Extension operator for the MIT bag model

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Extension operator for the MIT bag model

Naiara Arrizabalaga, Loïc Le Treust, Nicolas Raymond

To cite this version:

Naiara Arrizabalaga, Loïc Le Treust, Nicolas Raymond. Extension operator for the MIT bag model.

Annales de la Faculté des Sciences de Toulouse. Mathématiques., Université Paul Sabatier _ Cellule Mathdoc In press. �hal-01540149�

N. ARRIZABALAGA, L. LE TREUST, AND N. RAYMOND

Abstract. This paper is devoted to the construction of an extension operator for the MIT bag Dirac operator on aC^2,1 bounded open set of R³ in the spirit of the extension theorems for Sobolev spaces. As an elementary byproduct, we prove that the MIT bag Dirac operator is self-adjoint.

Contents

1. Introduction 1

1.1. The MIT bag Dirac operator 1

1.2. Main result 3

2. Proof of the main theorem 4

2.1. Extension operator in the half-space case 5

2.2. Proof of Lemma 2.1 7

2.3. Proof of Corollary 1.7 12

Appendix A. Some elementary properties 12

Acknowledgments 13

References 13

1. Introduction

1.1. The MIT bag Dirac operator. In the whole paper, Ω denotes a fixed bounded domain of R³ with C^2,1 boundary. The Planck constant and the velocity of light are assumed to be equal to 1. Let us recall the definition of the Dirac operator associated with the energy of a relativistic particle of mass m P R and spin ¹₂ (see [12]). The Dirac operator is a first order differential operator, acting on L²pΩ,C⁴q in the sense of distributions, defined by

(1.1) H“α¨D`mβ , D“ ´i∇,

2010Mathematics Subject Classification. 35J60,81Q10, 81V05.

Key words and phrases. Dirac operator, Hadron bag model, Relativistic particle in a box, MIT bag model.

1

where α“ pα1, α2, α3q,β andγ5 are the 4ˆ4 Hermitian and unitary matrices given by

β “

ˆ 12 0 0 ´12

˙

, γ5 “

ˆ 0 12

12 0

˙

, αk “

ˆ 0 σk

σk 0

˙

for k“1,2,3. Here, the Pauli matrices σ1, σ2 and σ3 are defined by

σ1 “

ˆ 0 1 1 0

˙

, σ2 “

ˆ 0 ´i i 0

˙

, σ3 “

ˆ 1 0 0 ´1

˙ , and α ¨X denotes ř³

j“1αjXj for any X “ pX1, X2, X3q. Let us now impose the boundary conditions under consideration in this paper and define the associated unbounded operator.

Notation 1.1. In the following , Γ :“ BΩ and for all x P Γ, npxq is the outward- pointing unit normal to the boundary.

Definition 1.2. The MIT bag Dirac operator pH_m^Ω,DompH_m^Ωqq is defined on the domain

DompH_m^Ωq “ tψ PH¹pΩ,C⁴q : Bψ “ψ on Γu, with B“ ´iβpα¨nq, byH_m^Ωψ “Hψ for allψ P DompH_m^Ωq. Note that the trace is well-defined by a classical trace theorem.

Notation 1.3. We will denote H “ H_m^Ω when there is no risk of confusion. We denote x¨,¨y the C⁴ scalar product (antilinear w.r.t. the left argument) and x¨,¨yU

the L² scalar product on the set U.

Remark 1.4. The operatorpH_m^Ω,DompH_m^Ωqqis symmetric (see Lemma A.2) and densely defined.

Remark 1.5. The operator B defined for all x P Γ is a Hermitian matrix which satisfies B² “ 14 so that its spectrum is t˘1u. Both eigenvalues have multiplicity two. Thus, the MIT bag boundary condition imposes the wavefunctions ψ to be eigenvectors of B associated with the eigenvalues `1 . This boundary condition is chosen by the physicists [8] so as to get a vanishing normal flow at the bag surface

´in¨j“0 at the boundary Γ where the current densityj is defined by j“ xψ, αψy.

Let us now describe our main result.

