Spectral Properties of the Dirac Operator coupled with δ-Shell Interactions
aa r X i v : . [ m a t h . SP ] F e b SPECTRAL PROPERTIES OF THE DIRAC OPERATOR COUPLED WITH δ -SHELL INTERACTIONS BADREDDINE BENHELLAL
Abstract.
Let Ω ⊂ R be an open set, we study the spectral properties of the free Dirac operator H := − iα · ∇ + mβ coupled with the singular potential V κ = ( ǫI + µβ + η ( α · N )) δ ∂ Ω , where κ = ( ǫ, µ, η ) ∈ R . In the first instance, Ω can be either a C -bounded domain or a locally deformedhalf-space. In both cases, the self-adjointness is proved and several spectral properties are given.In particular, we extend the result of [10] to the case of a locally deformed half-space, by givinga complete description of the essential spectrum of H + V κ , for the so-called critical combinationsof coupling constants. In the second part of the paper, the case of bounded rough domains isinvestigated. Namely, in the non-critical case and under the assumption that Ω has a VMO normal,we show that H + V κ is still self-adjoint and preserves almost all of its spectral properties. Moregenerally, under certain assumptions about the sign or the size of the coupling constants, we areable to show the self-adjointness of the coupling H + ( ǫI + µβ ) δ ∂ Ω , when Ω is bounded uniformlyrectifiable. Moreover, if ǫ − µ = − , 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 theanticommutator between ( α · N ) and the Cauchy operator associated to the free Dirac operator.Finally, we study the coupling H υ = H + iυβ ( α · N ) δ ∂ Ω . In particular, if Ω is a bounded C domain,then we show that H ± is essentially self-adjoint and generates confinement. Contents
1. Intoduction 22. Notations and Preliminaries 62.1. Integral operators associated to the Dirac operator 83. Self-adjointness of H κ δ -Shell interactions 143.2. The operators Λ a ± δ -interactions supported on compact Ahlfors regular surfaces 265.1. δ -interactions supported on the boundary of a Lipschitz domain with VMO normal 285.2. δ -interactions supported on the boundary of a bounded uniformly rectifiable domain 335.3. δ -interactions supported on the boundary of a C ,γ -domain 406. Quantum Confinement induced by Dirac operators with anomalous magnetic δ -shellinteractions 42Acknowledgement 48References 48 Date : February 18, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Dirac operators, self-adjoint extensions, shell interactions, critical interaction strength, Quan-tum confinement, Semmes-Kenig-Toro domains, Uniformly rectifiable domains. Intoduction
In this paper we investigate in R the self-adjointness character and the spectral properties of thecoupling H ǫ,µ + V η,υ , where H ǫ,µ is the Dirac operator with electrostatic and Lorentz scalar δ -shellinteractions, formally written as: H ǫ,µ := H + V ǫ,µ = H + ( ǫI + 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 , N is the unit normal vector field at ∂ Ω which points outwards of Ω and the δ -potential isthe 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- / in the externalpotential V ǫ,µ + V η,υ . From this physical point of view, the singular interactions given by the couplingconstants ǫ, µ, η and υ are called respectively electrostatic, Lorentz scalar, magnetic and anomalousmagnetic potential (we refer to [42] for more information on the classification of external fields). Thesurface ∂ Ω supporting the interactions is called a shell.Recently, Dirac operators with δ -shell interactions have been studied extensively. Namely, thecoupling H with the electrostatic and the Lorentz scalar δ -shell interactions (i.e H ǫ,µ ); we refer tothe survey [44] for a review on the topic. To our knowledge, the spectral study of the Dirac operator H ǫ,µ goes back to the papers [19] and [20], where the authors studied the spherical case (i.e ∂ Ω is asphere). Moreover, in [19] the authors point out that under the assumption ǫ − µ = − , the shellbecomes 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 \ Ω ), then during the evolution in time,it cannot cross the surface ∂ Ω to join the region R \ Ω (respectively Ω ) for all t > . Mathematically,this means that the Dirac operator in consideration decouples into a direct sum of two Dirac operatorsacting respectively on Ω and R \ Ω with appropriate boundary conditions. In particular, when ǫ = 0 ,this phenomenon has been known to physicists since the ’s (cf.[15] and [32] for example); andits mathematical model described by the Dirac operator with MIT boundary conditions has beenthe subject of several mathematical papers (we refer to the recent paper [9] as well as the referencescited 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 along period of silence. Indeed, apart from [46] where the authors studied the scattering theory and thenon-relativistic limit of H ǫ,µ (in the spherical case), the spectral study of H ǫ,µ has been forgotten fortwo decades. Since then, it has been relaunched in [2], where the authors developed a new techniqueto 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 ) supportedon the boundary of a bounded regular domain, and they proved that the perturbed operator is self-adjoint for all ǫ = ± . The same authors continue the spectral study of the electrostatic case; forinstance, the existence of point spectrum and related problems; see [3] and [4]. In [3] they add thescalar Lorentz interaction and they show that under the condition ǫ − µ = − , H ǫ,µ still generatesthe phenomenon of confinement.Subsequently, the concept of quasi-boundary triples and their Weyl functions were used in [6] tostudy the Dirac operators with electrostatic δ -shell interactions. In this paper, the authors prove IRAC OPERATORS WITH δ -SHELL INTERACTIONS 3 the self-adjointness for all ǫ = ± , and investigate several spectral properties, adding the scatteringtheory and asymptotic properties of the model. In all the above papers, the case ǫ = ± (known asthe critical interaction strengths) has not been considered. This gap has been covered in [43], then in[8] with different approaches. Indeed, in this particular case, it turns out that the Dirac operator withelectrostatic δ -shell interactions is essentially self-adjoint, and functions in the domain of the closureare less regular comparing to the non critical case. Moreover, the authors in [8] show that if ∂ Ω contains a flat part, then the point belongs to the essential spectrum of H ± , . Similar phenomenonappears when we study the Dirac operator H ǫ,µ . In fact, in this case, the critical combinations ofcoupling constants are ǫ − µ = 4 ; see [7] for example. The self-adjointness in the critical case ǫ − µ = 4 was proved for the two dimensional analogue of H ǫ,µ in [10], where the authors considered δ -interactions supported on a smooth closed curve. Furthermore, by making use of complex analysisand periodic pseudo-differential operators techniques, they show that Sp ess ( H ǫ,µ ) = (cid:0) − ∞ , − m (cid:3) ∪ n − mµǫ o ∪ (cid:2) m, + ∞ (cid:1) . (1.3)Of course, such techniques are no longer available in the three dimensional case. Nevertheless, at thisstage, one may ask the following question:(Q1) In the three dimensional setting, when ǫ − µ = 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 -shellinteractions have been done for Ω at least C -bounded domain (except in [45], where the particularcase 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 [44]:(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, (i.e υ = 0 ). Unlike most existing works, instead of treating the δ -interactions as atransmission problem, in this paper we made the choice to follow the strategy introduced in [2].Let us present the context we are considering and summarize the main results of our work. Weshall assume that the open set Ω satisfies (for instance) one of the following hypotheses:(1) Ω is a C -bounded domain.(2) Ω := Ω ν := { ( x, t ) ∈ R × R : t > νφ ( x ) } , where ν ∈ R and φ : R → R is a C -smooth,compactly supported function.We define the Dirac operators H κ := H ǫ,µ + V η, , on the domain dom( H κ ) = (cid:8) u + Φ[ g ] : u ∈ H ( R ) , g ∈ L ( ∂ Ω) , u | ∂ Ω = − Λ + [ g ] (cid:9) , κ := ( ǫ, µ, η ) ∈ R , where Φ is an appropriate fundamental solution of the unperturbed operator H , and Λ ± are boundedlinear operators acting on L ( ∂ Ω) (see Notation 2.1). We mention that the operator Λ ± appears inseveral works when the quasi-boundary triples theory is used to study the Dirac operator H ǫ,µ , see[8, Lemma 5.4] and [10, Proposition 4.3] for example. We point out that the consideration of thesecond assumption is motivated by [22], where the Schrödinger operator with δ -shell interaction wasconsidered.As a first step of the current paper, we study the self-adjointness character of H κ , when Ω satisfiesthe assumption (1) or (2) . We begin by proving that H κ is self-adjoint when ǫ − µ − η = 4 (i.ein the non-critical case), and we show that dom( H κ ) ⊂ H ( R \ ∂ Ω) , which means that functions in dom( H κ ) have a Sobolev regularity; cf. Theorem 3.1. To prove this result we develop a strategy veryclose to [43], it is based essentially on the fact that the anticommutators of Cauchy operator C ∂ Ω (see BADREDDINE BENHELLAL (2.17) for the definition) with β or with ( α · N ) have a regularizing effect. Indeed, as it was observedin several works (see [3] for example), the operator Λ ∓ Λ ± involves the above anticommutators and itturns 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 -Sobolev regularity. When ǫ − µ − η = 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 + , whichis 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 thesecond 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 spectrumof H κ when Ω satisfies the second assumption. More precisely, we show that Sp ess ( H κ ) = (cid:0) − ∞ , − m (cid:3) ∪ n − mµǫ o ∪ (cid:2) m, + ∞ (cid:1) , which answers positively to the question (Q1) , hence generalizing the result of [10] to this kind ofsurfaces. The proof is based on the use of compactness and localization arguments. We remark thateven after adding the perturbation by the potential V η, , the point which appears in the gap remainsthe same (see the discussion after Theorem 4.2 for more details). All these results will be proven usingan 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. [6] 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 -regularity is essential to use our technique, especially when ∂ Ω is unbounded, and the combination of the coupling constants is critical. Nevertheless, in the non-critical case, if Ω is bounded then one can do more. In fact, one of the reasons for choosing to workwith the strategy of [2] is that in the non-critical case, the fact that Λ + is Fredholm implies the self-adjointness of H κ (see [2, Theorem 2.11]). Moreover, the compactness of the anticommutators impliesthat Λ + is a Fredholm operator. Fredholm’s character and (or) invertibility of the boundary integraloperator 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 Neumannproblem on Lipschitz domains; cf. [31],[47] and [17]. As we will see later (see Lemma 3.1), { β, C ∂ Ω } isnothing else than the trace of the matrix valued Single-layer potential, which is therefore a compactoperator on L ( ∂ Ω) , 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 ( ∂ Ω) ?One of the main results of this article is the answer to this question, see Theorem 5.4. Looking closelyat the anticommutator { α · N , C ∂ Ω } , we observe that it involves a matrix version of the principal valueof the harmonic double-layer K , its adjoint K ∗ and the commutators [ N k , R j ] , where R j are the Riesztransforms. Hence the situation is more clear. In fact, from the harmonic analysis and geometricmeasure theory point of view, it is shown that the boundedness of Riesz transforms characterizesthe uniform rectifiability of ∂ Ω ; cf. [38], for example. In addition, functional analytic properties ofthe Riesz transforms (such as the identity P j =1 R j = − I ) and the analogue version of the stronglysingular part of ( α · N ) C ∂ Ω in the Clifford algebra C l , i.e. the Cauchy-Clifford operator (especiallyits self-adjointness and compactness character) are strongly related to the regularity and geometricproperties of the domain Ω , for more details we refer to [26] and [28]. The most important fact which IRAC OPERATORS WITH δ -SHELL INTERACTIONS 5 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 [28]. However, regular SKTdomains are not necessarily Lipschitz domains and vice versa. So, to stay in the context of compactLipschitz domains, we shall suppose that Ω satisfies the following property:(3) Ω is a bounded Lipschitz domain with normal N ∈ VMO( ∂ Ω , dS) .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 ( ∂ Ω) . (1.4)see Theorem 5.4. Once we have established that and hence proved the compactness of { α · N , C ∂ Ω } , theself-adjointness of H κ will be an easy consequence of [2, Theorem 2.11]. Moreover, we prove almostall the spectral properties as in C -smooth case. Another geometric type result that we establishin this article, is a characterization of the class of regular SKT domains via the compactness of theanticommutator { α · N , C ∂ Ω } in L ( ∂ Ω) , see Proposition 5.2. More precisely, using the materialprovided in [28], we show that if Ω is a two-sided NTA domain with a compact Ahlfors regularboundary, then it holds that Ω is a regular SKT domain ⇐⇒ { α · N , C ∂ Ω } is compact in L ( ∂ Ω) . (1.5)At this stage, beyond the two classes of domains which are characterized by (1.4) and (1.5), the com-pactness arguments mentioned previously are no longer valid. So, in order to go further in our study wechange the strategy and we turn to the invertibility arguments, which are rather valid in a more generalcontext. Indeed, we investigate the case of bounded uniformly rectifiable domains (see Section 5 for thedefinitions). In one direction, making the assumption that < | ǫ − µ | < / k C ∂ Ω k L ( ∂ Ω) → L ( ∂ Ω) ,we then show that H ǫ,µ is self-adjoint, cf. Theorem 5.6. In another direction, assuming that µ > ǫ or k W k L (Σ) → L (Σ) < ǫ − µ < / k W k L (Σ) → L (Σ) (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 thenrecover the same spectral properties as in the case of the assumption (3) . Moreover, we show that H ǫ,µ generates confinement when ǫ − µ = − , cf. Theorem 5.7 and Proposition 5.4.Having established the above results, and in order to enrich the knowledge on the connectionsbetween the smoothness of Ω and the Sobolev regularity of functions in dom( H κ ) , we consider theclass of Hölder’s domains C ,γ , with γ ∈ (0 , , and we prove that the functions in dom( H κ ) have the H s -Sobolev regularity, with s > / . In particular, we show that if γ ∈ (1 / , , then dom( H κ ) ⊂ H ( R \ ∂ Ω) . Moreover, the technique developed before for the C -smooth surfaces remains valid toprove such a result (cf. Remark 5.5).The last part of this paper is devoted to the spectral study of H υ := H + iυβ ( α · N ) δ ∂ Ω , the Diracoperator with anomalous magnetic δ -interactions. We mention that while preparing this manuscript,it turns out that the authors of the paper [14] (which will appear soon) worked on the two-dimensionalanalog of this problem at the same time, and our results intersect on this point (see Section 6 for moredetails). Assuming that Ω satisfies the assumption (1) , one of the most important properties that weshow for this operator is that, in the critical case υ = 4 , H υ is essentially self-adjoint and it decouplesin a direct sum of two Dirac operators acting respectively on Ω and R \ Ω , with boundary conditionsin H − / ( ∂ Ω) . Thus, H ± generates confinement and hence ∂ Ω becomes impenetrable. Moreover,the inner part of H ± which acts on Ω coincide with so-called Dirac operator with zig-zag boundarycondition, see Section 6. BADREDDINE BENHELLAL
Organisation of the paper.
The structure of the paper is as follows. In the second section, weset up the necessary notations and recall the relevant material from [2]. In Section 3, we study theself-adjointness of H κ , when ∂ Ω satisfies the first and the second assumption. Section 4 is devoted tothe spectral study of H κ . We focus namely on the case where Ω is a locally deformed half space and wegive a complete description of the essential spectrum of H κ , for the critical combinations of couplingconstants. 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, weinvestigate the case of a domain Ω satisfying assumptions (3) . After this, the general case of uniformlyrectifiable domains is considered in Subsection 5.2, for the Dirac operator H ǫ,µ . As a last step, theclass of Hölder’s domains C ,γ is considered in Subsection 5.3. Finally, in Section 6, we study thespectral properties of the Dirac operator H υ , for all possible combinations of interaction strengths.2. Notations and Preliminaries
We consider a surface Σ ⊂ R dividing the space into two regions Ω ± . More precisely, we assumethat Σ satisfies one of the hypotheses:(H1) Σ = ∂ Ω + with Ω + a C -bounded domain.(H2) Σ := Σ ν := { ( x , x , x ) ∈ R : x = νφ ( x , x ) } , where ν ∈ R + and φ : R → R is a C -smooth, compactly supported function. We denote by L φ the Lipschitz constant of φ andby F we denote the flat part of Σ ν i.e. F := { x = ( x , x , νφ ( x , x )) ∈ Σ ν : ( x , x ) / ∈ supp( φ ) } . (2.1)We parameterize Σ ν by the mapping(2.2) ( τ : R −→ R 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 )d x ,where J ν is the Jacobian given by J ν ( x ) = p ν |∇ φ ( x ) | . (2.3)Throughout the paper, we shall work on the Hilbert space L ( R d ) (respectivelly, L (Ω ± ) ) with re-spect to the Lebesgue measure. D (Ω ± ) denotes the usual space of indefinitely differentiable functionswith compact support, and D ′ (Ω ± ) is the space of distributions defined as the dual space of D (Ω ± ) .We define the unitary Fourier-Plancherel operator F : L ( R d ) → L ( R d ) as follows F [ u ]( ξ ) = 1(2 π ) d/ Z R d e − ix · ξ u ( x )d x, ∀ ξ ∈ R d , (2.4)and by F − we denote the inverse Fourier-Plancherel operator F − : L ( R d ) → L ( R d ) , given by F − [ u ]( x ) = 1(2 π ) d/ Z R d e iξ · x u ( ξ )d ξ, ∀ x ∈ R d . (2.5)Given x ∈ R d − , by F x we abbreviate the partial Fourier-Plancherel operator on the variable x . Given s ∈ [ − , , we denote by H s ( R d ) the Sobolev space of order s , defined as H s ( R d ) := { u ∈ L ( R d ) : Z R d (1 + | ξ | ) s |F [ u ]( ξ ) | d ξ < ∞} . (2.6) IRAC OPERATORS WITH δ -SHELL INTERACTIONS 7 The Sobolev space H (Ω ± ) is defined as follows: H (Ω ± ) = { ϕ ± ∈ L (Ω ± ) : there exists ˜ ϕ ± ∈ H ( R ) such that ˜ ϕ ± | Ω ± = ϕ ± } . (2.7)By L (Σ , d S ) := L (Σ) we denote the usual L -space over Σ . Given s ∈ [0 , , if Σ satisfies (H2) ,we then define the Sobolev spaces H s (Σ) in terms of the Sobolev spaces over R as usual. That isgiven g ∈ L (Σ) , we define g φ ( x ) = g ( x, νφ ( x )) , for x ∈ R . Then H s (Σ) := { g ∈ L (Σ) : g φ ∈ H s ( R ) } , for all s ∈ [0 , , (2.8)and then define H − s (Σ) to be the completion of L (Σ) with following norm: k g k H − s (Σ) := k g φ J ν k H − s ( R ) , for all s ∈ [0 , . (2.9)Recall that H − s (Σ) is a realisation of the dual space of H s (Σ) ; see [36] for example. Now, if Σ satisfies (H1) , we then define the Sobolev spaces H s (Σ) using local coordinates representation onthe surface Σ ; see [36]. By t Σ : H (Ω ± ) → H / (Σ) we denote the classical trace operator. For afunction u ∈ H ( R ) , with a slight abuse of terminology we will refer to t Σ u as the restriction of u on Σ .Let x ∈ Σ and a > , 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 > 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 > is fixed, we shall write Γ Ω ± ( x ) insteadof Let α = ( α , α , α ) and β be the × Hermitian and unitary matrices given by α k = σ k σ k ! for k = 1 , , β = I − I ! , (2.12)where σ = ( σ , σ , σ ) are the Pauli matrices defined by σ = ! , σ = − ii ! , σ = − ! . (2.13)We denote by N and δ Σ the unit normal vector field at Σ which points outwards of Ω + and the Diracdistribution supported on Σ respectively. Given m > , we consider the Dirac operator H κ = H + V κ = − iα · ∇ + mβ + ( ǫI + µβ + η ( α · N )) δ Σ , κ := ( ǫ, µ, η ) ∈ R . (2.14)in the Hilbert space L ( R ) , where H = − iα · ∇ + mβ is the free Dirac operator defined on H ( R ) .It is well known that ( H , H ( R ) ) is self-adjoint (see [42, subsection 1.4]) and its spectrum is givenby 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 andfor the convenience of the reader, we recall the relevant material from [2] (without detailed proofs),thus making our exposition self-contained. BADREDDINE BENHELLAL
Integral operators associated to the Dirac operator.
