Moreover, let G be another Hilbert space and Γ 0 , Γ 1 : dom A * → G be linear maps.The triple {G, Γ 0 , Γ 1 } is a boundary triple for A * if and only if (i) for all f, g ∈ dom A * there holds Let {G, Γ 0 , Γ 1 } be a boundary triple for the adjoint A * of a densely defined closed symmetric operator A on a Hilbert space H. Then B := A * ↾ ker Γ 0 is self-adjoint, and for any z ∈ ρ(B), one has the direct sum decomposition In particular, Γ 0 ↾ ker(A * − z) is bijective.We define the γ-field G z and the Weyl function M z associated to the triple {G, Γ 0 , Γ 1 }: For z ∈ ρ(B) the operators G z and M z are bounded, and z → G z and z → M z are holomorphic on ρ(B).Furthermore, the adjoints of G z and M z are given by For A a closed densely defined symmetric operator in a Hilbert space H, the knowledge of a boundary triple for the operator A * allows to move the study of its selfadjoint restrictions and their spectral properties to the (sometimes) easier setting of the Hilbert space G.This is shown in the next proposition, for which we need to introduce some notation.Let G Π be a closed subspace of G, viewed as a Hilbert space when endowed with the induced inner product.Denote the projection and the canonical embedding as respectively.Let Θ be a linear operator in G Π .We define the operator B Π,Θ := A * ↾ dom B Π,Θ , where 3.7.The free Dirac operator.Recall that the free Dirac operator D 0 is defined as follows: For any we have where the Green function φ z is given for x = 0 by the functions K j are the modified Bessel functions of the second kind of order j and we are taking the principal square root function, i.e. for z ∈ C \ (−∞, 0], Re √ z > 0.
We denote S the restriction of D 0 to the functions vanishing at Σ, i.e.
It is easy to see that S * is the maximal realization of D 0 in R 2 \Σ, i.e., for We finally recall some properties of the essential spectrum of any self-adjoint extension of S.
Then the following hold: for some s > 0, then the spectrum of A in (−|m|, |m|) is purely discrete and finite.

Auxiliary integral operators.
We introduce now several integral operators related to the Green function φ z .
Let us denote the Dirichlet trace operator in H 1 (R 2 ; C 2 ) on Σ by It is well known that T D is bounded, surjective, and ker [48,Theorems 3.37 and 3.40].For z ∈ ρ(D 0 ) we define and its anti-dual The potential operator Φ z is a bounded bijective operator from H − 1 2 (Σ; C 2 ) onto ker(S * − z).Moreover, for ϕ ∈ L 2 (Σ; C 2 ) one has the integral representation We denote C Σ the Cauchy transform on Σ.To define it, we identify R 2 ∼ C: writing where the complex line integral is understood as its principal value.Furthermore, let C ′ Σ be the operator which satisfies (3.29)The pseudodifferential operator C z belongs to Ψ 0 Σ , and, in particular, it gives rise to a bounded operator in H s (Σ; C 2 ), for any s ∈ R. Its realization in L 2 (Σ; C 2 ) it satisfies (C z ) * = C z .Furthermore, for the tangent vector field t = (t 1 , t 2 ) along Σ we denote (3.30) Then one has where ℓ is the length of Σ and Ψ ∈ Ψ −2 Σ , see [13,Proposition 3.4].The operators Φ z and C z are related to each other by the following relation, analogous in this context to the Plemelj-Sokhotskii formula (see [13, Proposition 3.5]):

UNITARY EQUIVALENCES
4.1.Reduction to ω = 0. Recall that the operator D η,τ,λ,ω is defined as in (2.2), (2.3).The purpose of this section is to show that, in many cases, D η,τ,λ,ω is unitarily equivalent to D η,τ , λ,0 for certain η, τ , λ : Σ → R defined in terms of η, τ, λ, and ω.This unitary equivalence is based on the following simple transformation.Given z ∈ C such that |z| = 1, let It is clear that Hence U z is a unitary operator in L 2 (R 2 ; C 2 ).With this at hand, we can introduce the operator which is unitarily equivalent to D η,τ,λ,ω by construction.
Before addressing the proof of Theorem 2.1, let us make some observations on the values of X and z, which were introduced in (2.7) and (2.8), respectively, depending on d and ω.
Therefore, we clearly have X ∈ R \ {0} and z ∈ C are constant.Also, to check that |z| = 1 is straightforward.