1.2. Main result. The aim of this paper is to construct a bounded extension op- erator from the domain of H_m^Ω into H¹pR³q⁴ in the spirit of extension operators for Sobolev spaces (see for instance [6, Section 9.2]). As we will see, a motivation to con- struct such an operator is to prove self-adjointness. Our main result is the following one.

Theorem 1.6. Let Ω be a nonempty, bounded and C^2,1 open set in R³ and m P R. There exist a constant Cą0 and an operator

P :DompHq ÑH¹pR³q⁴ such that P ψ_|Ω “ψ and

}P ψ}²H¹pR³q ďC´

}ψ}²L²pΩq` }α¨Dψ}²L²pΩq

¯ , for all ψ P DompHq.

Corollary 1.7. The operator pH,DompHqqis self-adjoint.

Remark 1.8. The proofs of Theorem 1.6 and Corollary 1.7 rely on the construction of an extension operator

P :DompH^‹q Ñ H¹pR³q⁴, where H^‹ is the adjoint ofH. Thus,

DompH^‹q ĂH¹pΩq⁴,

and then the inclusion DompH^‹q Ă DompHq easily follows. Since H is symmetric (see Lemma A.2), we get DompH^‹q “DompHq.

Remark 1.9. Note that the existence of an extension operator P :DompH^‹q Ñ H¹pR³q⁴

is a necessary condition forH to be self-adjoint. Indeed, ifH is self-adjoint, we have the bounded injections:

DompHq “DompH^‹qãÑH¹pΩq⁴ ãÑH¹pR³q⁴.

To see this, let us recall that, if Ω is C^1,1, we have (see [1, Theorem 1.5] and [7, p.379]):

(1.2) @ψ P DompHq, }α¨∇ψ}²L²pΩq“ }∇ψ}²L²pΩq` 1 2

ż

BΩ

κ|ψ|²ds ,

whereκis the trace of the Weingarten map. From this formula, we can show that the injection DompHq “ DompH^‹q ãÑ H¹pΩq⁴ is bounded. The embedding H¹pΩq⁴ ãÑ H¹pR³q⁴ is given by the extension theorem for Sobolev spaces (see for instance [9, Theorem 3.9]) which requires C^0,1 regularity on Ω.

Remark 1.10. Self-adjointness results have already been obtained in the case of C⁸- boundaries in [5] through Calder´on projections and sophisticated pseudo-differential techniques. In two dimensions, C²-boundaries are considered in [4] (see also [11]) by using Cauchy kernels and the Riemann mapping theorem. The recent paper [10] tackles the three dimensions case for C² boundaries via Calder´on projections.

The reader may also consult the survey [2] in the context of spin geometry or [3, Theorem 4.11] devoted to the smooth case. Let us also mention that more general local boundary conditions are considered in [5, 4].

2. Proof of the main theorem

We denote byLpE, Fqthe set of continuous linear applications fromE toF where E and F are Banach spaces. We recall that the domain ofH is independent of m:

DompHq “ tψ P H¹pΩq⁴, Bψ “ψ onBΩu, and that the domain of the adjoint H^‹ is defined by

DompH^‹q “ tψ PL²pΩq⁴, Lψ PLpL²pΩq⁴,Cqu, where

L_ψ :ϕ PDompHq ÞÑ xψ, Hϕy^Ω PC.

The proof is divided in several steps. First, we construct an extension map on the domain of the adjoint as follows.

Lemma 2.1. There exists an operator

P :DompH^‹q Ñ H¹pR³q⁴ such that P ψ_|Ω “ψ and

}P ψ}²H¹pR³qďC´

}ψ}²L²pΩq` }α¨Dψ}²L²pΩq

¯ , for all ψ P DompH^‹q.

We get as a consequence that

DompH^‹q ĂH¹pΩq⁴.

The second step in the proof of Theorem 1.6 relies on a study of the boundary conditions satisfied by the functions of DompH^‹q.

Let us remark that, without loss of generality, we can assume thatm“0 since the operator βm is bounded (and self-adjoint) from L²pΩq⁴ into itself so that DompH^‹q is independent of m.

2.1. Extension operator in the half-space case. In this section, we consider the case when Ω“R³

` and we establish the existence of an extension operator.