Here we list some well known resultsabout integral operators associated to the fundamental solution of the Dirac operator. Given z ∈ C \ (( −∞ , − m ] ∪ [ m, ∞ )) with the convention that Im √ z − m > , we recall that the fundamentalsolution of H − z is given by φ z ( x ) = e i √ z − m | x | π | x | (cid:18) z + mβ + (1 − i p z − m | x | ) iα · x | x | (cid:19) , for all x ∈ R \ { } , (2.15)see for example [42, Section 1.E]. Next, we define the following operators Φ z : L (Σ) −→ L ( R ) g Φ z [ g ]( x ) = Z Σ φ z ( x − y ) g ( y )dS( y ) , for all x ∈ R \ Σ , (2.16)then, Φ z : L (Σ) −→ L ( R ) is a bounded operator. Furthermore, ( H − z )Φ z [ g ] = 0 holds in D ′ (Ω ± ) , for all g ∈ L (Σ) . In particular, Φ z gives rise to a bounded operator from H / (Σ) onto H ( R \ Σ) ; cf. [8, Proposition 4.2]. Given x ∈ Σ and g ∈ L (Σ) , we set C z Σ [ g ]( x ) = lim ρ ց Z | 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 C z Σ , C z ± : L (Σ) → L (Σ) are linear bounded operators. Furthermore, the following hold:(i) C z ± = ∓ i ( α · N ) + C z Σ ,(Plemelj-Sokhotski jump formula).(ii) ( C z Σ ( α · N )) = − I . In particular, k C z Σ k > . Proof. If Σ satisfies (H1) , then the proof is analogous to the one of [3, Lemma 2.2], wherethe authors use essentially the Green’s theorem and the following well known result on the trace ofderivatives of a single-layer potential. Indeed, given g ∈ L (Σ) , then for dS -a.e. x ∈ Σ , we have lim Ω ± ∋ y −→ nt x Z y − w π | y − w | g ( w )dS( w ) = ∓ g ( x ) N ( x ) + lim ρ ց Z | x − w | >ρ x − w π | x − w | g ( w )dS( w ) . (2.18)Note that this result is also true if Σ satisfies (H2) , see [37, Theorem 5.4.7] for example. Thus, onecan adapt the proof of [2, Lemma 3.3] in this case. The detailed verification of items (i) and (ii) beingleft to the reader. (cid:3) Remark 2.1.
Note that in the same setting, Lemma 2.1 still holds true if for example Σ is a compactLipschitz surface or the graph of a Lipschitz function φ : R → R ; see [5] and [2, Remark 3.14] .Moreover, since (Φ ¯ z ) ∗ = ( H − z ) − ⇂ Σ , by duality arguments, it follows that the operator Φ z givesrise to a bounded operator from L (Σ) onto H / ( R \ Σ) ; cf. [9, Subsection 3.3] . Hence, thenon-tangential limit in Lemma 2.1 (i) coincides with the trace operator for all data in H / (Σ) . Corollary 2.1.
Let z ∈ C \ (( −∞ , − m ] ∪ [ m, ∞ )) . Then, the operator C z Σ is bounded from H / (Σ) onto itself. Moreover, it holds that ( C z Σ ) ∗ = C z Σ in L (Σ) . In particular, C z Σ is a self-adjoint operatorin L (Σ) , for all z ∈ ( − m, m ) . Proof.
Given g ∈ H / (Σ) . Since Φ z [ g ] ∈ H ( R \ Σ) , it follows that C z ± [ g ] ∈ H / (Σ) .Thus, from Lemma 2.1 (i) we deduce that C z Σ [ g ] = ( C z + + C z − )[ g ] ∈ H / (Σ) . This proves the firststatement. The second statement is a direct consequence of the fact that φ z ( y − x ) = φ z ( x − y ) . (cid:3) IRAC OPERATORS WITH δ -SHELL INTERACTIONS 9 Notation 2.1.
Let κ = ( ǫ, µ, η ) ∈ R such that sgn( κ ) := ǫ − µ − η = 0 . We define the operators Λ z ± as follows: Λ z ± = 1sgn( κ ) ( ǫI ∓ ( µβ + η ( α · N ))) ± C z Σ , ∀ z ∈ C \ (( −∞ , − m ] ∪ [ m, ∞ )) . (2.19) Since ( α · N ) is C -smooth and symmetric, it easily follows that Λ z ± are bounded (and self-adjoint for z ∈ ( − m, m ) ) from L (Σ) onto itself, and bounded from H / (Σ) onto itself. In the sequel, we shall write Φ , C Σ , C ± and Λ ± instead of Φ , C , C ± and Λ ± , respectively. Nowwe 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 such that sgn( κ ) = 0 . The Dirac operator coupled with acombination of electrostatic, Lorentz scalar and normal vector field δ -shell interactions of strength ǫ , µ and η respectively, is the operator H κ = H + V κ , acting in L ( R ) and defined on the domain dom( H κ ) = (cid:8) u + Φ[ g ] : u ∈ H ( R ) , g ∈ L (Σ) , t Σ u = − Λ + [ g ] (cid:9) , (2.20) where V κ ( ϕ ) = 12 ( ǫI + 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 ) = H u , 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 outthat 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. ( [43] , [8] ) 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 (Ω ± ) ) has a unique extension to abounded linear operator from L (Ω ± ) to H − / (Σ) .(ii) The operator Φ z admits a continuous extension from H − / (Σ) to L ( R ) , which we stilldenote Φ z .(iii) The operator C z Σ admits a continuous extension ˜ C z Σ : H − / (Σ) → H − / (Σ) . Moreover,we have ˜ C z ± [ h ] = ( ∓ i α · N ) + ˜ C z Σ )[ h ] , h ˜ C z Σ [ h ] , g i H − / , H / = h h, C z Σ [ g ] i H − / , H / , (3.1) for any g ∈ H / (Σ) and h ∈ H − / (Σ) . Proof.
Item (i) is the classical trace theorem, see [36, Theorem 3.38] for example. (ii) can be provedas much the same way as in [43, Theorem 2.2], see also [8, Proposition 4.4]. Since ( C z Σ ) ∗ = C z Σ , and C z Σ is bounded from H / (Σ) 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 [10, Proposition 3.5] and [8, Proposition4.4 (ii) ]. (cid:3) In the following, we denote by ˜Λ z ± the continuous extension of Λ z ± defined from H − / (Σ) ontoitself. Now, we can state the first main theorem of the paper, the remainder of this part will bedevoted 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 ) = n u + Φ[ g ] : u ∈ H ( R ) , g ∈ H / (Σ) , t Σ u = − Λ + [ g ] o . (3.2) (ii) If sgn( κ ) = 4 , then H κ is essentially self-adjoint and we have dom( H κ ) = n u + Φ[ g ] : u ∈ H ( R ) , g ∈ H − / (Σ) , t Σ u = − ˜Λ + [ g ] o . (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 alwaysadmits a closure, to prove the proposition it is suffices to show the following:(i) dom( H κ ) is dense in L ( R ) .(ii) H κ is symmetric on dom( H κ ) .First, observe that C ∞ ( R \ Σ) ⊂ dom( H κ ) ⊂ L ( R ) . Thus (i) follows from this and the fact that C ∞ ( R \ Σ) is a dense subspace of L ( R ) . Now we prove (ii) , let ϕ, ψ ∈ dom( H κ ) with ϕ = u + Φ[ g ] and ψ = v + Φ[ h ] . Then, we have hH κ ϕ, ψ i L ( R ) − h ϕ, H κ ψ i L ( R ) = hH u, v + Φ[ h ] i L ( R ) − h u + Φ[ g ] , H v i L ( R ) = hH u, Φ[ h ] i L ( R ) − h Φ[ g ] , H v i L ( R ) = h t Σ u, h i L (Σ) + h g, t Σ ν v i L (Σ) . Using the conditions t Σ u = − Λ + [ g ] and t Σ v = − Λ + [ h ] , and that Λ + is self-adjoint, we obtain hH κ ϕ, ψ i L ( R ) − h ϕ, H κ ψ i L ( R ) = h− Λ + [ g ] , h i L (Σ) + h g, − Λ + [ h ] i L (Σ) = 0 . (3.4)Thus, H κ is symmetric on dom( H κ ) and densely defined in L ( R ) . This finishes the proof. (cid:3) 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 ) = n u + Φ[ g ] : u ∈ H ( R ) , g ∈ H − / (Σ) , t Σ u = − ˜Λ + [ g ] o . (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 h ϕ, H κ ψ i L ( R ) = hH v, u i L ( R ) + h Φ[ h ] , H u i L ( R ) = hH v, u i L ( R ) + h h, t Σ u i H − / , H / = hH v, u i L ( R ) + h h, − Λ + [ g ] i H − / , H / = hH v, u i L ( R ) + h t Σ v, g i H − / , H / = hH v, ψ i L ( R ) . 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 := v + Φ[ h ] ∈ dom( H ∗ κ ) and let ψ ∈ C ∞ ( R \ Σ) . Then, there exists U ∈ L ( R ) such that hH ϕ, ψ i D ′ ( R ) , D ( R ) = h v + Φ[ h ] , H ψ i D ′ ( R ) , D ( R ) = h ϕ, H ψ i L ( R ) = h U, ψ i L ( R ) (3.6)Because H Φ[ h ] = 0 in D ′ (Ω ± ) , we get that H v = U in D ′ ( R ) and then in L ( R ) . Using this,it follows that v ∈ H ( R ) . Therefore, we deduce that Φ[ h ] = ϕ − v ∈ L ( R ) . Now, Proposition3.1 (iii) yields that h = i ( α · N )( ˜ C + − ˜ C − )[ h ] ∈ H − / (Σ) . Note that we actually proved that if ϕ := v + Φ[ h ] ∈ dom( H ∗ κ ) , then v ∈ H ( R ) and h ∈ H − / (Σ) . Next, let G ( H ∗ κ ) be the graph of IRAC OPERATORS WITH δ -SHELL INTERACTIONS 11 H ∗ κ , then G ( H ∗ κ ) : = (cid:8) ( ϕ, H ∗ κ ϕ ) : h ϕ, H ∗ κ ψ i L ( R ) = hH ∗ κ ϕ, ψ i L ( R ) , ∀ ψ ∈ dom( H κ ) (cid:9) (3.7) = (cid:8) ( ϕ, H ∗ κ ϕ ) : h Φ[ h ] , H u i L ( R ) = hH v, Φ[ g ] i L ( R ) , ∀ ψ ∈ dom( H κ ) (cid:9) (3.8) = (cid:8) ( ϕ, H ∗ κ ϕ ) : h h, t Σ u i H − / , H / = h t Σ v, g i H / , ∀ ψ ∈ dom( H κ ) (cid:9) (3.9) = n ( ϕ, H ∗ κ ϕ ) : h− ˜Λ + [ h ] , g i H − / , H / = h t Σ v, g i H / , ∀ ψ ∈ dom( H κ ) o . (3.10)Hence, t Σ v = − ˜Λ + [ h ] holds in H − / (Σ) , and then in H / (Σ) . This completes the proof of theproposition. (cid:3) Given z ∈ C \ (( −∞ , − m ] ∪ [ m, ∞ )) , it is well known that the fundamental solution of (∆+ m − z ) I is given by ψ z ( x ) = e i √ z − m | x | π | x | , for x ∈ R . (3.11)Moreover, the trace of the single-layer associated to (∆ + m − z ) I , denoted by S z , has the integralrepresentation S z [ g ]( x ) = Z Σ ψ z ( x − y ) g ( y )dS( y ) , ∀ x ∈ Σ and g ∈ L (Σ) . (3.12)If z = 0 , we simply write S := S .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 − / (Σ) onto H / (Σ) .In particular, if Σ satisfies (H1) , then { β, C a Σ } is a compact operator in L (Σ) .(ii) The anticommutator { α · N , C a Σ } extends to a bounded operator from H − / (Σ) to H / (Σ) .In particular, if Σ satisfies (H1) , then { α · N , C a Σ } is a compact operator in L (Σ) .(iii) If Σ satisfies (H2) , then { α · N , C Σ } is a compact operator in L (Σ) . Proof.
We are going to prove (i) . For that, observe that m − a ) ( m I − aβ ) { β, C a Σ } [ g ]( x ) = S a [ g ]( x ) . (3.13)Hence, the first statement of (i) follows by [36, Theorem 6.11] (see also [37]) for example. Furthermore,if Σ satisfies (H1) , then using that the embedding H / (Σ) ֒ → L (Σ) is compact, we then get that { β, C a Σ } is a compact operator in L (Σ) . This finishes the proof of (i) .Now we prove (ii) . Let x ∈ Σ and y ∈ R , a straightforward computation using the anticommutationrelations of the Dirac matrices yields ( α · N ( x ))( α · y ) = − ( α · y )( α · N ( x )) + 2( N ( x ) · y ) I . (3.14)Use (3.14) to obtain ( α · N ( x )) φ a ( y ) = − φ a ( y )( α · N ( x )) − e −√ m − a | y | iπ | y | (1 + m | y | )( N ( x ) · y ) I + 2 a ( α · N ( x )) ψ a ( y ) . Note that there are constants C and C such that, for all x, y ∈ Σ , it holds that | N ( x ) − N ( y ) | C | x − y | and | N ( x ) · ( x − y ) | C | x − y | , this can proved in similar way as in Proposition 5.7. Using this, for g ∈ L (Σ) , we have { α · N , C a Σ } [ g ]( x ) = Z Σ K a ( x, y ) g ( y ) dS ( y ) + 2 a ( α · N ( x )) S a [ g ]( x ):= T a, [ g ]( x ) + T a, [ g ]( x ) , (3.15)where the kernel K a is given by K a ( x, y ) = φ a ( x − y )( α · ( N ( y ) − N ( x )) − e −√ m − a | x − y | iπ | x − y | (1 + p m − a | x − y | )( N ( x ) · ( x − y )) I . Since Σ is C -smooth, from (i) it follows immediately that T a, is bounded from H − / (Σ) to H / (Σ) . Hence, it remains to prove that T a, is bounded from H − / (Σ) to H / (Σ) . Actually, if Σ satisfies (H1) , then the result follows with the same arguments as [43, Proposition 2.8], where theauthors prove the statement for a = 0 . Now, remark that if Σ satisfies (H2) , then K a ( x, y ) vanishes forall x, y ∈ F and it holds that | K a ( x, y ) | C | x − y | − . Moreover, it holds that T a, = T K a + T K a + ˜ K ∗ ,where ˜ K ∗ is the adjoint of the matrix valued harmonic double-layer defined by (5.17), and the kernels K a and K a are given by K a ( x, y ) = 14 π | x − y | ( α · ( x − y )) ( iα · ( N ( y ) − N ( x )) .K a ( x, y ) = e −√ m − a | x − y | π | x − y | (cid:20) (cid:18) a + mβ + i p m − a (cid:18) α · x − y | x − y | (cid:19)(cid:19) ( α · ( N ( y ) − N ( x ))+ 2 i p m − a ( N ( x ) · ( x − y )) | x − y | I (cid:21) + e −√ m − a | x − y | − π | x − y | ( iα · ( x − y )) ( α · ( N ( y ) − N ( x ))+ e −√ m − a | x − y | − iπ | x − y | ( N ( x ) · ( x − y )) I . Again, one can extend ˜ K ∗ and the integral operator with kernel K a to a bounded operators from H − / (Σ) to H / (Σ) as much the same way as in [43, Proposition 2.8]. Moreover, it is clear that K a is C -smooth and | K a ( x, y ) | = O (1) , when | x − y | tends to zero. Using this, it easily follows thatthe integral operator with kernel K a is bounded from L (Σ) to H (Σ) , and then one can extendit continuously to a bounded operator from H − / (Σ) to H / (Σ) by duality and interpolationarguments. The seconde statement is a direct consequence of the Sobolev injection, and this completesthe proof of (ii) .Now we turn to the proof of (iii) . Assume that Σ satisfies (H2) , let us prove that { α · N , C Σ } iscompact on L (Σ) . From (ii) , since a = 0 we have that { α · N , C Σ } coincides with T , which isgiven by (3.15). Let χ be a C ∞ (Σ) cutoff function vanishing out-side the deformation F . Using that K ( x, y ) vanishes for all x, y ∈ Σ , we then obtain that T , = χT , χ + χT , (1 − χ ) + (1 − χ ) T , χ. (3.16)Hence, the claimed result follows from (ii) and the compactness of the Sobolev embedding χ H / (Σ) ֒ → L (Σ) . This finishes the proof of the lemma. (cid:3) Remark 3.1.
Actually the above result is not surprising since the kernels associated to the anticom-mutators { α · N , C a Σ } and { β, C a Σ } behave locally like | x − y | − , when | x − y | tends to zero. Therefore,the operators in consideration are bounded from L (Σ) dans H (Σ) because Σ is C -smooth. We are now in position to prove Theorem 3.1.
IRAC OPERATORS WITH δ -SHELL INTERACTIONS 13 Proof of Theorem 3.1 (i)
Assume that sgn( κ ) = 4 . From the definition of ˜Λ a ± , a simplecomputation using Lemma 2.1 (ii) gives ˜Λ a ± ˜Λ a ∓ = 1sgn( κ ) − ( ˜ C a Σ ) + µ sng( κ ) { β, ˜ C a Σ } + η sgn( κ ) { α · N , ˜ C a Σ } = 1sgn( κ ) − − 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)Let g ∈ H − / (Σ) such that ˜Λ + [ g ] ∈ H / (Σ) . From (3.17), we have g = 4(sgn( κ ))4 − sng( κ ) (cid:18) 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 } (cid:19) [ g ] . Using Lemma 3.1, it follows that g ∈ H / (Σ) . Hence, given any ϕ = u + Φ[ g ] ∈ dom( H ∗ κ ) , since g ∈ H − / (Σ) and t Σ u = ˜Λ + [ g ] in H / (Σ) , we deduce that g ∈ H / (Σ) . Thus, dom( H ∗ κ ) = dom( H κ ) and it holds that dom( H κ ) = n u + Φ[ g ] : u ∈ H ( R ) , g ∈ H / (Σ) , t Σ u = − Λ + [ g ] o . (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 / (Σ) be a sequence of functions that converges to g in H − / (Σ) . Set g j := g + 2 ǫ ˜Λ − [ h j − g ] , ∀ j ∈ N . (3.19)Then ( g j ) j ∈ N , (Λ + [ g j ]) j ∈ N ⊂ H / (Σ) , and it holds that g j −−−→ j →∞ g in H − / (Σ) , Λ + [ g j ] −−−→ j →∞ ˜Λ + [ g ] , in H / (Σ) . (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 − / (Σ) to H / (Σ) by Lemma3.1 and (3.17). Now let ( v j ) j ∈ N ⊂ H ( R ) such that t Σ v j = 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 ( R ) and t Σ u j = − Λ + [ g j ] in H / (Σ) , hence ( ϕ j ) j ∈ N ⊂ dom( H κ ) . Moreover, since ( h j ) j ∈ N (respectively ( g j ) j ∈ N ) converges to g in H − / (Σ) as j −→ ∞ , using the continuity of ˜Λ ± ˜Λ ∓ it follows that ( ϕ j , H κ ϕ j ) −−−→ j →∞ ( ϕ, H ∗ κ ϕ ) in L ( R ) . Therefore H ∗ κ ⊂ H κ and the Theorem is proved. (cid:3) 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 (Ω ± ) . Now, wedefine δ Σ ϕ as the distribution h δ Σ ϕ, ψ i D ′ ( R ) , D ( R ) := 12 Z Σ h t Σ ϕ + + t Σ ϕ − , ψ i C dS( x ) , for all ψ ∈ D ( R ) . (3.21)Therefore, a simple computation in the sens of distributions yields ( H + ( ǫI + µβ + η ( α · N )) δ Σ ) ϕ =( − iα · ∇ + mβ ) ϕ + 12 ( ǫI + 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 + 12 ( ǫI + µβ + η ( α · N ))( t Σ ϕ + + t Σ ϕ − ) δ Σ . Using the Plemelj-Sokhotski formula, it easily follows that
12 ( ǫI + 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 − / (Σ) . Since ( − iα · ∇ + mβ ) ϕ + ( − iα · ∇ + mβ ) ϕ − ∈ L ( R ) , given ϕ = ( ϕ + , ϕ − ) ∈ L ( R ) such that ( α · ∇ ) ϕ ± ∈ L (Ω ± ) and satisfying (3.22), it holds that H κ ϕ ∈ L ( R ) . Inparticular, this leads to the following definition: Definition 3.1.