Let us address the proof of (4.2). The first step is to compute
and, similarly, From these calculations, we obtain Given a, ã ∈ R, a computation shows that , and using (2.7), we see that and Plugging (4.6) and (4.7) in (4.5), and combining then with (4.4), we conclude that where we used (2.8) in the last equality above.Therefore, (4.2) holds and the lemma follows.

Spectral relations.
From the proof of Theorem 2.1 we realize that, if ω = 0, we can also allow d to be variable in Σ in the conclusion of Theorem 2.1, as long as d(x) / ∈ {0, −4} for all x ∈ Σ.This is because (4.10) holds whenever z is constant in Ω − , and for ω = 0 and d = 0, −4 we can always take z = −1, as we explained below the statement of Theorem 2.1.Thus, we can take X = −4/d, which yields the isospectral transformation of parameters , −4} for all x ∈ Σ, with no more restrictions on d in Σ.We underline that this correspondence maps the set {(η, τ, λ Next, we apply the observation above on D η,τ,λ,0 = D η,τ,λ with constant parameters.Moreover, we investigate the spectral relation between D η,τ,λ and its charge conjugation.Proposition 4.2.Let η, τ, λ ∈ R, and D η,τ,λ be defined as in (2.4), (2.5).The following hold: (ii).Let C be the antilinear charge conjugation operator The operator C is an involution, i.e.
We finally mention that the three-dimensional analogue of D η,0,0,ω was investigated in [44], where the same transformation of the coefficients (η, ω) → (η, 0) by means of X and z was discovered.Since here we also admit τ, λ = 0, and, with a restriction, also non-constant coefficients, Theorem 2.1 can be understood as a generalization of [44] to the two-dimensional scenario for more general δ-shell interactions.
If d = η 2 − τ 2 − λ 2 = −4 everywhere on Σ, Lemma 5.1 (i) states that the values of f + and f − along Σ are related via the matrix R η,τ,λ : the presence of the δ-shell implies a transmission condition for the functions in the domain of D η,τ,λ across the surface Σ.

THE NON-CRITICAL CASE
In this section we prove Theorem 2.2 stated in Section 2. The operator D η,τ,λ is defined again as in (2.4), (2.5), where in general the condition in (2.4) is understood in the sense of H − 1 2 (Σ).We show the self-adjointness and some further properties of the operator D η,τ,λ : the strategy of the proofs mainly follows [13], but we need to modify the boundary triple that we use in order to include the magnetic interaction.
Recall that we assume that η, τ, λ ∈ C ∞ (Σ; R) and Σ = ∂Ω ∈ C ∞ in order to exploit the theory of pseudo-differential operators, but we expect that these assumptions can be greatly weakened.However, in the case that Σ has only Lipschitz regularity and in the case that η, τ, λ are not regular different phenomena are expected, see [42,53,24], see also [12,58] where coefficients with lower regularity are considered in the three dimensional case.
It will be convenient to introduce some extra notation.Recall the definition T := t 1 +it 2 ∈ C from (3.30), where t = (t 1 , t 2 ) is the tangent vector to Σ at the point x ∈ Σ.We define the following matrix-valued functions on Σ: For all x ∈ Σ, the matrix V (x) is unitary and In the following proposition we adapt to our setting the boundary triple constructed in [13,Proposition 3.6] for the operator S * , defined in (3.24).In the formulation of the below proposition we extend the operators Λ and C ζ defined in (3.17 ) and the Weyl function is Proof.In [13,Proposition 3.6] it is proved that {L 2 (Σ; C 2 ), Γ 0 , Γ 1 } is a boundary triple for S * , with Moreover, D 0 = S * ↾ ker Γ 0 and the γ-field G z and the Weyl function M z associated to the boundary triple {L 2 (Σ; C 2 ), Γ 0 , Γ 1 } are defined by We define Γ 0 , Γ 1 as in (6.3).The key observation of our proof is that and Λ −1 V Λ, ΛV Λ −1 are bounded and boundedly invertible on L 2 (Σ; C 2 ), since Λ : is surjective, i.e. the condition (ii) in Definition 3.4 of boundary triple is fulfilled.In order to verify the condition (i) of the definition, we observe that, for all φ, ψ ∈ dom S * , due to (3.18) and (6.7).The last term in the previous equation equals because V * V = I 2 .Combining (6.8) and (6.9), and using (3.18) again, we get This yields (i) in Definition 3.4.Therefore, we conclude that {L 2 (Σ; C 2 ), Γ 0 , Γ 1 } is a boundary triple for S * .Since ker Γ 0 = ker Γ 0 , it is true that D 0 = S * ↾ ker Γ 0 .From (3.19), we get that, for all z ∈ ρ(D 0 ), i.e. (6.4).Plugging this result together with (6.7) into (3.20),we obtain for all z ∈ ρ(D 0 ).This is just (6.5).
The following lemma is a regularity result concerning the boundary triple defined in Proposition 6.1.
Proof.The proof is analogous to [13,Lemma 3.7], reasoning as in the proof of Proposition 6.1.
Proof.From (6.3), we see that so the transmission condition in (2.4) rewrites as follows: Multiplying the last equation by V and then using (6.2) together with the identity We prove now (i).In the case that d(x) = 0 the matrix B = B(x) is invertible for all x ∈ Σ. Thanks to (6.15), we obtain the representation in (6.12).
We show (iii) only in the case that d(x) = 0 for all x ∈ Σ, the case d = 0 being similar.By Theorem 3.5 (i) combined with (6.26), z ∈ ρ(D 0 ) is an eigenvalue of D η,τ,λ if and only if there exists ψ ∈ dom Θ = H 1 (Σ; C 2 ) such that