Lemma 2.2. There exists an operator

P :DompH^‹q Ñ tψ PL²pR³q⁴, α¨DψPL²pR³q⁴u “H¹pR³q⁴ such that P ψ_|R³_` “ψ and

}P ψ}²H¹pR³q “ }P ψ}²L²pR³q` }∇P ψ}²L²pR³q“2´

}ψ}²L²pR³

`q` }α¨Dψ}²L²pR³

`q

¯ . Proof. The outward-pointing normal n is equal to ´e3 “ p0,0,´1q^T so that the boundary condition is

iβα3ψ “ψ , on BR³

`. Let us diagonalize the matrix iβα3 appearing in the boundary condition.

We introduce the matrix

T “ 1

?2

ˆ 12 i12

i12 12

˙ . We have

T βT^‹ “

ˆ 0 ´i1² i1² 0

˙

, T αkT^‹ “αk, Tpiβα3qT^‹ “

ˆ σ3 0 0 ´σ3

˙

“:B⁰. We consider Hr “ T HT^‹. The operator Hr is defined by Hψr “ α ¨Dψ for any ψ PDompHrq where

DompHrq “ ψ PH¹pR³

`q, B⁰ψ “ψ, onBR³

`

(

“ ψ PH¹pR³

`q, ψ² “ψ³ “0 on BR³

`

(2.1) (

and ψ “ pψ¹, ψ², ψ³, ψ⁴q^T. This unitarily equivalent representation of the Dirac operator is called the supersymmetric representation (see [12, Appendix 1.A]). This expression of the domain makes more apparent the fact that the MIT bag boundary condition is intermediary between the Dirichlet and Neumann boundary conditions.

Let us denote by S : R³ Ñ R³ and Π : R³ Ñ R³ the orthogonal symmetry with respect to BR³

` and the orthogonal projection onBR³

`. Based on (2.1), we define the extension operatorPr for ψ PDompHr^‹q as follows:

P ψr px, y, zq “

"

ψpx, y, zq, if z ą0

pψ¹,´ψ²,´ψ³, ψ⁴q^Tpx, y,´zq “B⁰pψ˝Sq px, y, zq, if z ă0 for px, y, zq P R³. In other words, we extend ψ¹, ψ⁴ by symmetry and ψ², ψ³ by antisymmetry.

Let us get back to the standard representation and define the extention operator P for ψ PDpH^‹q and px, y, zq PR³ as follows :

P ψpx, y, zq “T^‹P T ψr px, y, zq “

#ψpx, y, zq, if z ą0, pB˝Πq pψ˝Sq px, y, zq, if z ă0.

Since Bpsqis a unitary transformation of C⁴ for any sP BR³

`, we get that }P ψ}²L²pR³q“2}ψ}²L²pR³

`q.

Let us study α¨DP ψ in the distributional sense. We have forϕ PD“C₀⁸pR³q that xα¨DP ψ, ϕyD¹ˆD “ xP ψ, α¨Dϕy^R3 “ xψ, α¨Dϕy^R3<sub>`</sub> ` xpB˝Πqψ˝S, α¨Dϕy^R3_´ where x¨,¨yD¹ˆD is the distributional bracket on R³. Since B is Hermitian, commutes with α1,α2 and anti-commutes with α3, we obtain by a change of variables, that

xpB˝Πqψ˝S, α¨Dϕy^R3_´ “ xψ˝S,pB˝Πqα¨Dϕy^R3_´

“ xψ,´ipB˝Πq pα1B^x`α2B^y ´α3B^zqϕ˝SyR³

` “ xψ, α¨DppB˝Πqϕ˝SqyR<sup>3</sup><sub>`</sub>. Hence, we get

xα¨DP ψ, ϕyD¹ˆD “ xψ, α¨Dpϕ` pB˝Πqϕ˝SqyR<sup>3</sup><sub>`</sub>.