Given κ = ( ǫ, µ, η ) ∈ R such that sgn( κ ) = 0 and m > . The self-adjoint Diracoperator coupled with a combination of electrostatic, Lorentz scalar and normal vector field δ -shellinteractions of strength ǫ , µ and η respectively, is the operator H κ defined on the domain dom( H κ ) = (cid:8) ϕ = ( ϕ + , ϕ − ) ∈ L (Ω + ) ⊕ L (Ω − ) :( α · ∇ ) ϕ ± ∈ L (Ω ± ) and (3.22) holds in H − / (Σ) (cid:9) , (3.23) and it acts in the sens of distributions as H κ ( ϕ ) = ( H ϕ + ) ⊕ ( H ϕ − ) , for all ϕ ∈ dom( H κ ) . Remark 3.2.
Assume that sgn( κ ) = 0 , . Since the operator Φ is bounded from H / (Σ) to H ( R \ Σ) , it holds that ϕ ± := ϕ | Ω ± ∈ H (Ω ± ) . Moreover, following the same arguments as above, weconclude that the transmission condition (3.22) holds actually in H / (Σ) . Therefore, It follows that dom( H κ ) = n ϕ = ( ϕ + , ϕ − ) ∈ H (Ω + ) ⊕ H (Ω − ) : (3.22) holds in H / (Σ) o . (3.24)Let us make some comments on the technique developed here. As we have mentioned in theintroduction, the condition on Σ of being C -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 extensionof the operator Λ ± Λ ∓ from H − / (Σ) to H / (Σ) , or equivalently, the continuous extension ofthe anticommutators { β, C Σ } and { α · N , C Σ } . Since { β, C Σ } involves the trace of the single-layerpotential, we can always extend it to a bounded operator from H − / (Σ) to H / (Σ) , even if Σ isLipschitz. However, { α · N , C Σ } involves the principal value of the double-layer potential (or its adjointoperator), and it is well known that the C regularity is minimal to extend it to a continuous operatorfrom L (Σ) to H (Σ) . However, as we will see later, if sgn( κ ) = 0 and Σ is C ,γ -smooth and compactfor some γ ∈ (1 / , , then we can manage to prove the self-adjointness of H κ using the techniquedeveloped in this part, see Remark 5.5 for details.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 denoteit by H ǫ,µ . This operator is well known as the Dirac operator with electrostatic and Lorentz scalar δ -shell interactions, cf. [3],[7],[10]. If | ǫ | 6 = | µ | , from Theorem 3.1 we get immediately the followingresult. Proposition 3.4.
Given ǫ, µ ∈ R \ { } such that | ǫ | 6 = | µ | and define the operators Λ ± as follows Λ ± = 1 ǫ − µ ( ǫI ∓ µβ ) ± C Σ . (3.25) Then, the following hold:(i) If ǫ − µ = 4 , then H ǫ,µ is self-adjoint and we have dom( H ǫ,µ ) = n u + Φ[ g ] : u ∈ H ( R ) , g ∈ H / (Σ) , t Σ u = − Λ + [ g ] o . (3.26) (ii) If ǫ − µ = 4 , then H ǫ,µ is essentially self-adjoint and we have dom( H ǫ,µ ) = n u + Φ[ g ] : u ∈ H ( R ) , g ∈ H − / (Σ) , t Σ u = − ˜Λ + [ g ] o . IRAC OPERATORS WITH δ -SHELL INTERACTIONS 15 Now we turn to the special case ǫ = ± µ . Set P ± = ( I ± β ) / , then H ǫ, ± ǫ is given formally by H ǫ, ± ǫ = H + P ± V ǫ, ± ǫ = − iα · ∇ + mβ + 2 ǫP ± δ Σ . (3.27)Define Λ + = P ± (1 / ǫ + C Σ ) P ± and Λ − = P ± (1 / ǫ − C Σ ) P ± , ∀ ǫ = 0 . (3.28)Clearly , Λ ± are bounded and self-adjoint from P ± L (Σ) onto itself (respectively from P ± H / (Σ) onto itself). In order to define rigorously H ǫ, ± ǫ as in Definition 2.1, that is H ǫ, ± ǫ ϕ = H u in the senseof distributions for ϕ = u + Φ[ g ] , with u ∈ H ( R ) and g ∈ H / (Σ) . We shall take g ∈ P ± H / (Σ) and assume the condition P ± t Σ u = − P ± Λ + [ g ] . Indeed, if we set dom( H ǫ, ± ǫ ) = (cid:8) u + Φ[ g ] : u ∈ H ( R ) , g ∈ P ± L (Σ) and P ± t Σ u = − P ± Λ + [ g ] (cid:9) , (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 ) = n u + Φ[ g ] : u ∈ H ( R ) , g ∈ P ± H − / (Σ) , P ± t Σ u = − P ± ˜Λ + [ g ] o . (3.30)where ˜Λ ± denotes the bounded extension of Λ ± from P ± H − / (Σ) onto itself, and we get the anal-ogous 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 ) = n u + Φ[ g ] : u ∈ H ( R ) , g ∈ P ± H / (Σ) , P ± t Σ u = − P ± Λ + [ g ] o . (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 ± , asimple computation yields ˜Λ − ˜Λ + = 14 ǫ P ± − ǫ P ± ˜ C Σ P ± ˜ C Σ P ± = 14 ǫ P ± − m ǫ ( S ) P ± . (3.32)where S is given by (3.12). Recall that ˜Λ − ˜Λ + and SP ± are bounded from P ± H − / (Σ) into P ± H / (Σ) . Thus, from (3.32) it follows that, if g ∈ P ± H − / (Σ) such that ˜Λ + [ g ] ∈ P ± H / (Σ) ,then g ∈ P ± H / (Σ) . Which yields that dom( H ǫ, ± ǫ ) = dom( H ∗ ǫ, ± ǫ ) and the proposition is proved. (cid:3) The operators Λ a ± . Let a ∈ ( − m, m ) and let Λ a ± be as in the Notation 2.1. From the proofof Theorem 3.1, It is evident that the study of the self-adjointness character of H κ is related to theproperties 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 (here we assume that c is big enough if Σ satisfies (H2) , so that c is not in the spectrum of ∆ Σ ). Itis well known that L ± / is a bijective operator from H ± / (Σ) onto L (Σ) . Hence, one can writethe domain of H κ as follows: dom( H κ ) = n u + Φ L / [ g ] : u ∈ H ( R ) , g ∈ H / (Σ) and L / t Σ u = − L / Λ + L / [ g ] o , (3.33)which leads us to define the following operators L a ± := L / Λ ± L / with dom( L a ± ) = n g ∈ H / (Σ) : Λ a ± L / [ g ] ∈ H / (Σ) o . (3.34) Lemma 3.2.
Let κ ∈ R 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 (Σ) .(ii) If sgn( κ ) = 4 , then L a ± is essentially self-adjoint and we have dom( L a ± ) = n g ∈ L (Σ) : ˜Λ a ± L / [ g ] ∈ H / (Σ) o . (3.35) Proof.
Since L / and C a Σ are self-adjoint, it follows that L a ± is symmetric. Moreover, we have C ∞ (Σ) ⊂ dom( L a ± ) ⊂ L (Σ) , which yields that dom( L a ± ) is a dense subspace of L (Σ) , therefore L a ± is closable. Let h ∈ dom( L a ∗± ) and let g ∈ dom( L a ± ) , then there is f ∈ L (Σ) such that h h, L a ± [ g ] i L = h L / h, Λ a ± L / [ g ] i H − / , H / = h ˜Λ a ± L / [ h ] , L / [ g ] i D ′ , D = h L − / [ f ] , g i L . Thus, ˜Λ a ± L / [ h ] = L − / [ f ] in D ′ (Σ) , and therefore ˜Λ a ± L / [ h ] ∈ H / (Σ) . Hence we get theinclusion dom( L a ∗± ) ⊂ n g ∈ L (Σ) : ˜Λ a ± L / [ g ] ∈ H / (Σ) o . Now, one can check easily the other inclusion and we thus get the equality. Therefore, item (i) is animmediate 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 ˜Λ a ± and ˜Λ a ∓ ) and use the fact that ˜Λ a ± ˜Λ a ∓ are continuous from H − / (Σ) to H / (Σ) , we omit thedetails. This finishes the proof of the lemma. (cid:3) 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 studythe self-adjoint extension of H κ from the point of view of the boundary triples theory (see [8] and[10], for a more general view of the theory we refer to [11] and [13] for example). Indeed, denote by S := H ⇂ H ( R \ Σ) and let T := H with dom( T ) = { u + Φ[ g ] : u ∈ H ( R ) , g ∈ L (Σ) } , and define the linear mappings Γ , Γ : dom( T ) −→ L (Σ) by Γ ( ϕ ) = g and Γ ( ϕ ) = t Σ u + C Σ [ g ] , (3.37)Then, { L (Σ) , Γ , Γ } is a quasi-boundary triples for T = S ∗ (adapt the arguments of [8] or [10]).Moreover, if we denote by ˜Γ the extension of Γ , that is ˜Γ : T −→ H − / (Σ) , we then get that { L (Σ) , L − / ˜Γ , L / Γ } is an ordinary boundary triple for T = S ∗ . Now it is easy to check that H κ = T ⇂ Kr(( ǫ I + µβ + η ( α · N ))Γ + Γ ) = T ⇂ Kr(Γ − C Σ Γ + Λ + Γ ) . (3.38)Thus, after transforming the quasi-boundary triples to an ordinary boundary triples (see [8, Theorem4.5] 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 [8, Corollary 2.8].4. Spectral properties
In this section, we examine the spectral properties of the operator H κ . First, we give a necessarycondition for the existence of the points spectrum in the gap ( − m, m ) and a Krein-type resolventformula. More precisely, we have the following. Proposition 4.1.
Let H κ be as in the definition 2.1. The following hold:(i) Given a ∈ ( − m, m ) , then Kr( H κ − a ) = 0 ⇐⇒ Kr(˜Λ a + ) = 0 (Birman-Schwinger principle).(ii) For all z ∈ C \ R the operator ˜Λ z + is bounded invertible from H − / (Σ) to H / (Σ) and wehave ( H κ − z ) − = ( H − z ) − − Φ z (˜Λ z + ) − (Φ z ) ∗ . (4.1) IRAC OPERATORS WITH δ -SHELL INTERACTIONS 17 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 H u = aϕ = a ( u + Φ[ g ]) . (4.2)From this we deduce that ( H − a ) H u = ag holds in D ′ ( R ) , and therefore H u = a ˜Φ a [ g ] , (4.3)it is clear that if a = 0 , then u = 0 which yields Kr(˜Λ a + ) = 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 − ˜ C Σ )[ g ] = − ˜Λ + [ g ] , therefore Kr(˜Λ a + ) = 0 . We turnnow to prove the implication ( ⇐ ) , so fix a ∈ ( − m, m ) such that ˜Λ a + [ g ] = 0 for some g ∈ H − / (Σ) .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 = a H − Φ a [ g ] ∈ H ( R ) and set ϕ = u + Φ[ g ] . Then H u = a Φ a [ g ] and ( H − a ) u = a Φ[ g ] in D ′ ( R ) , this amounts to saying that H κ ϕ = H u = 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 − ˜ C Σ )[ g ] = − ˜Λ + [ g ] . Thus ϕ ∈ dom( H κ ) and H κ ϕ = aϕ , which yields Kr( H κ − a ) = 0 . This ends the proof of (i) . (ii) Fix z ∈ C \ R . Since H κ is self-adjoint it follows that ( H κ − z ) − is well defined and bounded.Moreover, from (i) it follows that Kr(˜Λ z + ) = ∅ , as otherwise z would be a non-real eigenvalue of H κ .Let u ∈ L ( R ) and set ϕ := ( H κ − z ) − u ∈ dom( H κ ) . Taking into account the decomposition dom( H κ ) = dom( H ) + Kr( H λ,µ − z ) and the fact that ˜Φ z : H − / (Σ) −→ Kr( H κ − z ) is a boundedbijective operator (this can be proved as much as the same way as see behrendt 2018), we deduce thatthere is a unique functions v ∈ H ( R ) and g ∈ H − / (Σ) such that ϕ = v +Φ z [ g ] . Moreover one has ( H κ − z ) ϕ = ( H − z ) v , and thus v = ( H − z ) − u , which means actually that ϕ = ( H − z ) − u + Φ z [ g ] .Next, remark that iα · N ( t Σ ϕ + − t Σ ϕ − ) = g and
12 ( t Σ ϕ + + t Σ ϕ − ) = ( H − z ) − u ⇂ Σ + ˜ C z Σ [ g ] . (4.4)Using that ( H − z ) − u ⇂ Σ = (Φ z ) ∗ u and the transmission condition (3.22), we obtain that ˜Λ z + [ g ] =(Φ z ) ∗ u ∈ H / (Σ) . Since this is true for all u ∈ L ( R ) , therefore Rn(˜Λ z + ) = H / (Σ) . Hence ˜Λ z + : H − / (Σ) −→ H / (Σ) is a bounded bijective operator. Summing up, we have proved that ∈ ρ (˜Λ z + ) and ( H κ − z ) − u = ( H − z ) − u + Φ z [(˜Λ z + ) − (Φ z ) ∗ u ] holds for all u ∈ L ( R ) , which provesthe identity (4.1). (cid:3) Remark 4.1.
A careful inspection of the argument used above reveals that a ∈ ( − m, m ) is an isolatedpoint of Sp( H κ ) if and only if is an isolated point of Sp(˜Λ a + ) , and dimKr( H κ ) = dimKr(˜Λ a + ) .Furthermore, item (ii) holds true for all z ∈ ρ ( H κ ) ∩ ρ ( H ) . As 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:(i) For all a ∈ ( − m, m ) , one has a ∈ Sp p ( H κ ) ⇐⇒ ∈ Sp p ( L a + ) ,a ∈ Sp ess ( H κ ) ⇐⇒ ∈ Sp ess ( L a + ) . (ii) For all z ∈ ρ ( H κ ) ∩ ρ ( H ) , the operator L z + is bounded invertible from L (Σ) to L (Σ) andwe have ( H κ − z ) − = ( H − z ) − − Φ z L ( L z + ) − L (Φ z ) ∗ . (4.5) In order to avoid repetitions, we focus namely in the remainder of this section on the spectralproperties of H κ when Σ satisfies the assumption (H2) . When Σ is compact we prove them in generalframework, see Section 5 for more details. Notation 4.1.
For all ν > , we denote by H νκ (respectively Φ zν , ˜Λ z + ,ν and (Φ z ν ) ∗ ) the operator H κ (respectively Φ z , ˜Λ z + and (Φ z ) ∗ ) whenever Σ = Σ ν , i.e Σ satisfies (H2) , and we write H k (respectively Φ z , ˜Λ z + and (Φ z ) ∗ ) instead of H κ (respectively Φ z , ˜Λ z + , and (Φ z ) ∗ ). Non-critical case.
This part deal with the basic spectral properties of H κ in the non-criticalcase. The results are mainly known for the Dirac operator coupled with a combination of electrostaticand Lorentz scalar δ -interactions (i.e η = 0 ) supported on a closed, bounded and sufficiently smoothsurface (see [7] for example ), and they are still true when Σ satisfies (H2) and η = 0 . Theorem 4.1.