THE PURELY MAGNETIC CRITICAL INTERACTION
In this section we give the proof of Theorem 2.4: we consider the case λ = 2 only, since the case λ = −2 can be treated analogously.Recall that by Theorem 2.3 the operator D 0,0,2 can be decomposed into the orthogonal sum Hence, D ± have the following representations Using the Cauchy-Riemann differential expressions we can represent D ± as follows Lemma 18] (see also [13,Lemma 3.1]) the domains of D ± can alternatively be given by The operators D ± can be viewed as bounded symmetric perturbations of the respective massless Dirac operators.Since by [61, Proposition 1] the unperturbed massless Dirac operators are self-adjoint, we conclude that D ± are self-adjoint as well.
Next, we will show the symmetry of the spectrum of D ± .For ν ∈ C such that ν 2 − m 2 = 0 we introduce the matrix Clearly, T ν is invertible and T −1 ν = T −ν .Moreover, for any ν ∈ C \ {−m, m} we have T ν (dom D ± ) = dom D ± and Let ν ∈ σ(D ± ) \ {−m, m}, then there exists a sequence Then we get Hence, we conclude that −ν ∈ σ(D ± ).Moreover, if ν = ±m is an eigenvalue of D ± , then in view of identity (7.1) −ν is also an eigenvalue of D ± .Now we perform the spectral analysis of D + .First of all, we notice the inclusion Indeed, Since the space of square-integrable anti-holomorphic functions on Ω + is infinitedimensional, −m is an eigenvalue of infinite multiplicity in the spectrum of D + .In particular, dom D + ⊂ H s (Ω + ; C 2 ) for any s > 0, as otherwise the spectrum of D + would be purely discrete, due to the compactness of embedding of the Sobolev spaces Consider the auxiliary operators The adjoints of A ± are characterised in the spirit of [61, Proposition 1] as The quadratic form for (D + ) 2 is given by Next, we compute The domain of h + can be written as Therefore, we end up with the orthogonal decomposition ), from which we deduce, using [61, Propositions 2 and 3], that where −∆ Ω+ D is the Dirichlet Laplacian on Ω + .Using the symmetry of the spectrum shown above we obtain that Hence, (iii) of Theorem 2.4 follows.Moreover, we observe that ) is a core for D + .The latter follows from the density of H 1 (Ω + ) in dom A * + ; cf.[3, Lemma 14].Now we perform the spectral analysis of D − .As in the analysis of D + we notice the inclusion We observe that the space of square-integrable holomorphic functions on Ω − is also infinite-dimensional. Indeed, for an arbitrary z 0 ∈ Ω + the family of linear independent functions {(z − z 0 ) −k } k≥2 is square-integrable and holomorphic in Ω − Hence, m is an eigenvalue of infinite multiplicity in the spectrum of D − and thus combining with the fact that −m is an eigenvalue of infinite multiplicity in the spectrum of D + shown above the claim of (ii) of Theorem 2.4 follows.Next we consider the quadratic form Repeating the same type of computation as we did for D + we get The domain of h − can be written as Hence, we have the orthogonal decomposition Thus the proof is concluded.
Remark 7.1.The above spectral analysis of D ± is reminiscent of the spectral analysis of three-dimensional Dirac operators with zig-zag boundary conditions on general open sets performed in [38], which has appeared while the present paper was under preparation.

APPROXIMATION OF δ-SHELL INTERACTIONS BY REGULAR POTENTIALS
In this section we prove Theorem 2.5 on approximation of the Dirac operator with δ-shell interaction by a sequence of Dirac operators with regular scaled potentials.