Let us remark that the function ϕ` pB˝Πqϕ˝S belongs to DompHq. Indeed, we have that

pB˝Πq pϕ` pB˝Πqϕ˝Sq px, y,0q “ pϕ` pB˝Πqϕ˝Sq px, y,0q for all px, yq PR². Since ψ PDompH^‹q, by a change of variables, we have that

xα¨DP ψ, ϕy^D1ˆD “ xα¨Dψ,pϕ` pB˝Πqϕ˝Sqy<sup>R3</sup><sub>`</sub>

“ xα¨Dψ, ϕy^R3<sub>`</sub>` xpB˝Πq pα¨Dψq ˝S, ϕy^R3_´ . Thus, we obtain that in the distributional sense

α¨DP ψ “χR³

`pα¨Dψq `χR³

´pB˝Πq pα¨Dψq ˝S PL²pR³q so that

}∇P ψ}²L²pR³q “ }α¨DP ψ}²L²pR³q “2}α¨Dψ}²L²pR³

`q.

2.2. Proof of Lemma 2.1. Let us now consider the case of our general Ω. Let us remark that the understanding of the case of the half-space is not sufficient to conclude since curvature effects have to be taken into account (see for instance (1.2)).

The proof of Lemma 2.2 will be used as a guideline for the proof of Lemma 2.1.

Proof. Using a partition of unity and the fact that

tuPL²pR³q⁴ : α¨DuPL²pR³q⁴u “H¹pR³q⁴,

we are reduced to study the case of a deformed half-space. Let us recall the standard tubular coordinates near the boundary of Ω :

η:pU X BΩq ˆ p´T, Tq ÝÑU, px0, tq ÞÑx0´tnpx0q

whereT ą0 andU is a suitable bounded open set ofR³. Since Ω is C², without loss of generality, we can assume that η is a C¹-diffeomorphism such that

ηppUX BΩq ˆ p0, Tqq “ΩXU , ηppU X BΩq ˆ t0uq “ BΩXU . The rest of the proof is divided into four steps:

(a) we introduce a bounded extension operator P :L²pU XΩq ÑL²pUq,

(b) we introduce a map α˜ which extends the α-matrices on U so that, we have }α˜¨DP ψ}L²pUq ďC´

}ψ}²L²pΩXUq` }α¨Dψ}²L²pΩXUq

¯ ,

for any function ψ PDompH^‹q whose support is a compact subset of U XΩ, (c) we show that the norm } ¨ }^V defined on

V “ tv PL²pUq, α˜¨DvPL²pUq, suppv ĂĂUu by

}v}^2V “ }v}²L²` }α˜¨Dv}²L²

is equivalent to the H¹ norm on C₀⁸pUq,

(d) we deduce by a density argument that V ĂH0¹pUq.

Note that the parts of the proof that are almost immediate in the cases of Sobolev spaces have to be studied carefully. Here, the presence of the Dirac matrices introduce some additional difficulties. We tried to stress where the differences occur and where the regularity on Ω is needed.

Step (a). Let us define the symmetry φ_s “ η˝S ˝η^´1 and the projection φ_p “ η˝Π˝η^´1, where S :px, tq ÞÑ px,´tq and Π : px, tq ÞÑ px,0q. For allx0 P BΩXU, let us denote byPpx0qthe matrix of the identity map ofR³ from the canonical basis pe1, e2, e3q to the orthonormal basis pǫ1px0q, ǫ2px0q,npx0qqdefined by

Ppx0q “ MatpId,pe1, e2, e3q,pǫ1px0q, ǫ2px0q,npx0qqq,

where pǫ1px0q, ǫ2px0qq is a basis of the tangent space T_x0BΩ. Up to taking a smaller T, we have, for all x0 P BΩXU,

Jacφspx0q “Ppx0q^´1

¨

˝ 1 0 0 0 1 0 0 0 ´1

˛

‚Ppx0q,

and, for all xPU,

(2.2) 3

2 ě |Jacφspxq|:“ |detJacφspxq| ě 1 2.