Let κ ∈ R such that sgn( κ ) = 0 , and suppose that Σ satisfies (H2) . Then thefollowing hold:(i) Sp ess ( H νκ ) = ( −∞ , − m ] ∪ [ m, + ∞ ) . In particular, we have Sp( H κ ) = Sp ess ( H κ ) .(ii) Sp disc ( H νκ ) ∩ ( − m, m ) is finite, for all ν > . When ν = 0 , Theorem 4.1 gives us a simple way to construct functions of dom( H κ ) . Indeed, since / ∈ Sp( H κ ) it follows that Λ + is invertible and we get then dom( H κ ) = (cid:8) u + Φ[ − Λ − [ t Σ u ]] : u ∈ H ( R ) (cid:9) . Proof. (i)
First, we show the result for ν = 0 , the case ν > is an immediate consequence ofthe Proposition 4.1. Fix a ∈ ( − m, m ) and set Γ ± m, ± a ( ξ ) = [ α · ( ξ , ξ , ± mβ ± a ] . Since the α j ’santicommute with β , by a simple computation we get the following properties Γ m,a ( ξ ) = | ξ | + m − a + 2 a Γ m,a ( ξ ) , Γ − m, − a ( ξ )Γ m,a ( ξ ) = | ξ | + m − a − mβ Γ m,a ( ξ ) , Γ m, − a ( ξ )Γ m,a ( ξ ) = | ξ | + m − a . (4.6)Using the Fourier-Plancherel operator, it is not hard to prove that Λ a + is unitary equivalent to thefollowing multiplication operator: Π a + := 1sgn( κ ) ( ǫI − µβ − η ( α · N )) + 12 p | ξ | + m − a Γ m,a ( ξ ) . (4.7)Moreover, take into account the properties (4.6), a simple computation shows that Π a + is invertibleand it’s inverse is given explicitly by (Π a + ) − = C − ǫa + µm p | ξ | + m − a − ( ǫ + µβ + η ( α · N ))2 p | ξ | + m − a Γ m,a ( ξ ) ! ( ǫ + µβ + η ( α · N )) , (4.8)where C = 4 − sgn( κ )4 + ǫa + µm p | ξ | + m − a . 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 −→ C ( x, x ) (cid:18) ξ − iξ a − m , , , (cid:19) t e ix · ξ , IRAC OPERATORS WITH δ -SHELL INTERACTIONS 19 here ξ = ( ξ , ξ ) and | ξ | = a − m . Observe that we have ( − iα · ∇ + mβ − a ) ϕ = 0 . Let R > , χ ∈ C ∞ ( R ) and θ ∈ C ∞ ([0 , ∞ [) such that θ = ( for R > | x | > R., for R | x | R/ . For n ∈ N ⋆ , we define the sequences of functions ϕ + ,n ( x, x ) = ϕ ( x, x ) χ ( x/n ) θ ( x /n ) for x > ,ϕ − ,n ( x, x ) = ϕ ( x, x ) χ ( x/n ) θ ( − x /n ) for x < . (4.9)It’s clear that ϕ ± ,n ∈ H (Ω ± ) and t Σ ϕ ± ,n = 0 , thus ϕ n := ( ϕ + ,n , ϕ − ,n ) ∈ dom( H κ ) . Moreover, wehave k ϕ n k L ( R ) = k ϕ + ,n k L (Ω + ) + k ϕ − ,n k L (Ω − ) = n aa − m k χ k L ( R ) k θ k L ( R + ) and k ( − iα · ∇ + mβ − a ) ϕ n k L = k ( − iα · ∇ + mβ − a ) ϕ + ,n k L (Ω + ) + k ( − iα · ∇ + mβ − a ) ϕ − ,n k L (Ω − ) n aa − m (cid:0) k∇ η k L ( R ) k θ k L ( R + ) + k χ k L ( R ) k θ ′ k L ( R + ) (cid:1) Thus, we get k ( − iα · ∇ + mβ − a ) ϕ n k L ( R ) k ϕ n k L ( R ) −−−−→ n →∞ . Summing up, we have proved ( −∞ , − m ) ∪ ( m, ∞ ) ⊂ Sp( H κ ) ⊂ ( −∞ , − m ] ∪ [ m, ∞ ) . Since thespectrum 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 essentialbecause (non-degenerate) intervals have no isolated points, this yields Sp( H κ ) = Sp ess ( H κ ) , for ν = 0 .Now assume that ν > , let’s determine the essential spectrum of H νκ . For that, recall that F is theflat part of Σ ν given by (2.1); fix z ∈ C \ R and let T : L ( R ) → L ( R ) be the bounded operatordefined by T = Φ zν (Λ z + ,ν ) − (Φ zν ) ∗ − Φ z (Λ z + ) − (Φ z ) ∗ . (4.10)Then T is a compact operator in L ( R ) . Indeed, observe that T can be written as follows: T = (cid:0) Φ zν (Λ z + ,ν ) − − Φ z (Λ z + ) − (cid:1) ( H − z ) − ⇂ F +Φ zν (Λ z + ,ν ) − ( H − z ) − ⇂ Σ ν \ F − Φ z (Λ z + ) − ( H − z ) − ⇂ Σ \ F := T + T + T . Since Σ ν \ F is compact for all ν > , it follows that the injection H / (Σ ν \ F ) ֒ → L (Σ ν ) iscompact. Using this and the fact that ( H − z ) − ⇂ Σ ν \ F is bounded from L ( R ) to H / (Σ ν \ F ) , itholds that ( H − z ) − ⇂ Σ ν \ F is a compact operator from L ( R ) to L (Σ) . As Φ zν (Λ z + ,ν ) − is boundedfrom L (Σ) to L ( R ) , we get therefore that T and T are compact operators on L ( R ) . Now, let χ be a C ∞ (Σ ν ) cutoff function vanishing out-side the deformation F , then we write T as follows T = (cid:0) Φ zν χ (Λ z + ,ν ) − − Φ z χ (Λ z + ) − (cid:1) ( H − z ) − ⇂ F + (cid:0) Φ zν (1 − χ )(Λ z + ,ν ) − − Φ z (1 − χ )(Λ z + ) − (cid:1) ( H − z ) − ⇂ F := T + T . From the definition of Φ zν , it is clear that T = 0 . Moreover, as (Λ z + ,ν ) − ( H − z ) − ⇂ F is boundedfrom L ( R ) to H / (Σ ν ) , using again the compactness of the Sobolev embedding, we obtain that T is also a compact operator on L ( R ) . Therefore T is a compact operator in L ( R ) . Now, using Proposition 4.1 (ii) it follows that T = ( H νκ − z ) − − ( H κ − z ) − . Therefore, by Weyl’s theorem weconclude that H νκ 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 [30] and [7]. Fix ν > and let Q be the quadraticform associated to ( H νκ ) with domain dom( H νκ ) , that is Q [ ϕ ] = k ( α · ∇ ) ϕ + k L (Ω + ) + k ( α · ∇ ) ϕ − k L (Ω − ) + m k ϕ k L ( R ) + h ( − iα · N ) t Σ ν ϕ + , mβt Σ ν ϕ + i L (Σ ν ) − h ( − iα · N ) t Σ ν ϕ − , mβt Σ ν ϕ − i L (Σ ν ) . Given
R > such that (Σ ν \ F ) ⊂ B (0 , R ) , and define the closed and semi-bounded sesquilinear forms Q ext [ ϕ ] = k ( α · ∇ ) ϕ + k L (Ω + \ B (0 ,R )) + k ( α · ∇ ) ϕ − k L (Ω − \ B (0 ,R )) + m k ϕ k L ( R \ B (0 ,R )) + h ( − iα · N ) t Σ ν ϕ + , mβt Σ ν ϕ + i L ( F ) − h ( − iα · N ) t Σ ν ϕ − , mβt Σ ν ϕ − i L ( F ) . with domain dom( Q ext ) = (cid:8) ϕ = ( ϕ + , ϕ − ) ∈ H (Ω + \ B (0 , R )) ⊕ H (Ω − \ B (0 , R )) : t Σ ν ϕ = Θ λ,µ t Σ ν ϕ − (cid:9) , and Q int [ ϕ ] = k ( α · ∇ ) ϕ + k L (Ω + ∩ B (0 ,R )) + k ( α · ∇ ) ϕ − k L (Ω − ∩ B (0 ,R )) + m k ϕ k L ( B (0 ,R )) + h ( − iα · N ) t Σ ν ϕ + , mβt Σ ν ϕ + i L (Σ ν ∩ B (0 ,R )) − h ( − iα · N ) t Σ ν ϕ − , mβt Σ ν ϕ − i L (Σ ν ∩ B (0 ,R )) . with domain dom( Q int ) = (cid:8) ϕ = ( ϕ + , ϕ − ) ∈ H (Ω + ∩ B (0 , R )) ⊕ H (Ω − ∩ B (0 , R )) : t Σ ν ϕ = Θ λ t Σ ν ϕ − (cid:9) , Set ˜ Q = Q ext ⊕ Q int . Then, dom( Q ) ⊂ dom( ˜ Q ext ) and ˜ Q ext [ ϕ ] Q [ ϕ ] , hold for all ϕ ∈ dom( Q ) . Sincethe injection H (Ω + ∩ B (0 , R )) ⊕ H (Ω − ∩ B (0 , R )) ֒ → L ( B (0 , R )) is compact, we conclude that theresolvent of the operator H int associated to Q int is compact. Hence, H int has a finite purely discretespectrum in the gap ( − m, m ) . Moreover, from (i) we deduce that inf Sp( Q ext ) = inf Sp ess ( Q ext ) > m .Therefore, the min-max principle gives us inf Sp ess ( Q ) > inf Sp ess ( Q ext ) > m and { Sp disc ( H νλ ) ∩ ( − m, m ) } < ∞ . (4.11)Thus, Sp disc ( H νκ ) ∩ ( − m, m ) is finite and the theorem is finally proved. (cid:3) Critical case.
From now, we assume that sgn( κ ) = 4 . The goal of this subsection is to provethe following result. Theorem 4.2.
Given κ = ( ǫ, µ, η ) ∈ R such that sgn( κ ) = 4 and let H κ be as in Theorem 3.1. If Σ satisfies (H2) , then for all ν > it holds that Sp ess ( H νκ ) = (cid:0) − ∞ , − m (cid:3) ∪ n − mµǫ o ∪ (cid:2) m, + ∞ (cid:1) . (4.12) In particular, we have the equality
Sp( H κ ) = Sp ess ( H κ ) . A few comments are in order. Actually, one can imagine that the operator H κ is unitary equivalentto H ǫ ,µ , for some ǫ , µ ∈ R , such that ǫ − µ = 4 and ǫ /ǫ = µ /µ . Indeed, in [35] and [14] it hasbeen shown that the potential η ( α · N ) δ Σ can always be absorbed as a change of gauge. So the existenceof such a unitary transformation is not excluded. Another way to understand Theorem 4.2 comes fromthe 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 + 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 IRAC OPERATORS WITH δ -SHELL INTERACTIONS 21 ( ǫI + µβ ) δ Σ . The reason is very simple. Indeed, for all η ∈ R , the operator H + η ( α · N ) δ Σ is self-adjointeven if Σ is Lipschitz, cf. Proposition 5.5. Moreover, Sp( H + η ( α · N ) δ Σ ) = (cid:0) − ∞ , − m (cid:3) ∪ (cid:2) m, + ∞ (cid:1) .As in the non-critical case, from Theorem 4.2 we get a simple way to describe functions belongingto the domain of H κ when Σ = R × { } . Indeed, we have the following result. Corollary 4.2.
Assume that
Σ := R × { } and let H κ be as above. The following hold:(i) If µ = 0 , then dom( H κ ) = n u + Φ[ − ˜Λ − [ t Σ u ]] : u ∈ H ( R ) o . (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. (cid:3)
The main properties of the operators L a ± which are relevant for us to prove Theorem 4.2 are collectedin the following proposition. Proposition 4.2.
Let κ = ( ǫ, µ, η ) ∈ R such that sgn( κ ) = 0 and let L a ± ,κ := L a ± be as in Lemma3.2. Then, for all a ∈ ( − m, m ) , it holds that ∈ Sp( L a + ,κ ) ⇐⇒ ∈ Sp( L − a + , ˜ κ ) ⇐⇒ ∈ 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 such that sgn( κ ) = 0 . Following [7, Proposition 4.2], for f ∈ L (Σ) we define C ( f ) = iβα f c , T ( f ) = γ βf, γ := − iα α α = I I ! , (4.14)where f c is the the complex conjugate of f . Remark that α = − α , γ β = − βγ and γ ( α · x ) =( α · x ) γ , for all x ∈ R . Using this, it easily follows that C ( f ) = f and T ( f ) = − f . Moreover, asimple 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 ∈ Sp( L a + ) . Then, there exists a sequence of functions ( g j ) j ∈ N ⊂ dom( L a + ) ⊂ L (Σ) , such that || g j || L (Σ) = 1 and (cid:12)(cid:12)(cid:12)(cid:12) L a + ,κ g j (cid:12)(cid:12)(cid:12)(cid:12) L (Σ) −−−→ j →∞ . Hence, if we set f j = C ( g j ) and h j = T ( g j ) , then it is clear that || h j || L (Σ) = || f j || L (Σ) = 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 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L − a + , ˜ κ [ f j ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L (Σ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L − a − ,κ [ h j ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L (Σ) = (cid:12)(cid:12)(cid:12)(cid:12) L a + ,κ [ g j ] (cid:12)(cid:12)(cid:12)(cid:12) L (Σ) . Therefore ∈ Sp( L − a + , ˜ κ ) and ∈ Sp( L − a − ,κ ) . The reverse implications follow in the same way and thisproves the first statement. The last statement is a direct consequence of the first one and Corollary4.1. This completes the proof. (cid:3) Proposition 4.3.
Let a ∈ ( − m, m ) and let L a ± be as in Lemma 3.2. Assume that ν = 0 , then it holdsthat ∈ Sp( L a + ) ⇐⇒ a = − mµǫ and ∈ Sp( L a − ) ⇐⇒ a = mµǫ . (4.16) Moreover, is an isolated eigenvalue of L − mµ/ǫ + and L mµ/ǫ − with infinite multiplicity. Proof.
Given a ∈ ( − m, m ) , once the claimed statement is shown for L a + , by Proposition 4.2 we getthe result for L a − . As in Theorem 4.1, on the Fourier side, one can check that L a + is unitary equivalentto the following multiplication operator: f Π a + := h ξ i κ ) ( ǫI − ( µβ + η ( α · N ))) + 12 p | ξ | + m − a Γ m,a ( ξ ) ! . (4.17)Since sgn( κ ) = 4 , from (4.8) it follows that f Π a + is invertible for all a = − mµ/λ , and we have ( f Π a + ) − = 1 h ξ i p | ξ | + m − a ǫa + µm − ( ǫ + ( µβ + η ( α · N )))2( ǫa + µm ) Γ m,a ( ξ ) ! ( ǫ + ( µβ + η ( α · N ))) . Furthermore it holds that h ξ i f Π a + − ( λ + µβ + η ( α · N ))2 p | ξ | + m − a Γ m,a ( ξ ) ! = 0 , for a = − mµǫ . From this, it follows that is an eigenvalue of the operators L − mµ/ǫ + with infinite multiplicity, andthereby ∈ Sp ess ( L − mµ/ǫ + ) . Thus, we conclude that ∈ Sp( L a + ) if and only if aǫ = − mµ . Now weturn 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( f Π a + − θ ) = (cid:20) θ (cid:18) θ − h ξ i (cid:18) a p | ξ | + m − a + ǫ (cid:19)| {z } θ (cid:19)(cid:21) , (4.18)where det( f Π a + − θ ) is the determinant of ( f Π a + − θ ) . By studying the variations of the non-trivial root θ , we obtain that Sp( L − mµ/ǫ + ) = { } ∪ θ ([0 , ∞ )) = { } ∪ " ǫ − µ p ǫ − µ , ∞ if ǫ > , Sp( L − mµ/ǫ + ) = θ ([0 , ∞ )) ∪ { } = " −∞ , ǫ µ p ǫ − µ ∪ { } if ǫ < . Take into account that sgn( κ ) = 4 , we then get that is an isolated eigenvalue of L − mµ/ǫ + with infinitemultiplicity, which completes the proof of the proposition. (cid:3) Remark 4.2.
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 (Σ) with closed range. 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 > . The result will follow fromthe following statements:(a) (cid:0) − ∞ , − m (cid:1) ∪ (cid:0) m, + ∞ (cid:1) ⊂ Sp ess ( H νκ ) .(b) {− mµ/ǫ } ∈ Sp ess ( H νκ ) and { mµ/ǫ } / ∈ Sp ess ( H νκ ) .(c) Sp ess ( H νκ ) ∩ [( − 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 sequenceof 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 νκ and a . Therefore we get the inclusion ( −∞ , − m ) ∪ ( m, + ∞ ) ⊂ Sp ess ( H νκ ) ,which yields (a) .Now, we turn to the proof of (b) . Actually from Proposition 4.3 and Corollary 4.1, we knowthat item (b) holds true for ν = 0 . Since Kr(˜Λ − mµ/ǫ + ,ν ) ∩ Kr(˜Λ − mµ/ǫ − ,ν ) = { } holds, it sufficient to IRAC OPERATORS WITH δ -SHELL INTERACTIONS 23 prove that (cid:8) − mµǫ (cid:9) ∈ Sp ess ( H νκ ) . So fix ν > and assume that (cid:8) − mµǫ (cid:9) / ∈ Sp ess ( H νκ ) . Then, byCorollary 4.1 and Proposition 4.2 it follows that / ∈ Sp ess ( L − mµ/ǫ + ,ν ) and / ∈ Sp ess ( L mµ/ǫ − ,ν ) . Now we set B ν := ˜Λ − mµ/ǫ + ,ν ˜Λ mµ/ǫ − ,ν , D ν := Λ mµ/ǫ − ,ν Λ − mµ/ǫ + ,ν and we consider the operator Υ ν : L (Σ ν ) −→ L (Σ ν ) defined by: Υ − mµ/ǫν := L ν D ν B ν L ν = L ν (Λ mµ/ǫ − ,ν Λ − mµ/ǫ + ,ν )(˜Λ − mµ/ǫ + ,ν ˜Λ mµ/ǫ − ,ν ) L ν . (4.19)Observe that B ν = C − mµ/ǫ Σ ( α · N ) { α · N , ˜ C − mµ/ǫ Σ } − mµǫ C − mµ/ǫ Σ S − mµ/ǫ + mµη ǫ ( α · N ) S − mµ/ǫ + η { α · N , ˜ C − mµ/ǫ Σ } , (4.20)and D ν = C − mµ/ǫ Σ ( α · N ) { α · N , C − mµ/ǫ Σ } − mµǫ S − mµ/ǫ C − mµ/ǫ Σ + mµη ǫ ( α · N ) S − mµ/ǫ + η { α · N , C − mµ/ǫ Σ } . (4.21)As L ν is an isomorphism, using Lemma 3.1 it easily follows that Υ − mµ/ǫν is a bounded, self-adjointoperator on L (Σ ν ) . Moreover, taking into account the above assumption, by Proposition 4.2 it holdsthat dim (cid:0) Kr(Υ − mµ/ǫν ) (cid:1) < ∞ , since dim (cid:0) Kr(˜Λ mµ/ǫ − ,ν ) (cid:1) = dim (cid:0) Kr(˜Λ mµ/ǫ − ,ν ) (cid:1) < ∞ , for all ν > . Now weintroduce the unitary transformation U : L (Σ ν ) −→ L ( R ) , defined by U g ( x ) = J / ν ( x ) g ( τ ( x )) .We claim that U Υ − mµ/ǫν U − − Υ − mµ/ǫ is a compact operator on L (Σ ν ) . Indeed, let χ be a C ∞ (Σ) cutoff function vanishing out-side the deformation F , we then get U Υ − mµ/ǫν U − − Υ − mµ/ǫ := ( U Υ − mµ/ǫν U − − Υ − mµ/ǫ ) χ + ( U Υ − mµ/ǫν U − − Υ − mµ/ǫ )(1 + χ ) . Since the embedding χ L ( R ) ֒ → H − / ( R ) is compact, and U Υ − mµ/ǫν U − is bounded from H − / ( R ) to L ( R ) , for all ν > , it follows that ( U Υ − mµ/ǫν U − − Υ − mµ/ǫ ) χ is a compact operatoron L ( R ) . Next, observe that ( U Υ − mµ/ǫν U − − Υ − mµ/ǫ )(1 + χ ) := T + T , where T = U L ν D ν B ν χL ν U − (1 + χ ) − L D B χL (1 + χ ) ,T = (cid:16) U L ν D ν B ν − L D B (cid:17) (1 − χ ) L (1 + χ ) . Recall that L ν U − is bounded from L ( R ) to H − / (Σ ν ) , for all v > , and the embedding χ H − / (Σ ν ) ֒ → H − (Σ ν ) is compact. Since B ν is bounded from H − (Σ ν ) to L (Σ ν ) (see Remark3.1) it follows that B ν χL ν U − is a compact operator from L ( R ) to L (Σ ν ) . Now, as U L ν D ν isbounded from L (Σ ν ) to L ( R ) , we then get that T is a compact operator on L ( R ) . We nowapply this argument again to the operator T , and we get T = (cid:16) U L ν D ν χB ν − L D χB (cid:17) (1 − χ ) L (1 − χ )+ (cid:18) U L ν D ν (1 − χ ) B ν − L D (1 − χ ) B (cid:19) (1 − χ ) L (1 − χ ) := T + T . Then in the same manner, it is easy to check that T is a compact operator on L ( R ) . Now set ˜ B ν := (1 − χ ) B ν (1 − χ ) = (1 − χ )(˜Λ − mµ/ǫ + ,ν ˜Λ mµ/ǫ − ,ν )(1 − χ ) , Observe that ˜ B ν = (cid:18) C − mµ/ǫ Σ χ ( α · N ) { α · N , ˜ C − mµ/ǫ Σ } − mµǫ C − mµ/ǫ Σ χS − mµ/ǫ + mµη ǫ χ ( α · N ) S − mµ/ǫ + η χ { α · N , ˜ C − mµ/ǫ Σ } − mµǫ χ C − mµ/ǫ Σ (1 + χ ) S − mµ/ǫ − χ mµǫ C − mµ/ǫ Σ (1 + χ ) S − mµ/ǫ (cid:19) (1 − χ )+ (1 + χ ) (cid:18) − mµǫ C − mµ/ǫ Σ (1 + χ ) S − mµ/ǫ + mµη ǫ ( α · N ) S − mµ/ǫ (cid:19) (1 − χ ) := B ν, + B . Here we used the fact that (1 + χ )( α · N ) { α · N , ˜ C − mµ/ǫ Σ } (1 + χ ) vanishes identically. Therefore weobtain that T = (cid:18) U L ν D ν B ν, − L D B , (cid:19) L (1 − χ ) + (cid:18) U L ν D ν − L D (cid:19) B L (1 − χ ) := T + T . Again, using the compactness of the Sobolev injection, one can show that T is a compact operatoron L ( R ) . Next, remark that D ν (1 − χ ) = (cid:18) C − mµ/ǫ Σ χ ( α · N ) { α · N , C − mµ/ǫ Σ } − mµǫ χS − mµ/ǫ C − mµ/ǫ Σ + mµη ǫ ( α · N ) S − mµ/ǫ + η χ { α · N , C − mµ/ǫ Σ } (cid:19) (1 + χ )+ (1 + χ ) (cid:18) − mµǫ S − mµ/ǫ (1 + χ ) C − mµ/ǫ Σ + mµη ǫ ( α · N ) S − mµ/ǫ (cid:19) (1 + χ ):= D ν, (1 + χ ) + D (1 + χ ) . Note that ( U (1 − χ ) L ν − (1 − χ ) L ) D B L (1 − χ ) = 0 . Therefore, we obtain that T = (cid:18) U L ν D ν, − L D , (cid:19) B L (1 − χ ) + (cid:18) U χL ν − χL (cid:19) D B L (1 − χ ) . Then, we conclude as above that T is a compact operator on L ( R ) . Therefore, U Υ − mµ/ǫν U − − Υ − mµ/ǫ is compact in L ( R ) . As ∈ Sp ess (Υ − mµ/ǫ ) because ∈ Sp ess ( L mµ/ǫ − ) by Proposi-tion 4.3, then by Weyl’s theorem we get that ∈ Sp ess (Υ − mµ/ǫν ) . This contradicts the fact that dim (cid:0) Kr(Υ − mµ/ǫν ) (cid:1) < ∞ , which proves (b) .We now show (c) , so assume that a ∈ ( − m, m ) \ {− mµ/ǫ, mµ/ǫ } . We introduce the operator G aν : L (Σ ν ) −→ L (Σ ν ) defined by: G aν := L ν (Λ a − ,ν Λ a + ,ν )(˜Λ a − ,ν ˜Λ a + ,ν ) L ν . Clearly, G aν is bounded self-adjoint in L (Σ ν ) , since ˜Λ a − ,ν ˜Λ a + ,ν = ˜Λ a + ,ν ˜Λ a − ,ν . Moreover, by definitionwe have ∈ Sp ess ( L a ± ,ν ) = ⇒ ∈ Sp ess ( G aν ) . (4.22)As ˜Λ a + , and ˜Λ a − , are bounded, invertible operators for all a ∈ ( − m, m ) \ {− mµ/ǫ, mµ/ǫ } , fromProposition 4.3 it follows that ∈ Sp ess ( G a ) if and only if a = ∓ mµ/ǫ . Next, we claim that: if a = ∓ mµ/ǫ , then / ∈ Sp ess ( G aν ) . To prove this, let U be the unitary transformation defined inthe proof of (b) , and set T = U G aν U − − G a . Then, T is a compact operator in L ( R ) . Indeed,this may be handled in much the same way as in the proof of the previous statement, we omit thedetails. Therefore Sp ess ( G aν ) = Sp ess ( G a ) holds by Weyl’s theorem. This proves the claim because ∈ Sp ess ( G a ) if and only if a = ∓ mµ/ǫ . Using this, from (4.22) it follows that, if a = ∓ mµ/ǫ then / ∈ Sp ess ( L a ± ,ν ) . Therefore, Corollary 4.1 yields that Sp ess ( H νκ ) ∩ [( − m, m ) \ {− mµ/ǫ, mµ/ǫ } ] = ∅ ,which yields (c) . IRAC OPERATORS WITH δ -SHELL INTERACTIONS 25 Summing up, from (a) and (b) we obtain that (cid:0) − ∞ , − m (cid:1) ∪ {− mµ/ǫ } ∪ (cid:0) m, + ∞ (cid:1) ⊂ Sp ess ( H νκ ) .From (b) and (c) we get the inclusion Sp ess ( H νκ ) ⊂ (cid:0) − ∞ , − m (cid:3) ∪ {− mµ/ǫ } ∪ (cid:2) m, + ∞ (cid:1) . Since theessential spectrum of a self-adjoint operator is closed, we get then the equality (4.12). This completesthe proof of the theorem. (cid:3) Actually in the case
Σ = R × { } , 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 ( R ) . (4.23)A simple computation yields the following relations (cid:20)
12 ( ǫI − µβ + ηα ) + iα (cid:21) (cid:20)
12 ( ǫI + µβ − ηα ) − iα (cid:21) = (2 − iη ) I , (cid:20)
12 ( ǫI − µβ + ηα ) + iα (cid:21) (cid:20)
12 ( ǫI + µβ − ηα ) − iα (cid:21) = iα ( 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 = 0 , (2 − iη ) t Σ ϕ + = − iα ( ǫ + µβ )] t Σ ϕ − for x = 0 . (4.25)Since ( H + a )( H − a ) = ( − ∆ + m − a ) I , one get that ϕ is also solution of the following equation ( − ∆ + m − a ) I ϕ = 0 , for all x = 0 Thus, applying Fourier-Plancherel operator on x = ( x , x ) , we get that(4.26) F x [ ϕ ] ( ξ, x ) = e − x √ | ξ | + m − a F x [ ψ + ] ( ξ ) for x > ,e x √ | ξ | + m − a F x [ ψ − ] ( ξ ) for x < , for some ψ ± ∈ H ( R ) . Since ( H + a ) ϕ = 2 aϕ , by applying the inverse Fourier-Plancherel operator,we obtain that ϕ ( x, x ) = π Z R e ix · ξ e − x √ | ξ | + m − a Γ i ( ξ ) F x [ ψ + ] ( ξ )d ξ for x > , π Z R e ix · ξ e x √ | ξ | + m − a Γ − i F x [ ψ − ] ( ξ )d ξ for x < , (4.27)where Γ ± i ( ξ ) = h α · ( ξ , ξ , ± i p | ξ | + m − a ) + mβ + a i . From this, it is clear that ϕ ± , ( α ·∇ ) ϕ ± ∈ L (Ω ± ) . Now, if we set ψ − = − η − i ǫ − µ ( ǫ + µβ ) α ψ + , then we get (2 − iη )Γ + i ( ξ ) F x [ ψ + ] ( ξ ) = − iα ( ǫI − µβ )Γ − i ( ξ ) F x [ ψ − ] ( ξ ) . (4.28)Thus ( ϕ + , ϕ − ) satisfies the transmission condition. Therefore, for all ψ ∈ H ( R ) the function ϕ ( x, x ) = π Z R e ix · ξ e − x √ | ξ | + m − a Γ i ( ξ ) F x [ ψ + ] ( ξ ) for x > , − i π Z R e ix · ξ e x √ | ξ | + m − a Γ − i ( λ − µβ ) α F x [ ψ + ] ( ξ ) for x < , (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 roughdomain Ω + . More precisely, we assume that Ω + is uniformly rectifiable (see the definitions below),as usual ∂ Ω + = Σ ⊂ R divide the space into two regions Ω ± . Before stating the main results, weneed to give a few preliminary definitions. Namely, we recall the necessary definitions related to thenotion of bounded regular Semmes-Kenig-Toro domains developed by S. Hofmann, M. Mitrea, andM. Taylor in [28]. Let Ω ⊂ R be open with locally finite perimeter, that is the characteristic functionof Ω , denoted by Ω , satisfies the following: ω := ∇ Ω , (5.1)with ω a locally finite R -valued measure and the equality (5.1) can be understood in the sense ofdistributions as follows: Z Ω div f d x = Z ∂ Ω h N , f i d ι, ∀ f ∈ C ∞ ( R , R ) , (5.2)here ω = − N ι , ι is a locally finite positive measure, supported on ∂ Ω , and N ∈ L ∞ ( ∂ Ω) is an R -valued 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 r ց ι ( B ( x, r )) Z B ( x,r ) N d ι = N ( x ) . (5.3) Definition 5.1 (Ahlfors regularity) . We say that a closed set E ⊂ R is -dimensional Ahlfors regularif there exist < a b < ∞ such that ar H ( B ( x, r ) ∩ E ) br , ∀ x ∈ E, r ∈ (0 , diam( E )) , (5.4) where H 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 is an open set such that ∂ Ω is Ahlfors regular, then we saythat Ω is an Ahlfors regular domain. Definition 5.2 (uniformly rectifiable domains) . We say that a compact set E ⊂ R is uniformlyrectifiable provided that it is Ahlfors regular and the following holds. There exist ρ , M ∈ (0 , ∞ ) (calledthe uniform rectifiability constants of E ) such that for each x ∈ E , r ∈ (0 , , there is a Lipschitz map φ : B r → R (where B r is a ball of radius r in R ) with Lipschitz constant L φ M , such that H ( E ∩ B ( x, r ) ∩ φ ( B r )) > ρr . (5.5) A nonempty, proper and bounded open subset Ω ⊂ R is called uniformly rectifiable, provided that ∂ Ω is uniformly rectifiable and also H ( ∂ 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 satisfies the Corkscrewcondition if there are constants C > and r > such that for all x ∈ ∂ Ω and r ∈ (0 , r ) there exists y ∈ Ω (which depends on x and r ), such that | x − y | < r and dist( y, ∂ Ω) > r/C . Also Ω satisfies theexterior corkscrew condition if R \ Ω satisfy the interior corkscrew condition. Finally, Ω satisfies thetwo-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 satisfies the HarnackChain condition if for every ǫ > and every pair of points x , x ∈ Ω ∩ B ( y, r/ for some y ∈ ∂ Ω , IRAC OPERATORS WITH δ -SHELL INTERACTIONS 27 r ∈ (0 , r ) (with reference to C and r as in Definition 5.3), and if dist( x j , ∂ Ω) > ǫ and | x − x | < k ǫ ,there is a chain of open balls B , . . . , B n ⊂ Ω , n Ck such that each B j has a radius r j , with x ∈ B , x ∈ B n and B j ∩ B j +1 = ∅ for j n and having the following property r j C < dist(
B, ∂ Ω) < Cr j and diam( B j ) > C min (dist( x , ∂ Ω) , dist( x , ∂ Ω)) . One remark is in order here. Generally speaking, the Corkscrew condition is a quantitative, scaleinvariant version of openness, and the Harnack Chain condition is a scale invariant version of pathconnectedness.
Definition 5.5 (two-sided NTA domains) . We say that a nonempty, proper open set Ω of R is anNTA (non-tangentially accessible) domain if Ω satisfies both the two-sided Corkscrew and HarnackChain conditions. Furthermore, we say that Ω is a two-sided NTA domain if both Ω and R \ Ω arenon-tangentially accessible domains. Definition 5.6 (Separation property) . A nonempty, bounded proper subset Ω ⊂ R is said to satisfythe separation property if there exists r > such that for each x ∈ ∂ Ω and r ∈ (0 , r ] there existsa -dimensional plane P ( x, r ) in R passing through x and a choice of unit normal vector N x,r to P ( x, r ) such that (cid:26) y + tN x,r ∈ B ( x, r ) : y ∈ P ( x, r ) , t < − r/ (cid:27) ⊂ Ω , and (cid:26) y + tN x,r ∈ B ( x, r ) : y ∈ P ( x, r ) , t > r/ (cid:27) ⊂ R \ Ω . Moreover, if Ω is unbounded, we also require that ∂ Ω divides R into two distinct connected componentsand that R \ Ω has a non-empty interior. Definition 5.7 (Reifenberg flatness) . Let E ⊂ R be a compact set and let δ ∈ (0 , / √ . We saythat E is δ -Reifenberg flat if there exists r > such that for every x ∈ E and every r ∈ (0 , r ] thereexists a -dimensional plane P ( x, r ) which contains x such that r max (cid:26) sup (cid:8) dist( y, E ∩ P ( x, r )) : y ∈ E ∩ B ( x, r ) (cid:9) , sup (cid:8) dist( y, E ∩ B ( x, r )) : y ∈ E ∩ P ( x, r ) (cid:9)(cid:27) < δ. Definition 5.8 ( δ -Reifenberg flat domains) . Given a bounded open set Ω ⊂ R and δ ∈ (0 , δ ) . We saythat Ω is δ -Reifenberg flat domain provided Ω satisfies the separation property and ∂ Ω is δ -Reifenbergflat. Definition 5.9 ( δ -SKT Domains) . Let δ ∈ (0 , δ ) , where δ is a fixed, sufficiently small number.Call a bounded set Ω ⊂ R 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 unitnormal ν is such that || ν || BMO( ∂ Ω , dS) < δ, (5.7) where dS = H ⇂ ∂ Ω and BMO( ∂ Ω , dS) stands for the space of functions with bounded mean oscilla-tion, relative to the surface measure dS = H ⇂ ∂ Ω . Definition 5.10 (Regular SKT Domains) . We say that a bounded open set Ω ⊂ R is a regular SKTdomain provided Ω is δ -Reifenberg flat domain, for some δ ∈ (0 , δ ) , and whose geometric measuretheoretic outward unit normal ν ∈ VMO( ∂ Ω , dS) . Here VMO(Σ , dS) stands for the Sarason space offunctions with vanishing mean oscillation on Σ , relative to the surface measure dS = H ⇂ ∂ Ω . In what follows, we always suppose that Ω + is a bounded, uniformly rectifiable domain, and we set ∂ Ω + := Σ . Hence Σ is compact, -dimensional Ahlfors regular. Moreover, if we let Ω − = R \ Ω + ,then Ω − is uniformly rectifiable (see [28, Proposition 3.10]) and we have R = Ω + ∪ Σ ∪ Ω − . As wementioned in the introduction, in this section, the self-adjointness of the Dirac operator H κ will bederived using the main result of [2]. However, the way the result [2, Theorem 2.11] is stated doesn’ttake into account the case when Σ is Ahlfors regular, therefore, a few comments on how to extendit should be in order. In fact, the only thing left to do, is to give the "good" trace theorem for thefunctions of the Sobolev space H ( R ) in the case of Ahlfors regular surfaces, and this is the purposeof the next proposition. Of course [2, Theorem 2.11] remains valid in a more general context, but togo further in our analysis we restrict ourselves to the class of bounded uniformly rectifiable domains.Let dS = H ⇂ Σ , recall that the Besov space B / (Σ) (see [34, Chapter V] for example), consists ofall functions g ∈ L (Σ) for which Z Z | x − y | < | g ( x ) − g ( y ) | | x − y | dS( y )dS( x ) < ∞ . (5.8)The Besov space is equipped with the norm k g k B / (Σ) := Z Σ | g ( x ) | dS ( x ) + Z Z | x − y | < | g ( x ) − g ( y ) | | x − y | dS( y )dS( x ) . (5.9)In the following, we use the notation H B ( x,r ) U ( y )d y for the mean of U in B ( x, r ) . Given U ∈ H ( R ) ,we set T Σ u ( x ) := lim r ց I B ( x,r ) U ( y )d y, (5.10)at every point 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 ]. Proposition 5.1.
Let Σ be as above. Then the trace operator t Σ (which until now was defined on D ( R ) ) extend to a bounded linear operator T Σ from H ( R ) to B / (Σ) (where T Σ is given by (5.10) )with a bounded linear inverse operator E from B / (Σ) to H ( R ) . In other words, B / (Σ) is thetrace to Σ of H ( R ) and T Σ E is the identity operator. 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 / (Σ) is the trace to Σ of H ( R ) ). Now, using Proposition 5.1 insteadof [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 [2] (more precisely [2, Theorem 2.11 (iii) ]) reads as follows: Theorem 5.1.
Let Ω + and Σ be as above. Let Λ : L (Σ) −→ L (Σ) be a bounded linear self-adjointoperator. Define T = H + V with dom( T ) = (cid:8) u + Φ[ g ] : u ∈ H ( R ) , g ∈ L (Σ) and T Σ u = Λ[ g ] (cid:9) (5.11) and T ( u + Φ[ g ]) = H u , 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 Lipschitzdomains with
VMO normals, where we can discuss the spectral properties of the Dirac operator H κ ,for any κ ∈ R with sgn( κ ) = 0 , . This is the purpose of the next subsection.5.1. 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: IRAC OPERATORS WITH δ -SHELL INTERACTIONS 29 (H3) Σ = ∂ Ω + with Ω + a bounded Lipschitz domain with normal N ∈ VMO( ∂ Ω , dS) .Roughly speaking, the above assumption implies smallness of the Lipschitz constant of Ω + . Anotherway to reformulate the assumption (H3) is to say that Ω + belongs to the intersection of the class ofbounded 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 such that sgn( κ ) = 0 , , and assume that Σ satisfies (H3) , and let H κ beas in Definition 2.1. Then H κ is self-adjoint and dom( H κ ) ⊂ H / ( R \ Σ) . Let g ∈ L (Σ) , then the harmonic double layer K and the Riesz transforms ( R k ) k on Σ aredefined by K [ g ]( x ) = lim ρ ց Z | x − y | >ρ N ( y ) · ( x − y )4 π | x − y | g ( y )dS( y ) ,R k [ g ]( x ) = lim ρ ց Z | x − y | >ρ x k − y k π | x − y | g ( y )dS( y ) . (5.12)The following theorem is implicitly contained in [28], but we state and prove it here for the sake ofcompleteness. In the proof, we use the notation R ± = { x ∈ R : ± x > } for the upper (respectivelythe lower) half space. Theorem 5.3.
Assume that Σ satisfies (H3) . Then, the harmonic double layer K and the commuta-tors [ N j , R k ] , j, k , are compact operators on L (Σ) . Proof.
The result follows from the fact that Ω + is a bounded regular SKT domain. To see thisindeed, note that bi-Lipschitz mappings preserve the class of two-sided NTA domains with Ahlforsregular boundaries, and the class of regular SKT domains is invariant under continuously differentiablediffeomorphisms, see [27]. Now, by definition Ω + is locally the region above the graph of a Lipschitzfunction φ : R −→ R . Therefore, one may assume (via a partition of unity and a local flattening ofthe boundary) that Ω + = { x = ( x, x ) ∈ R × R : x > φ ( x ) } . Let F : R −→ R be defined for all ( x, x ) ∈ R × R as F ( x, x ) := ( x, x + φ ( x )) . Then it easilyfollows that F is a bijective function with inverse F − : R −→ R given by F − ( y, y ) := ( y, y − φ ( y )) for all ( y, y ) ∈ R × R . Moreover, F and F − are both Lipschitz functions with constants L F , L F − (1 + ||∇ φ || L ∞ ) . It is clear that Ω + (respectively Ω − ) is the image of R (respectively R − ) under thebi-Lipschitz homeomorphism F , which also maps R × { } onto Σ . From this, it follows that Ω + is atwo-sided NTA domain and Σ is Ahlfors regular (because R is a two sided NTA domain and R × { } is Ahlfors regular). Since N ∈ VMO(Σ) by assumption, thanks to [28, Theorem 4.21], we know that Ω + is a regular SKT domain. Therefore the claimed result follows by [28, Theorem 4.47]. (cid:3) Lemma 5.1.
Assume that Σ satisfies (H3) . Then, { α · N , C z Σ } is a compact operator on L (Σ) , forall z ∈ C \ (( −∞ , − m ] ∪ [ m, ∞ )) . Proof.