Following the idea of the proof of Lemma 2.2, we define the extension operator P :L²pU XΩq Ñ L²pUq

for ψ PL²pU XΩq and xPU as follows:

P ψpxq “

#ψpxq, if xPU XΩ, pB˝φppxqqψ˝φspxq, if xPU XΩ^c. By (2.2) and a change of variables, we get that

}P ψ}^L2pUqďC}ψ}^L2pUXΩq. Step (b). Let us extend the α-matrices as follows:

r αpxq “

#pα1, α2, α3q^T , if xPU XΩ,

|Jacφspxq|B˝φppxq`

Jacφspφspxqqpα1, α2, α3q^T˘

B˝φppxq, if xPU XΩ^c. Let us remark that αrpxqis a column-vector of three matrices and the above matrix product makes sense as a product in the modulus on the ring of the 4ˆ4 Hermitian matrices. For instance, the first matrix αr1pxq is given for xPU XΩ^c by

r

α1pxq “ |Jacφspxq|B˝φppxq

˜ ₃ ÿ

k“1

b1,kαk

¸

B˝φppxq

where Jacφspφspxqq “ pbi,jq^i,j“1,3 PR^3ˆ3. We get for almost every x0 P BΩXU that

|Jacφspx0q|B˝φppx0q`

Jacφspφspx0qqpα1, α2, α3q^T˘

B˝φppx0q

“Bpx0q

¨

˝Ppx0q^´1

¨

˝ 1 0 0 0 1 0 0 0 ´1

˛

‚Ppx0q

¨

˝ α1

α2

α3

˛

‚

˛

‚Bpx0q

“Bpx0q

¨

˝Ppx0q^´1

¨

˝ 1 0 0 0 1 0 0 0 ´1

˛

‚

¨

˝ α¨ǫ1px0q α¨ǫ2px0q α¨npx0q

˛

‚

˛

‚Bpx0q

“Ppx0q^´1Bpx0q

¨

˝ α¨ǫ1px0q α¨ǫ2px0q

´α¨npx0q

˛

‚Bpx0q

“Ppx0q^´1

¨

˝ α¨ǫ1px0q α¨ǫ2px0q α¨npx0q

˛

‚“

¨

˝ α1

α2

α3

˛

‚.

Hence, the application ˜αis continuous onU. Since it is also aC¹-map on both ΩXU and Ω^cXU, we get that ˜αis Lipschitzian. This choice for the extension ofαis made in order to get

r

α¨DP ψ PL²pUq,

in the sense of distributions. Indeed, since ˜αis Lipschitz, we get that, forϕPH0¹pUq, xαr¨DP ψ, ϕyH^´1pUqˆH0¹pUq “ xP ψ,αr¨DϕyU` xP ψ,´idivprαqϕyUXΩ^c.

For xP UXΩ, we also have

prα¨∇ϕqpφspxqq “ |Jacφspφspxqq| pB˝φpαB˝φpq ¨∇pϕ˝φsq pxq and thus

prα¨∇ϕqpφ_spxqq “ |Jacφ_spφ_spxqq|B˝φ_ppα¨∇ppB˝φ_pqϕ˝φ_sqq pxq

´ |Jacφ_spφ_spxqq|B˝φ_ppα¨∇pB˝φ_pqqϕ˝φ_spxq. We deduce that

xP ψ,αr¨DϕyUXΩ^c “ xψ, α¨DppB˝φpqϕ˝φsqyUXΩ

´ xψ,pα¨DpB˝φpqqϕ˝φsyUXΩ . Since ψ P DompH^‹q and the function ϕ` pB˝φpqϕ˝φs : ΩXU Ñ C⁴ belongs to DompHq (since φs and φp are C¹), we get that

xrα¨DP ψ, ϕyH^´1pUqˆH¹0pUq“ xα¨Dψ, ϕ` pB˝φ<sub>p</sub>qϕ˝φ<sub>s</sub>yUXΩ` xP ψ, R ϕyUXΩ^c ,

where R PL⁸pU XΩ^c,C^4ˆ4q is defined by R “ ´idivprαq `i|Jacφs|B˝φp

`Jacφspφsp¨qqpα1, α2, α3q^T˘

¨∇pB˝φpq. By the Riesz theorem, we get αr¨DP ψP L²pUq and

}α˜¨DP ψ}^L2pUq ďC´

}ψ}²L²pΩq` }α¨Dψ}²L²pΩq

¯ , where C ą0 does not depend on ψ.