Given g ∈ L (Σ) , similar computation as in the proof of Lemma 3.1 give { α · N , C z Σ } [ g ]( x ) = T K [ g ]( x ) + T K [ g ]( x ) , (5.13) where the kernels K j , j = 1 , , are given by K ( x, y ) = e i √ z − m | x − y | π | x − y | ( α · N ( x )) (cid:18) z + mβ + p z − m (cid:18) α · x − y | x − y | (cid:19)(cid:19) + e i √ z − m | x − y | π | x − y | (cid:18) z + mβ + p z − m (cid:18) α · x − y | x − y | (cid:19)(cid:19) ( α · N ( y ))+ e i √ z − m | x − y | − π | x − y | [( α · N ( x ))( iα · ( x − y )) + ( iα · ( x − y )) ( α · N ( y ))] .K ( x, y ) = i π | x − y | (( N ( x ))( α · ( x − y )) + α · ( x − y ))( N ( y ))) . Using the estimate (cid:12)(cid:12)(cid:12) e i √ z − m | x | − (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)p z − m (cid:12)(cid:12)(cid:12) | x | , (5.14)it easily follows that sup k,j | K ( x − y ) | = O ( | x − y | − ) when | x − y | −→ . (5.15)Once (5.15) has been established, working component by component and using [24, Lemma 3.11], onecan show that T K is a compact operator in L (Σ) . Now it is straightforward to check that T K [ g ]( x ) = ˜ K [ g ]( x ) + ˜ K ∗ [ g ]( x ) + X j =1 3 X k =1 k = j α j α k [ N j , R k ] [ g ]( x ) . (5.16)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 (Σ) , we have ˜ K [ g ]( x ) = lim ρ ց Z | x − y | >ρ N ( y ) · ( x − y )4 π | x − y | I g ( y )dS( y ) , ˜ K ∗ [ g ]( x ) = lim ρ ց Z | x − y | >ρ N ( x ) · ( x − y )4 π | x − y | I g ( y )dS( y ) , R k [ g ]( x ) = lim ρ ց Z | x − y | >ρ x k − y k π | x − y | I g ( y )dS( y ) . (5.17)Since the adjoint of a compact operator is a compact operator and α j ’s are constants matrices, usingTheorem 5.3 and working component by component, we get that T K is a compact operators in L (Σ) .Therefore { α · N , C z Σ } is a compact operator in L (Σ) and this finishes the proof of the lemma. (cid:3) Note that Lemma 5.1 is not valid for general Lipschitz surfaces. In fact, it turns out that assum-ing (H3) means that we are excluding the special class of corner domains. Indeed, from Theorem5.3 we know that any bounded Lipschitz domain Ω is an NTA domain and ∂ Ω is Ahlfors regular.However, in our context (i.e R = Ω + ∪ Σ ∪ Ω − ), the presence of any angle θ = 0 implies that dist( N , VMO( ∂ Ω , dS) ) > , where the distance is taken in BMO( ∂ Ω , dS) , cf. [28, Proposition4.38] and the discussion that precedes it. Hence, Ω + is not a regular SKT domain and then by [28,Theorem 4.47], the principale value of the harmonic double layer K and the commutators [ N j , R k ] , j, k , are not compact on L (Σ) . So, { α · N , C Σ } is not a compact operator on L (Σ) , andthus the assumption (H3) is sharp. To make this clearer, we have the following result: Theorem 5.4.
Let Ω + be a bounded Lipschitz domain, such that the decomposition R = Ω + ∪ Σ ∪ Ω − holds, where ∂ Ω + = Σ . Then, Σ satisfies (H3) if and only if { α · N , C Σ } is compact in L (Σ) . IRAC OPERATORS WITH δ -SHELL INTERACTIONS 31 Proof.
The first implication follows from Lemma 5.1 by taking z = 0 . Let us prove the reverseimplication, so assume that { α · N , C Σ } is compact in L (Σ) . Given g ∈ L (Σ) , we define thebounded, linear operator W : L (Σ) −→ L (Σ) as follows: W [ g ]( x ) = lim ρ ց Z | x − y | >ρ iσ · ( x − y )4 π | x − y | g ( y )dS( y ) , (5.18)where σ = ( σ , σ , σ ) are the Pauli matrices defined by (2.13). That W is bounded in L (Σ) is aconsequence of Lemma 2.1 and Remark 2.1. Now, from (5.13) it holds that { α · N , C Σ } = T K + { σ · N , W } { σ · N , W } ! , (5.19)where T K is a compact operator in L (Σ) . From this, it follows that { α · N , C Σ } is compact in L (Σ) ⇐⇒ { σ · N , W } is compact in L (Σ) . (5.20)Hence, it remains to show that { σ · N , W } is compact in L (Σ) = ⇒ 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 uniformlyrectifiable by [18] (see also [28, Corollary 3.9]). Next, we claim that there exists C > , dependingonly on the dimension, the uniform rectifiability and Ahlfors regularity constants of Σ , such that dist (cid:0) N , VMO( ∂ Ω , dS) (cid:1) C dist (cid:0) { σ · N , W } , L c ( L (Σ) ) (cid:1) , (5.22)where the distance in the right-hand side is measured in L ( L (Σ) ) . Here L ( L (Σ) ) (respectively L c ( L (Σ) ) ) denotes the set of bounded operators (respectively bounded and compact operators) from L (Σ) into itself. Now, assume for instance that (5.22) holds true. Since { σ · N , W } is compact in L (Σ) , by (5.22), it holds that N ∈ VMO( ∂ Ω , dS) . Therefore, Σ satisfies (H3) , which proves thetheorem. Now, let us come back to the proof of (5.22). Given x, y ∈ R , we define the followingmultiplication 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 | , x ⊙ y + y ⊙ x = − x · y ) I , ∀ x, y ∈ R . Now, we make the observation that the multiplication operator defined by (5.23) has the same prop-erties as the multiplication operator in the Clifford algebra C l (see [28, Section 4.6] for the precisedefinition). Moreover, W ( σ · N ) plays the same role as the Cauchy-Clifford operator defined on L (Σ) ⊗ C l (i.e. it acts on C l -valued functions), cf. [28, Section 4.6]. Thus, one can adapt the samearguments of [28, Theorem 4.46] and show that the claim (5.22) holds true, we omit the details. Thiscompletes the proof of the theorem. (cid:3) As it was done in [28, Theorem 4.47], one can also characterize the class of bounded regular SKTdomains via the compactness of the anticommutators { σ · N , W } in L (Σ) , or equivalently via thecompactness of the anticommutator { α · N , C Σ } in L (Σ) . This is the purpose of the followingproposition. Note that the algebra generated by the Pauli spin matrices σ = ( σ , σ , σ ) (as an algebra on the real field) is isomorphicto C l . Thus, W ( σ · N ) can be viewed as the restriction of the Cauchy-Clifford operator on L (Σ) ⊗ C . Proposition 5.2.
Let Ω ⊂ R be a bounded two-sided NTA domain with a compact, Ahlfors regularboundary. Then the following statements are equivalent:(i) Ω is a regular SKT domain.(ii) The harmonic double layer K and the commutators [ N j , R k ] , j, k , are compactoperators on L ( ∂ Ω) .(iii) { α · N , C ∂ Ω } is a compact operator on L ( ∂ Ω) .(iv) { σ · N , W } is a compact operator on L ( ∂ Ω) . Proof. (i) ⇒ (ii) is a consequence of [28, Theorem 4.21] and [28, Theorem 4.47]. (ii) ⇒ (iii) readilyfollows from (5.13) and (5.16). (iii) ⇒ (iv) is an immediate consequence of (5.20). Finally, (iv) ⇒ (i) follows from (5.22) and [28, Theorem 4.21]. (cid:3) Corollary 5.1.
Let z ∈ C \ (( −∞ , − m ] ∪ [ m, ∞ )) , then Λ z ± is a Fredholm operator on L (Σ) . Proof.
Fix z ∈ C \ (( −∞ , − m ] ∪ [ m, ∞ )) . Taking into account (3.13), from the proof of Theorem3.1 we have that Λ z ∓ Λ z ± = 1sgn( κ ) − − C z Σ ( α · N ) { α · N , C z Σ } + 2 µ sgn( κ ) ( m I + zβ ) S z + η sgn( κ ) { α · N , C z Σ } , (5.24)recall that S z is defined by (3.12). As the injection H / (Σ) into L (Σ) is compact, cf. [36, Theorem6.11] for example, it follows that S z is a compact operator in L (Σ) . Using that C z Σ ( α · N ) is boundedin L (Σ) and that { α · N , C z Σ } is a compact operator on L (Σ) by Lemma 5.1, we thus obtain that C z Σ ( α · N ) { α · N , C z Σ } is a compact operator on L (Σ) . Hence Λ z ∓ Λ z ± is Fredholm operator andtherefore Λ z ± is Fredholm operator by [1, Theorem 1.46 ( iii )]. (cid:3) Proof of Theorem 5.2
The proof is a straightforward application of Theorem 5.1. Indeed, bydefinition if we set ( T, dom( T )) = ( H κ , dom( H κ ) , V = V κ , Λ = − Λ + . (5.25)Then, for all ϕ = u + Φ[ g ] ∈ dom( T ) , it holds that V ( ϕ ) = ( ǫI + µβ + η ( α · N ))( t Σ u + C Σ [ g ]) = ( ǫI + µβ + η ( α · N ))( − Λ + + C Σ )[ g ]= − κ ) ( ǫI + µβ + η ( α · N ))( ǫI − µβ − η ( α · N ))[ g ] = − g. Therefore, V ( ϕ ) = − g and hence T ( ϕ ) = H u 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 self-adjoint operator on L (Σ) . 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. (cid:3) 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:(i) Given a ∈ ( − m, m ) , then Kr( H κ − a ) = 0 ⇐⇒ Kr(Λ a + ) = 0 (Birman-Schwinger principle).(ii) For all z ∈ C \ R , it holds that ( H κ − z ) − = ( H − z ) − − Φ z (Λ z + ) − (Φ z ) ∗ . (5.26) (iii) Sp ess ( H κ ) = ( −∞ , − m ] ∪ [ m, + ∞ ) .(iv) a ∈ Sp p ( H κ ) if and only if − a ∈ Sp p ( H ˜ κ ) , where ˜ κ = ( − ǫ, µ, − η ) . 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 IRAC OPERATORS WITH δ -SHELL INTERACTIONS 33 Corollary 5.1, we conclude that Λ z + : L (Σ) −→ L (Σ) is bijective and thus (5.26) makes sense. Nowgiven v ∈ L ( R ) , we set ϕ = ( H − z ) − v − Φ z (Λ z + ) − (Φ z ) ∗ [ v ] . To prove item (ii) , it remains to show that ϕ ∈ dom( H κ ) . For this, remark that ϕ = u + Φ[ g ] where u = ( H − z ) − v − (Φ z − Φ)(Λ z + ) − (Φ z ) ∗ [ v ] and g = − (Λ z + ) − (Φ z ) ∗ [ v ] . Note that (Λ z + ) − (Φ z ) ∗ is abounded operator from L ( R ) to L (Σ) and ( H − z ) u = v + z Φ[ g ] ∈ L ( R ) . Consequently, weget that g ∈ L (Σ) and u ∈ H ( R ) . Moreover, using Lemma 2.1 (ii) , we obtain t Σ u = (cid:0) (Φ z ) ∗ − ( C z Σ − C Σ )(Λ z + ) − (Φ z ) ∗ (cid:1) [ v ] = (cid:0) (Φ z ) ∗ − (Λ z + − Λ + )(Λ z + ) − (Φ z ) ∗ (cid:1) [ v ] = − Λ + [ g ] . Thus ϕ ∈ dom( H κ ) , which yields (ii) . Item (iii) is a consequence of item (ii) . Indeed, since (Φ z ) ∗ is a bounded operator from L ( R ) to H / (Σ) , using the Sobolev injection it follows that Φ z (Λ z + ) − (Φ z ) ∗ is a compact operator on L ( R ) . Thus ( H κ − z ) − − ( H− z ) − is a compact operatoron L ( R ) , therefore Weyl’s theorem yields Sp ess ( H κ ) = Sp ess ( H ) = ( −∞ , − m ] ∪ [ m, + ∞ ) . Finally,item (iv) follows by Proposition des equivalences. (cid:3) The reader interested on confinement may wonder if the Dirac operator H κ generates this phenom-enon under the assumption that sgn( κ ) = − . Our aim in what follows is to clarify and provide an an-swer to this question. Let Φ ± : L (Σ) −→ L (Ω ± ) be the operators defined by Φ Ω ± [ g ]( x ) = Φ[ g ]( x ) ,for g ∈ L (Σ) and x ∈ Ω ± . Let H κ be as in Theorem 5.2 and assume that sgn( κ ) = − . Given 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 α · N ))[ g ]= (cid:18)
14 ( ǫ − µβ − η ( α · N )) ∓ i α · N ) (cid:19) g, (5.28)where in the last equality we used that t Σ u = − Λ + [ g ] . Now, multiplying the identity (5.28) by (cid:18)
12 ( ǫ + µβ + η ( α · N )) ± i ( α · N ) (cid:19) , we get (cid:18)
12 ( ǫ + µβ + η ( α · N )) ± i ( α · N ) (cid:19) lim nt ϕ ± = ∓ iηg. (5.29)As consequence, if sgn( κ ) = − and η = 0 , then H κ cannot generate confinement. Hence Σ ispenetrable. Clearly, if we set η = 0 in (5.29), then H κ generates confinement, but we postpone thiscase to the next subsection, where we establish that in a more general context.5.2. 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 boundeduniformly rectifiable and η = 0 . The idea is to identify some situations where the operator Λ + givesrise to a Fredholm operator, and thereby use Theorem 5.1 to get the self-adjointness of H κ . So inthis subsection the domain Ω + is bounded uniformly rectifiable unless stated otherwise. Before goingany further, we remind the reader that, since Ω + is bounded uniformly rectifiable, all the singularintegral operators defined in this article are bounded in their corresponding L -spaces; cf [28]. Given g ∈ L (Σ) , we define the Hardy-Littlewood maximal operator by M Σ g ( x ) = sup r> I B ( x,r ) ∩ Σ | g ( y ) | dS , x ∈ Σ . (5.30)Then, by [16, p. 624], there is C > such that k M Σ g k L (Σ) C k g k L (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 thenontangential 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 holdstrue. Moreover, it holds that (( σ · N ) W ) = ( W ( σ · N )) = − I . (5.33) Proof.
Given z ∈ C \ (( −∞ , − m ] ∪ [ m, ∞ )) . Set k ( x ) := φ z ( x ) − i ( α · x | x | ) , for all x ∈ R \ { } , (5.34)Then there is a constant C such that | k ( ω, y ) | C/ | ω − y | / := ˜ k ( ω, y ) , for ω, y ∈ Ω + . Define T [ g ]( x ) = Z Σ ˜ k ( x, y ) g ( y )dS( y ) , (5.35)Clearly, T is bounded in L (Σ) . Now, recall the definition of Γ Ω ± ( x ) from (2.10). Let x ∈ Σ and ω ∈ Γ Ω ± ( x ) , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ( ω, | x − ω | ) ∩ Σ k ( ω, y ) g ( y )dS( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z B ( ω, | x − ω | ) ∩ Σ C (cid:18) a | x − ω | (cid:19) / | g ( y ) | dS( y ) . (5.36)Using the Ahlfors regularity, it follows that there is C depending only on the Ahlfors regularityconstant of Ω + such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ( ω, | x − ω | ) ∩ Σ k ( ω, y ) g ( y )dS( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C | x − ω | / M Σ g ( x ) . (5.37)Let y ∈ Σ \ B ( x, | x − ω | ) , then | ω − y | | x − y | , and thus | k ( ω, y ) | ˜ k ( ω, y ) ˜ k ( x, y ) . Therefore (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Σ \ B ( ω, | x − ω | ) k ( ω, y ) g ( y )dS( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T [ | g | ]( x ) . (5.38)Thus, (5.37), (5.38) and the dominate convergence theorem yield that lim Γ Ω+ ( x ) ∋ ω −→ nt x Z Σ k ( ω, y ) g ( y )dS( y ) = Z Σ k ( x, y ) g ( y )dS( y ) , (5.39)holds for all g ∈ L (Σ) and dS -a.e. x ∈ Σ . Similarly, one can show that lim Γ Ω − ( x ) ∋ ω −→ nt x Z Σ k ( ω, y ) g ( y )dS( y ) = Z Σ k ( x, y ) g ( y )dS( y ) , (5.40) IRAC OPERATORS WITH δ -SHELL INTERACTIONS 35 holds for all g ∈ L (Σ) and dS -a.e. x ∈ Σ . Now, let ˜ φ be the fundamental solution of the masslessDirac operator − iσ · ∇ , that is ˜ φ ( x ) = iσ · x | x | , for all x ∈ R \ { } . (5.41)Now, we define the bounded operator ˜Φ : L (Σ) −→ L ( R ) as follows ˜Φ[ g ]( x ) = Z Σ ˜ φ ( x − y ) g ( y )dS( y ) , for all x ∈ R \ Σ , (5.42)Observe that ( − iσ · ∇ ) ˜Φ[ g ] = 0 holds in D ′ (Ω ± ) , for all g ∈ L (Σ) . We set W ± [ g ]( x ) := lim Γ Ω ± ( x ) ∋ y −→ nt x ˜Φ[ g ]( y ) . (5.43)Then, by [28, Proposition 3.30] it follows that W ± [ g ]( x ) exist for dS -a.e. x ∈ Σ , and W ± : L (Σ) → L (Σ) are linear bounded operators. Moreover, the following holds true: W ± = ∓ i σ · N ) + W. (5.44)Therefore, Lemma 2.1 (i) follows from the above considerations and this completes the proof the firststatement. The proof of the second statement is a relatively straightforward modification of thetechnique used in the proof of [2, Lemma 3.3] (ii) . Indeed, by [28, p. 2659] it follows that kN Ω ± a ( ˜Φ[ g ]) k L (Σ) C k g k L (Σ) , (5.45)for some C > depending only on a as well as the Ahlfors regularity and the uniform rectifiabilityconstants of Σ . Using that ( − iσ · ∇ ) ˜Φ[ g ] = 0 in Ω ± , by [28, Theorem 4.49] it holds that ˜Φ[ g ] = Z Σ ˜ φ ( x, y )( ± iσ · N ( y )) g ( y )dS( y ) , x ∈ Ω ± . (5.46)Although [28, Theorem 4.49] was stated in the case of tow-sided NTA domains it also holds foruniformly rectifiable domains by the discussion on [28, p. 2758]. Now, given x ∈ Ω + and g ∈ L (Σ) .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). Thiscompletes the proof of the proposition. (cid:3) Given z ∈ C \ (( −∞ , − m ] ∪ [ m, ∞ )) , recall that Λ z ± are defined in Notation 2.1. Let κ ∈ R suchthat sgn( κ ) = 0 , . Then, the Dirac operator H κ = H + V κ acting in L ( R ) , is defined on the domain dom( H κ ) = (cid:8) u + Φ[ g ] : u ∈ H ( R ) , g ∈ L (Σ) , T Σ u = − Λ + [ g ] (cid:9) , (5.48)where T Σ is the trace operator defined in Proposition 5.1, and V κ ( ϕ ) = 12 ( ǫI + 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 ) = H u , 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 electrostaticand Lorentz scalar δ -shell interactions supported on Σ . The first main result on the spectral propertiesof the Dirac operator H ǫ,µ reads as follows: Theorem 5.6.
Let λ, µ ∈ R such that < | ǫ − µ | < / k C Σ k L (Σ) → L (Σ) , then H ǫ,µ is self-adjoint.In particular, if Ω + is Lipschitz, then dom( H ǫ,µ ) ⊂ H / ( R \ Σ) . If we assume in addition thatthere is z ∈ C \ R , such that | ǫ − µ | < / k C z Σ k L (Σ) → L (Σ) , then it holds that Sp ess ( H ǫ,µ ) = ( −∞ , − m ] ∪ [ m, + ∞ ) . (5.51) Proof . Fix ǫ, µ ∈ R such that < | ǫ − µ | < / k C z Σ k L (Σ) → L (Σ) , holds for some z ∈ C \ (( −∞ , − m ] ∪ [ m, ∞ )) . Then, from the proof of Corollary 5.1 we have Λ z ∓ Λ z ± = 1 ǫ − µ − ( C z Σ ) + 2 µǫ − µ ( m I + zβ ) S z , (5.52)Recall that C z Σ is bounded in L (Σ) . Using Neumann’s lemma, it follows that M z := ( I − ( ǫ − µ )( C z Σ ) ) is a bounded invertible operator in L (Σ) . Now, since ( m I + zβ ) is bounded and S is acompact on L (Σ) , we therefore get that K z := µǫ − µ ( m I + zβ ) S z is compact on L (Σ) . Combiningthis, from (5.52) it holds that I − ( ǫ − µ ) M − z Λ z − Λ z + = − ( ǫ − µ ) M − z K z , I − ( ǫ − µ )Λ z + Λ z − M − z = − ( ǫ − µ ) K z M − z . (5.53)As M − z Λ z − and Λ z − M − z are bounded operators on L (Σ) , M − z K z and K z M − z are compact on L (Σ) , then [1, Theorem 1.50 and Theorem 1.51] yields that Λ z + is Fredholm. Hence, the firststatement is a direct consequence of Theorem 5.1 and the fact that Λ + is a self-adjoint, Fredholmoperator on L (Σ) . Now, assume that Ω + is Lipschitz, then dom( H ǫ,µ ) ⊂ H / ( R \ Σ) is aconsequence of Remark 2.1. The last statement follows by the same method as in Theorem 5.5 usingthe Fredholm property of Λ z + , we omit the details. (cid:3) Remark 5.1.
From Lemma 2.1( ii ) it easily follows that k C z Σ k > / (cf. [3, Remark 3.5] ), whichimplies that | ǫ − µ | < . Hence, the combination of coupling constants ǫ and µ is not critical. Ofcourse, we already know that the above result is false in the case ǫ − µ = 4 . Note that Theorem 5.6remains valid if one control the norm of the Cauchy operator instead of controlling the combinationof interactions. However, this can influence the geometrical characterization of Σ which can imply anincrease in terms of regularity. As it was mentioned in the introduction, under the assumption that m = 0 and µ ∈ ( − , , it wasproved in [45] the existence of a unique self-adjoint realization of the two dimensional Dirac operatorwith pure Lorentz scalar δ -shell interactions, where Σ is a closed curve with finitely many corners. itseems 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 − µ | so that H ǫ,µ is self-adjoint, this is notsatisfactory in the sense that this bound involves k C Σ k L (Σ) → L (Σ) , which is not easy to quantify.In what follow, we are going to remove this restriction by imposing a sign hypothesis on the couplingconstants, and give a better quantitative assumption than the one of Theorem 5.6. The next theoremmakes this more precise. Theorem 5.7.
Let λ, µ ∈ R such that | ǫ | 6 = | 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: IRAC OPERATORS WITH δ -SHELL INTERACTIONS 37 (a) µ > ǫ .(b) ǫ > µ and k W k L (Σ) → L (Σ) < ǫ − µ < / k W k L (Σ) → L (Σ) .Then H ǫ,µ is self-adjoint. In particular, if Ω + is Lipschitz, then dom( H ǫ,µ ) ⊂ H / ( R \ Σ) . More-over, the following statements hold true:(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 ) − = ( H − z ) − − Φ z (Λ z + ) − (Φ z ) ∗ . (iii) Sp ess ( H ǫ,µ ) = ( −∞ , − m ] ∪ [ m, + ∞ ) .(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 − ǫǫ − µ , − µǫ − µ ) , for all a ∈ ( − m, m ) .(vi) C := sup a ∈ [ − m,m ] k C a Σ k < ∞ . Moreover, Sp disc ( H ǫ,µ ) ∩ ( − m, m ) = ∅ either if | ǫ − µ | < /C and | ǫ + µ | < /C , or if | ǫ − µ | > C and | ǫ + µ | > C . 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-adjointoperator, and we conclude by using Theorem 5.1 and Remark 2.1 to obtain the first statement of thetheorem. So, fix λ, µ ∈ R such that | ǫ | 6 = | µ | , and let z ∈ C \ (( −∞ , − m ] ∪ [ m, ∞ )) . Then, from thedefinition of C z Σ it follows that C z Σ = T zK + WW ! := T zK + ˜ W , (5.54)where the kernel K satisfies sup k,j | K ( x − y ) | = O ( | x − y | − ) when | x − y | −→ . (5.55)Hence, T zK is compact in L (Σ) . Therefore, in the same way as in (5.52) we get that Λ z ∓ Λ z ± = 1 ǫ − µ − ˜ W + ( T zK ) + { T zK , ˜ W } + 2 µǫ − µ ( m I + zβ ) S z := 1 ǫ − µ − ˜ W + K , (5.56)where K is compact in L (Σ) . Now observe that ˜ W = W W ! . Now, if ǫ < µ holds. Then, using that W is a bounded self-adjoint operator in L (Σ) , it followsthat ˜ W is a nonnegative, self-adjoint operator on L (Σ) . From this, it follows that / ( ǫ − µ ) belongs to the resolvent set of ˜ W and hence I − ( ǫ − µ ) ˜ W in invertible on L (Σ) . In anotherhand, assume that ǫ > µ and ǫ − µ < / k W k L (Σ) → L (Σ) hold, then Neumann’s lemma yieldsthat I − ( ǫ − µ ) ˜ W is invertible on L (Σ) . In both cases, similar arguments to those of the proofof Theorem 5.6 yield that Λ z + is Fredholm, which proves the first statement of the theorem for thistwo cases. Now we deal with the case k W k L (Σ) → L (Σ) < ǫ − µ . From Proposition 5.33, we havethat W is invertible on L (Σ) and W − = − σ · N ) W ( σ · N ) . Thus, from (5.56) it follows that ( ǫ − µ )( ˜ W − ) Λ z − Λ z + = ( ˜ W − ) − ( ǫ − µ ) I + ( ǫ − µ )( ˜ W − ) K , ( ǫ − µ )Λ z + Λ z − ( ˜ W − ) = ( ˜ W − ) − ( ǫ − µ ) I + ( ǫ − µ ) K ( ˜ W − ) . (5.57)As k ˜ W k L (Σ) → L (Σ) = k W k L (Σ) → L (Σ) , from this and Proposition ?? (ii) we get that k ˜ W − k L (Σ) → L (Σ) k ˜ W k L (Σ) → L (Σ) = 4 k W k L (Σ) → L (Σ) (5.58) Hence, if ǫ − µ > k W k L (Σ) → L (Σ) , then ǫ − µ > k ˜ W − k L (Σ) → L (Σ) . Thus ǫ − µ is not inthe spectrum of ( ˜ W − ) . Thereby ˜ W − − ( ǫ − µ ) I is invertible on L (Σ) . Now, from (5.57) itfollows that I − ( ǫ − µ ) (cid:16) ( ˜ W − ) − ( ǫ − µ ) I (cid:17) − ( ˜ W − ) Λ z − Λ z + = K , I − ( ǫ − µ )Λ z + Λ z − ( ˜ W − ) (cid:16) ( ˜ W − ) − ( ǫ − µ ) I (cid:17) − = K . (5.59)where K and K are compact operators on L (Σ) . 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 proofof items (i) , (ii) , (iii) and (iv) runs as in the proof of Theorem 5.5. Item (v) is a direct consequenceof (i) . Indeed, denote by Λ a + ,ǫ ′ ,µ ′ the Λ a + operator associated to the Dirac operator H − ǫǫ − µ , − µǫ − µ .Observe that Λ a + ,ǫ ′ ,µ ′ is given by Λ a + ,ǫ ′ ,µ ′ = −
14 ( ǫI − 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 = g ∈ L (Σ) : − ǫ − µ ( ǫI − µβ ) g = C a Σ [ g ] ⇐⇒ ǫ − µ ( ǫI − µβ )(( α · N ) C a Σ ) [ g ] = C a Σ [ g ] ⇐⇒ C a Σ (( α · N ) C a Σ )[ g ] = 14 ( ǫI − µβ )(( α · N ) C a Σ )[ g ] ⇐⇒ there is = f = (( α · N ) C a Σ )[ g ] ∈ L (Σ) : Λ a + ,ǫ ′ ,µ ′ [ f ] = 0 ⇐⇒ a ∈ Sp disc ( H − ǫǫ − µ , − µǫ − µ ) ∩ ( − m, m ) . Which proves (v) . Now we turn to the proof of item (vi) , the first claim of statement is contained in[3, Lemma 3.2] and [6, Proposition 3.5], for a C -compact surface Σ , and the same arguments holdtrue in the Lipschitz case. To see the last claim, note that for all a ∈ ( − m, m ) , we have ∈ Sp disc (Λ a + ) ⇐⇒ − ∈ Sp disc (( ǫ I + µβ ) C a Σ ) . (5.61)Using the first claim, it follows that if | ǫ − µ | < /C and | ǫ + µ | < /C , then k ( ǫ I + µβ ) C a Σ k L (Σ) → L (Σ) < . Therefore, − / ∈ Sp disc (( ǫ I + µβ ) C a Σ ) . Hence, (5.61) and (i) yield that Sp disc ( H ǫ,µ ) ∩ ( − m, m ) = ∅ .Using (v) , the case | ǫ − µ | > C and | ǫ + µ | > C , follows by iterating the same arguments, whichgives (vi) . This completes the proof of theorem. (cid:3) Remark 5.2.
Assume that Ω + is Lipschitz. Then, following essentially the same arguments as inTheorem 5.7, one can show that , if for all a ∈ ( − m, m ) the following holds: k C a Σ k L (Σ) → L (Σ) < ǫ − µ < / k C a Σ k L (Σ) → L (Σ) , then H ǫ,µ is self-adjoint with dom( H ǫ,µ ) ⊂ H / ( R \ Σ) . However, if µ = 0 , then from Theorem 5.7 (vi) it follows that Sp disc ( H ǫ, ) ∩ ( − m, m ) = ∅ , see also [3, Theorem 3.3] and [6, 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-adjointnesscharacter of H ǫ,µ , if ǫ − µ ∈ J . IRAC OPERATORS WITH δ -SHELL INTERACTIONS 39 Next, we discuss the particular case ǫ − µ = − . Assume that Ω + is Lipschitz, given ϕ = u + Φ[ g ] ∈ dom( H ǫ,µ ) , recall the definition of ϕ ± from (5.27). Define P ± = (cid:0) ( ǫ + µβ ) ± i ( α · N ) (cid:1) ,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 − µ = − , 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 ) = n ϕ ± := u Ω ± + Φ Ω ± [ g ] , u Ω ± ∈ H (Ω ± ) , g ∈ L (Σ) : P ± lim nt ϕ ± = 0 o . Proof.
That Σ is impenetrable is consequence of [3, Theorem 5.4. and Theorem 5.5.] and Theorem5.7 (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 ] = −
14 ( ǫI − µβ ) g ⇐⇒
12 (lim nt ϕ + + lim nt ϕ − ) = − i ǫI − 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 ] ⇐⇒ (cid:18)
12 ( ǫI − µβ ) + i ( α · N )) (cid:19) lim nt ϕ + = − (cid:18)
12 ( ǫI − µβ ) − i ( α · N )) (cid:19) 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. (cid:3) Remark 5.4.
By Taking ǫ = 0 in Theorem 5.7 (i) , we conclude that the free Dirac operator coupledwith Lorentz scalar δ -shell interactions is always self-adjoint and Σ becomes impenetrable for µ = ± .In particular, if Ω + is Lipschitz, then dom( H ǫ,µ ) is included in H / (Σ) , for any compact Lipschitzsurface Σ . The reason we assumed that η = 0 is due to the technique that we have use above. However, aswe have already commented in Subsection 4.2, the coupling H + η ( α · N ) δ Σ is always self-adjoint. Infact, we have the following proposition. Proposition 5.5.
Assume that Ω + is Lipschitz. Let η ∈ R \ { } , set κ = (0 , , η ) and let H κ be asabove. Then H κ is self-adjoint and we have dom( H κ ) = (cid:8) u + Φ[ − η ( η + 4) − ( α · N )Λ − ( α · N )[ t Σ u ]] : u ∈ H ( R ) (cid:9) . Moreover, dom( H κ ) ⊂ H ( R \ Σ) 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 \ { } and fix z ∈ C \ (( −∞ , − m ] ∪ [ m, ∞ )) . Recall that Λ z ± are givenby Λ z ± = 1 η ( α · N ) ± C z Σ . Now, using Lemma 2.1, a simple computation yields ( η ( α · N ))Λ z − ( η ( α · N ))Λ z + = Λ z + ( η ( α · N ))Λ z − ( η ( α · N )) = 1 + η . Therefore, Λ z + is invertible with (Λ z + ) − = 4 η ( η +4) − ( 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 (Σ) , using Theorem 5.1we then get the first statement. Now, by Remark 2.1 it follows that (Φ ¯ z ) ∗ is bounded from L ( R ) onto H / (Σ) . That Sp( H κ ) is characterized by (5.64) is a consequence of the Birman-Schwingerprinciple and the resolvent formula. This completes the proof of the proposition. (cid:3) To finish this part, we discuss briefly the particular case µ = ± ǫ . Given ǫ ∈ R \ { } and let H ǫ, ± ǫ be as in Proposition 3.5. Note that from (3.32) we have Λ − Λ + = 14 ǫ P ± − m ǫ ( S ) P ± . (5.65)Since S is bounded from H − / (Σ) to H / (Σ) and hence compact on L (Σ) (see [36, Theorem6.11] for example), we get the following result. Proposition 5.6.
Let ǫ ∈ R \ { } and assume that Σ is a compact Lipschitz surface. Then H ǫ, ± ǫ isself-adjoint and dom( H ǫ, ± ǫ ) = n u + Φ[ g ] : u ∈ H ( R ) , g ∈ P ± H / (Σ) , P ± t Σ u = − P ± Λ + [ g ] o . Proof.
This readily follows from the compactness and regularization property of the operator S and Theorem 5.2. (cid:3) δ -interactions supported on the boundary of a C ,γ -domain. In this part, we discuss howthe smoothness of the surface supporting the singular perturbation influence the regularity of thedomain of the Dirac operator H κ in the non-critical case. As we have already seen in section 3, when Σ is a C -smooth compact surface, from Theorem 3.1 and Remark 3.2 we know that the functions in dom( H κ ) are indeed in H ( R \ Σ) . However, such a result can fails if Σ is less regular. Indeed, thereare two obstacles which prevent us from obtaining such a result. The first one is that ( α · N )Λ + [ g ] should belongs to H / (Σ) , which clearly fails if for example Σ is C -smooth. The second reasonis that we need also to extend the anticommutator { α · N , C Σ } to a bounded operator from L (Σ) to H / (Σ) . Although again, we know that behind this operator there are components of the Riesztransforms as well as the principale value of the harmonic double layer operator and its adjoint, whichdo not have this property, even if Σ is C , / -smooth.In the following, we assume that Ω + is a bounded C ,γ -smooth domain with γ ∈ (0 , , and we set Σ = ∂ Ω + . For all s ∈ (0 , γ ) , we denote by H s (Σ) the Sobolev space endowed with norm k g k H s (Σ) := Z Σ | g ( x ) | dS ( x ) + Z Σ Z Σ | g ( x ) − g ( y ) | | x − y | s ) dS( y )dS( x ) . (5.66)Note that this definition is equivalent to the one given in Section 2; cf. [36]. The main result of thissubsection reads as follows: Theorem 5.8.
Let κ ∈ R such that sgn( κ ) = 0 , and let H κ be as in Definition 2.1. Then H κ isself adjoint and the following hold:(i) If γ / , then for all s < γ we have dom( H κ ) ⊂ (cid:8) u + Φ[ g ] : u ∈ H ( R ) , g ∈ H s (Σ) , t Σ u = − Λ + [ g ] (cid:9) ⊂ H / s ( R \ Σ) . (ii) If γ > / , then dom( H κ ) = n u + Φ[ g ] : u ∈ H ( R ) , g ∈ H / (Σ) , t Σ u = − Λ + [ g ] o ⊂ H ( R \ Σ) . IRAC OPERATORS WITH δ -SHELL INTERACTIONS 41 Proposition 5.7.
There is a constant
C > such that for all x, y ∈ Σ , it hold that | N ( x ) · ( x − y ) | C | x − y | γ . (5.67) Proof.
Since Σ is C ,γ -smooth, it suffies to prove the statement for | x − y | < . Without loss ofgenerality (after translation and rotation if necessary), we may assume that x = 0 and N ( x ) = (0 , , .There is a C ,γ -smooth function φ : B (0 , ⊂ R −→ R such that φ (0) = 0 , |∇ φ (0) | = 0 and B (0 , ∩ Σ = { x = ( x , x , x ) : x = φ ( x , x ) } . Then we get | N ( x ) · ( x − y ) | = | y | = | φ ( y , y ) | C | y | γ . (5.68)Therefore the statement is proven since Σ is compact. (cid:3) In the following proposition, we prove that the anticommutator of the Cauchy operator C Σ andthe multiplication operator by ( α · N ) is bounded from L (Σ) to H s (Σ) , for all s ∈ (0 , γ ) . Thisresult should be compared to [9, Proposition 3.10], where the authors showed that for Σ a C -smoothcompact surface, the commutator of the Cauchy operator C Σ with a Hölder continuous function oforder a ∈ (0 , is bounded from L (Σ) to H s (Σ) , for all s ∈ (0 , a ) . In fact, both results are identicalmodulo a slight change of the assumptions. Lemma 5.2.
Suppose that Σ is C ,γ . Then, for all s ∈ (0 , γ ) , the anticommutator { α · N , C Σ } is abounded operator from L (Σ) to H s (Σ) . Proof.
Let g ∈ L (Σ) , in the same manner as in the proof of Lemma 3.1, we can see that { α · N , C Σ } [ g ]( x ) = Z 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 | iπ | x − y | ( N ( x ) · ( x − y )) I − e − m | x − y | − iπ | x − y | ( N ( x ) · ( x − y )) I . As Σ is C ,γ -smooth, there is a constant C > 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 | − . Hence the integraloperator T K ′ is not singular, and since N is in the Hölder class C ,γ (Σ) , we then can adapt the proofof [43, Proposition 2.8] (see also [9, Proposition 3.10]) and show that T K ′ is bounded from L (Σ) to H s (Σ) for all s ∈ (0 , γ ) . Finally, the fact that ˜ K ∗ is bounded from L (Σ) onto H s (Σ) , for all s ∈ (0 , γ ) , follows by [40, p. 165]. (cid:3) 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 secondstatement follows by the same method as in Theorem 3.1 . Indeed, fix γ ∈ (0 , and assume that Σ is C ,γ , and let g ∈ L (Σ) such that Λ + [ g ] ∈ H / (Σ) . Note that multiplication by N is bounded in H s (Σ) for all s ∈ [0 , γ ) (cf. [9, Lemma A.2]) and C Σ is bounded from H / (Σ) into itself. Therefore,we obtain that Λ − Λ + [ g ] ∈ H s (Σ) , for all s ∈ [0 , γ ) . Now, note that Λ − Λ + = 1sgn( κ ) − − { α · N , C Σ } ( α · N ) C Σ + 2 mµ sgn( κ ) S + η sgn( κ ) { α · N , C Σ } , here we used the fact that { α · 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 ) (cid:18) Λ − Λ + + { α · N , C Σ } ( α · N ) C Σ − η sgn( κ ) { α · N , C Σ } − mµ sgn( κ ) S (cid:19) [ g ] , (5.70)As C Σ ( α · N ) is bounded from L (Σ) into itself and S is bounded from L (Σ) to H / (Σ) , usingLemma 5.2, from (5.70) it follows that g ∈ H s (Σ) , for all s ∈ [0 , γ ) . Since for all s ∈ [0 , / theoperator Φ gives rise to a bounded operator Φ : H s (Σ) −→ H / s ( R \ Σ) (this follows by adaptingthe same arguments as [9, Proposition 3.6]), we get then the inclusions in (i) . In particular, if γ > / ,we then obtain that g ∈ H / (Σ) and therefore Φ[ g ] ∈ H ( R \ Σ) . This gives the equality in (ii) and the theorem is shown. (cid:3) Remark 5.5.
Note that if Σ is C ,γ with γ ∈ (1 / , and sgn( κ ) / ∈ { , } , then using the sametechnique 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 (Σ) to H / (Σ) , by duality, we can extend it to a bounded operator from H − / (Σ) to L (Σ) . Hence, by iterating twice the same argument to those of the proof of Theorem3.1, we then get that dom( H ∗ κ ) = n u + Φ[ g ] : u ∈ H ( R ) , g ∈ H / (Σ) , t Σ u = − Λ + [ g ] o , 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 interactionswhich 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 fieldis given by (see [42]): ˜ H = α · ( − i ∇ − eA ( x )) + mβ + eφ el ( x ) I , (6.1)where e is the charge of the particle, φ el ( x ) is the electric field, and A ( x ) is the magnetic vectorpotential. 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 (cid:18) iβ ( α · E ( x )) − β (( α × α ) · B ( x )) (cid:19) . (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 > , if x ∈ S R = { x ∈ R : | 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 satisfying the assumption (H1) , we can consider the following Diracoperator H + V υ = H + V υ δ Σ , υ ∈ R . (6.4) IRAC OPERATORS WITH δ -SHELL INTERACTIONS 43 and called it Dirac operator with anomalous magnetic δ -shell interactions of strength υ . As we alreadymentioned in the introduction, when finalizing the current manuscript, it turns out that the authorsof the article [14] considered this problem in dimension two, and their work was also in the final phaseof preparation. However, instead of deriving the potential V υ as we had done here, they rigorouslyproved that the two-dimensional analog of V υ can be approximated by regular shrinking potentials ofmagnetic type, and hence they justified the fact that V υ is a "magnetic" δ -shell interactions.Recall the matrix γ defined in (4.14), we set V ζ = ζγ , for all ζ ∈ R . To our knowledge, thepotential V ζ does not seem to have a physical interpretation, but mathematically, it has the samecharacteristics as the electrostatic potential when ζ = ± ; 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 + 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, becauseof the presence of anomalous magnetic potential, several commutativity properties are no longer truein this case. In addition, H ,υ (i.e ζ = 0 ) has the particularity of combining two important phenomenathat we have seen before. In fact, as it was indicated in the introduction, in the critical case, H , ± 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: Λ z ± = 1 ζ + υ ( ζγ + iυβ ( α · N )) ± C z Σ . (6.6)Since iυβ ( α · N ) is C -smooth and symmetric, it follows that Λ z ± are bounded from L (Σ) onto itself(respectively from H / (Σ) onto itself). Moreover, Λ z ± are self-adjoint on L (Σ) , for all z ∈ ( − m, m ) .Now, using the same notations as before, the Dirac operator H ζ,υ (acting in L ( R ) ) is defined onthe domain dom( H ζ,υ ) = (cid:8) u + Φ[ g ] : u ∈ H ( R ) , g ∈ L (Σ) , t Σ u = − Λ + [ g ] (cid:9) , (6.7)and the potential V ζ,υ is defined by: V ζ,υ ( ϕ ) = 12 ( ζγ + 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 ) = H u , for all ϕ = u + Φ[ g ] ∈ dom( H ζ,υ ) .We remind the reader that ˜Λ z ± denotes the continuous extension of Λ z ± defined from H − / (Σ) onto itself.. Using the same method as in Section 3, one can show that H ζ,υ is closable and the domainof the adjoint is given by dom( H ∗ ζ,υ ) = n u + Φ[ g ] : u ∈ H ( R ) , g ∈ H − / (Σ) , t Σ u = − ˜Λ + [ g ] o . (6.9)In the following, we briefly discuss the basic spectral properties of H ζ,υ in the non-critical case, i.e ζ + υ = 4 . The following theorem gathers the most important properties of H µ,υ . Theorem 6.1.
Let ζ, υ ∈ R such that ζ + υ = 0 , . Then H ζ,υ is self adjoint and we have dom( H ζ,υ ) = n u + Φ( g ) : u ∈ H ( R ) , g ∈ H / (Σ) , t Σ u = − Λ + [ g ] o ⊂ H ( R \ Σ) . (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 ) − = ( H − z ) − − Φ z (Λ z + ) − (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 theabove theorem, we need the following proposition. Proposition 6.1.
Let z ∈ C \ (( −∞ , − m ] ∪ [ m, ∞ )) . Then, the commutator [ β ( α · N ) , C z Σ ] gives riseto a bounded operator [ β ( α · N ) , C z Σ ] : H − / (Σ) → H / (Σ) . (6.12) In particular, [ β ( α · N ) , C z Σ ] is compact in L (Σ) . Proof.
Let x, y ∈ Σ , using (3.14) a trivial verification shows that β ( α · N ( x ))( α · ( x − y )) − ( α · ( x − y )) β ( α · N ( y )) =( α · ( x − y )) β ( α · ( N ( y ) − N ( x ))+ ( N ( x ) · ( x − y )) β. (6.13)Now, let g ∈ L (Σ) , 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 by T z [ g ]( x ) = Z Σ K z ( x, y ) g ( y ) dS ( y ) , (6.15)with K z ( x, y ) = β e i √ z − m | x − y | π | x − y | (1 − i p z − m | x − y | ) (cid:18) ( α · ( x − y ))( α · ( N ( x ) − N ( y )) − N ( x ) · ( x − y )) I (cid:19) . As N is C -smooth and S z is bounded from H − / (Σ) to H / (Σ) , it follows that β [( α · N ) , S z ] and { ( α · N ) , S z } are bounded from H − / (Σ) to H / (Σ) . Now, that T z is bounded from H − / (Σ) to H / (Σ) 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. (cid:3) 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 + υ − − C z Σ ( α · N ) { α · N , C z Σ } ± ζζ + υ [ γ , C z Σ ] ± iυζ + υ [ β ( α · N ) , C z Σ ] . A straightforward computation yields [ γ , C z Σ ] = 2 mγ βS . Using this, it follows that Λ z ∓ Λ z ± = 1 ζ + υ − − C z Σ ( α · N ) { α · N , C z Σ } ± mζζ + υ γ βS z ± iυζ + υ [ β ( α · N ) , C z Σ ] . (6.16)As ζ + υ = 0 , , 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 directconsequence of Theorem 5.1. That dom( H ζ,υ ) is given by (6.10) follows using the same argumentsas 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. (cid:3) IRAC OPERATORS WITH δ -SHELL INTERACTIONS 45 Remark 6.1.
Note that if Σ satisifies the assumption (H3) , then using the same arguments as inSubsection 5.1 yield that H ζ,υ is self-adjoint with dom( H ζ,υ ) ⊂ H / ( R \ Σ) . If Σ satisifies theassumption (H2) , then Theorem 6.1 is still holds true. In the following theorem, we discuss the self-adjointness of H µ,υ in the critical case, i.e ζ + υ = 4 .We mention that assertions (a) and (c) have already been proven in [29], where the author studied theinner part of H , ± which acts on Ω + , known as the Dirac operator with zig-zag boundary conditions(see also [14] for the two-dimensional case). Theorem 6.2.
Let ζ, υ ∈ R such that ζ + υ = 4 , then H ζ,υ is essentially self -adjoint and we have dom( H ζ,υ ) = n u + Φ[ g ] : u ∈ H ( R ) , g ∈ H − / (Σ) , t Σ u = − ˜Λ + [ g ] o , Moreover, the following hold true:(i) a ∈ Sp( H ζ,υ ) ⇐⇒ − a ∈ Sp( H ζ,υ ) .(ii) for all z ∈ C \ R the operator ˜Λ z + is bounded invertible from H − / (Σ) to H / (Σ) and wehave ( H ζ,υ − z ) − = ( H − z ) − − Φ z (˜Λ z + ) − (Φ z ) ∗ . (6.17) (iii) if ζ = 0 , then Σ becomes impenetrable and it holds that H ,υ = H Ω + υ ⊕ H Ω − υ = ( − iα · ∇ + mβ ) ⊕ ( − iα · ∇ + mβ ) , (6.18) where H Ω ± υ are the self-adjoint Dirac operators defined on dom( H Ω ± υ ) = (cid:8) ϕ ± ∈ L (Ω ± ) : ( α · ∇ ) ϕ ± ∈ L (Ω ± ) and P ∓ ,υ t Σ ϕ ± = 0 (cid:9) , where the boundary condition has to be understood as an equality in H − / (Σ) , and P ± ,υ arethe projectors defined by P ± ,υ = 12 (cid:16) I ± υ β (cid:17) . (6.19) Furthermore, we have(a) − m and m are eigenvalues of H ,υ with infinite multiplicities.(b) Sp( H ,υ ) = ( −∞ , − m ] ∪ [ m, + ∞ ) .(c) There is a sequence ( λ j ( m )) j ∈ N ⊂ Sp( H ,υ ) such that each λ j ( m ) is an eigenvalue withfinite multiplicity, with λ j ( m ) > m for all j ∈ N , and λ j ( m ) −→ ∞ as j −→ ∞ . Proof.
Let us show the first statement. The proof is a relatively straightforward modification ofthe technique used in the proof of Theorem 3.1. Indeed, as H ζ,υ is closable the only thing left to proveis 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 / (Σ) bea sequence of functions that converges to g in H − / (Σ) . Set g j := 12 ( ζγ + iυβ ( α · N )) (cid:16) ˜Λ + [ g ] + Λ − [ h j ] (cid:17) , ∀ j ∈ N . (6.20)Clearly, ( g j ) j ∈ N ⊂ H / (Σ) and g j −−−→ j →∞ g in H − / (Σ) . Since Λ + is bounded from H / (Σ) ontoitself, we then get that (Λ + [ g j ]) j ∈ N ⊂ H / (Σ) . Now, remark that −
12 ( ζγ + iυβ ( α · N )) ˜Λ − [ g ] = − g + 12 ( ζγ + iυβ ( α · N )) ˜Λ + [ g ] (6.21)Using this, it follows that g j := g −
12 ( ζγ + iυβ ( α · N )) ˜Λ − [ h j − g ] , ∀ j ∈ N . (6.22) Therefore we obtain ˜Λ + [ g j − g ] = −
12 ˜Λ + ( ζγ + iυβ ( α · N )) ˜Λ − [ g − h j ] = (cid:16) ˜Λ + ˜Λ − ˜Λ − + Λ + ˜Λ + ˜Λ − (cid:17) [ h j − g ] . (6.23)From Lemma 3.1, Proposition 6.1 and (6.16) it follows that ˜Λ ± ˜Λ ∓ are bounded from H − / (Σ) to H / (Σ) . Therefore, (6.23) yields that Λ + [ g j ] −−−→ j →∞ ˜Λ + [ g ] , in H / (Σ) . (6.24)Now, let ( v j ) j ∈ N ⊂ H ( R ) such that t Σ v j = ˜Λ + ( ζγ + iυβ ( α · N )) ˜Λ − [ h j − g ] , for all j ∈ N (thiscan be easily done by taking the average of linear extensions of t Σ v j on Ω ± , since t Σ v j ∈ H / (Σ) by definition). Set u j = u − v j and define ϕ j := u j + Φ[ g j ] . It is clear that u j ∈ H ( R ) and t Σ u j = − Λ + [ g j ] in H / (Σ) , 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 − / (Σ) as j −→ ∞ , using the continuity of ˜Λ ± ˜Λ ∓ it follows that ( ϕ j , H ζ,υ ϕ j ) −−−→ j →∞ ( ϕ, H ∗ ζ,υ ϕ ) in L ( R ) . Therefore H ∗ ζ,υ ⊂ H ζ,µ and hence H ζ,υ is self-adjoint with dom( H ζ,υ ) = n u + Φ[ g ] : u ∈ H ( R ) , g ∈ H − / (Σ) , t Σ u = − ˜Λ + [ g ] o , (6.25)this finishes the proof of the first statement. In order to continue the proof of the theorem we use thedefinition of dom( H ζ,υ ) with transmission condition. As in Definition 3.1, using the Plemelj-Sokhotskiformula, 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 ) = (cid:26) ϕ = ( ϕ + , ϕ − ) ∈ L (Ω + ) ⊕ L (Ω − ) : ( α · ∇ ) ϕ ± ∈ L (Ω ± ) and (cid:18)
12 ( ζγ + iυβ ( α · N )) + i ( α · N ) (cid:19) t Σ ϕ + = − (cid:18)
12 ( ζγ + iυβ ( α · N )) − i ( α · N ) (cid:19) t Σ ϕ − (cid:27) , where the transmission condition holds in H − / (Σ) . Let us show item (ii) , for that recall that theoperator 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 ( R ) . (6.27)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 ,υ ) = (cid:26) ϕ = ( ϕ + , ϕ − ) ∈ L (Ω + ) ⊕ L (Ω − ) : ( α · ∇ ) ϕ ± ∈ L (Ω ± ) and i ( α · N ) P − ,υ t Σ ϕ + = i ( α · N ) P + ,υ t Σ ϕ − (cid:27) , Since P ± ,υ are projectors, it easily follows that a function ϕ = ( ϕ + , ϕ − ) ∈ L (Ω + ) ⊕ L (Ω − ) with ( α · ∇ ) ϕ ± ∈ L (Ω ± ) belongs to dom( H ,υ ) if and only if P ∓ ,υ t Σ ϕ ± = 0 (this should be understoodas an equality in H − / (Σ) ). 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 ofProposition 4.1, it is clear that the Birman-Schwinger principle applies also for a = ± m (if g ∈ Kr(Λ m ± ) ,then ϕ = m Φ m [ g ]+ Φ[ g ] is an eigenfunction of H ,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 (Σ) . Let g ∈ L (Σ) ,using [2, Lemma 4.1] it easily follows that Λ m + ( iυβ ( α · N ))Λ m − ( iυβ ( α · N )) = (1 − υ g ! = 0 . (6.28) IRAC OPERATORS WITH δ -SHELL INTERACTIONS 47 Therefore m is an eigenvalue of H ,υ with infinite multiplicity. By (i) it follows that − m is also aneigenvalue of H ,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 D Ω ± υ ψ = ( − iα · ∇ + mβ ) ψ, ψ ∈ dom( D Ω ± υ ) = (cid:8) ϕ ± ∈ H (Ω ± ) : P ∓ ,υ t Σ ϕ ± = 0 (cid:9) . (6.29)Then, one can check easily that D Ω ± υ are symmetric, closable operators. Moreover, it holds that D Ω ± υ = H Ω ± υ . Indeed, denote by Q Ω ± υ the quadratic form associated to ( D Ω ± υ ) . Given ϕ ∈ dom( D Ω ± υ ) ,using the Green’s formula and the boundary conditions, it easily follows that: Q Ω ± υ [ ϕ ] = k ( α · ∇ ) ϕ k L (Ω ± ) + m k ϕ k L (Ω ± ) . (6.30)Hence, we get Q Ω ± υ [ ϕ ] > m k ϕ k L (Ω ± ) . (6.31)Thus ( D Ω ± υ ) is lower semi-bounded. Therefore, by [21, Theorem 6.3.2] it follows that ( H Ω ± υ ) is theFriedrichs extension of ( D Ω ± υ ) and it holds that Sp( H Ω ± υ ) ⊂ ( −∞ , − m ] ∪ [ m, + ∞ ) . (6.32)Now, let ( − ∆ Ω ± ) be the Dirichlet realization of ( − ∆) in Ω ± , with domain H (Ω ± ) ∩ H (Ω ± ) . Usingthe Weyl’s theorem and the fact that H (Ω + ) is compactly embedded in L (Ω + ) , it is not hard toshow that Sp( − ∆ Ω − + m ) = [ m , + ∞ ) , Sp( − ∆ Ω + + m ) = Sp disc ( − ∆ Ω + + m ) = { m + λ j , j ∈ N } , (6.33)with λ j > for all j ∈ N , and λ j −→ ∞ as j −→ ∞ . Now assume that υ = 2 (the case υ = − canbe recovered with the same arguments), using the boundary condition it follows that ϕ = ϕ ϕ ! ∈ dom( D Ω + ) = ⇒ ϕ ∈ H (Ω + ) , ϕ = ϕ ϕ ! ∈ dom( D Ω − ) = ⇒ ϕ ∈ H (Ω − ) . (6.34)Denote by ˜ Q Ω ± the quadratic form associated to ( − ∆ Ω ± + m ) I . Using (6.34) and the Green’sformula, from (6.30) it follows that Q Ω + [ ϕ ] = k ( σ · ∇ ) ϕ k L (Ω + ) + m k ϕ k L (Ω + ) + ˜ Q Ω + [ ϕ ] , ∀ ϕ ∈ dom( D Ω + ) , Q Ω − [ ϕ ] = k ( σ · ∇ ) ϕ k L (Ω − ) + m k ϕ k L (Ω − ) + ˜ Q Ω − [ ϕ ] , ∀ ϕ ∈ dom( D Ω − ) . (6.35)Using this and assertion (i) , it follows that λ j ( m ) = ± p m + λ j is an eigenvalue of H ,υ with finitemultiplicity, and we have Sp( H Ω ± υ ) = ( −∞ , − m ] ∪ [ m, + ∞ ) , (6.36)which yields (b) and (c) . This achieves the proof of theorem. (cid:3) Remark 6.2.
Let ζ = ± and let H ± , be as in Theorem 6.2. Given ( ϕ + , ϕ − ) ∈ dom( H ζ, ) , wewrite ϕ ± = ( ϕ ± , , ϕ ± , ) . Then, one can write the transmission condition as follows: t Σ ϕ + , = iζ σ · N ) t Σ ϕ − , , t Σ ϕ + , = iζ σ · N ) t Σ ϕ − , . (6.37) Thus, we deduce that H ζ, coincide with the Dirac operator coupled with the electrostatic δ -interactionsof strength − ζ . Thus, in this sense, one can consider the potential V ζ as an electrostatic potential for ζ = ± . Remark 6.3.
If one assume that Σ satisfies the assumption (H2) , then H ζ,υ is essentially self-adjoint,when ζ + υ = 4 . In particular, if υ = 0 , then Remark 6.2 and Theorem 4.2 yield that Sp ess ( H ± , ) = (cid:0) − ∞ , − m (cid:3) ∪ { } ∪ (cid:2) m, + ∞ (cid:1) . (6.38) However, if ζ = 0 , then there is no embedded eigenvalues in the essential spectrum of H , ± , and wehave Sp( H , ± ) = Sp ess ( H , ± ) = (cid:0) − ∞ , − m (cid:3) ∪ (cid:2) m, + ∞ (cid:1) . (6.39) 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 tothank Mihalis Mourgoglou for various fruitful discussions on topics related to the analysis of PDEs inrough domains. I am particularly grateful to him for suggesting several references related to Section 5his careful reading of a preliminary version of that section. This project is based upon work supportedby the Government of the Basque Country under Grant PIFG / "ERC Grant: Harmonic Analysisand Differential Equations", researcher in charge: Luis Vega. References [1] P. Aiena:
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Departamento de Matemáticas, Universidad del País Vasco, Barrio Sarriena s/n 48940 Leioa, SPAIN,and Université de Bordeaux, IMB, UMR 5251, 33405 Talence Cedex, FRANCE.
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