Step (c). Letϕ PC₀⁸pUq, we have

} ´iαr¨∇ϕ}²L²pUq “ xϕ,p´iαr¨∇q²ϕyU ´ xϕ, divprαq prα¨∇ϕqyUXΩ^c

and

p´iαr¨∇q² “ ´ ÿ3

j,k“1

r

α_jαr_kB²jk` pαr_jB^jαr_kq B^k. Let us define the matrix-valued function A for all xPU by

Apxq “ |Jacφspxq|pJacφspφspxqqqχUXΩ^cpxq `13χUXΩpxq “ pajkpxqq^jk and denote by Ajpxqthe j-th line ofApxq.We get that, for all xPU,

r

α_jpxqrα_kpxq “B˝φ_ppa_j1α1`a<sub>j</sub>2α2`a_j3α3q pa_k1α1`a<sub>k</sub>2α2`a_k3α3qB˝φ_p

“

˜ ₃ ÿ

l“1

ajlakl

¸

14`B˝φp

˜ ÿ

1ďlăsď3

αlαspajlaks´ajsaklq

¸ B˝φp

and

ÿ3

j,k“1

r

αjαrkBjk² “14

ÿ3

j,k“1

AjA^T_kBjk² .

Since, AA^Tpxq “ 1³ for all x P U X BΩ, we get that x ÞÑ AA^Tpxq is a Lipschitz mapping on U and

ÿ3

j,k“1

r

αjαrkBjk² “14div`

AA^T∇˘

´14

ÿ3

j,k“1

`B^jAA^T˘ B^k. Integrating by parts yields

} ´iαr¨∇ϕ}²L²pUqě }A^T∇ϕ}²L²pUq´C}ϕ}L²pUq}∇ϕ}L²pUq

ěc}∇ϕ}²L²pUq´C}ϕ}^L2pUq}∇ϕ}^L2pUq, where

c“mintinf sppAA^Tpxqq, xPUu.

Note that c ą 0 by (2.2). This ensures that the H¹-norm and the } ¨ }^V-norm are equivalent on C₀⁸pUq.

Step (d). Letv P V and pρ_εq^ε a mollifier defined for xPR³ by ρ_εpxq “ 1

ε³ρ1

´x ε

¯ ,

whereρ1 PC₀⁸pR³q,supp ρ1 ĂBp0,1q,ρ1 ě0 and}ρ1}^L1 “1. Let us definevε “v˚ρε

for anyεą0. There exists ε0 ą0 such that for allεP p0, ε0s, the functionv_ε belongs toC₀⁸pUq. Let us temporarily admit that there exists Cindependent of v and εsuch that

}v_ε}^V ďC}v}^V. (2.3)

Then, Step (c) and the fact that vε converges to v inL²pUqensure that V ĂH0¹pUq and the result follows.

It remains to prove (2.3). There exists a constant C ą0 such that }vε}^L2 ďC}v}^L2

and

}rα¨Dvε}^L2 ď }rα¨∇vε´ pαr¨∇vq ˚ρε}^L2 ` } prα¨∇vq ˚ρε}<sup>L2</sup> ď }rα¨∇v<sub>ε</sub>´ pαr¨∇vq ˚ρ<sub>ε</sub>}<sup>L2</sup> `C}rα¨∇v}^L2. By integration by parts, we get, for xPU,

r

α¨∇vεpxq ´ prα¨∇vq ˚ρεpxq

“ ż

R³

r

αpxq ¨ pvpyq∇ρ_εpx´yqqdy´ ż

R³

r

αpyq ¨∇vpyqρ_εpx´yqdy

“ ż

R³prαpxq ´αrpyqq ¨ pvpyq∇ρεpx´yqq dy` ż

R³ pdivαrpyqqvpyqρεpx´yqdy, and by a change of variable

ż

R³prαpxq ´αrpyqq ¨ pvpyq∇ρεpx´yqq dy

“ ż

R³

r

αpxq ´αrpx´εzq

ε ¨ pvpx´εzq∇ρ1pzqq dz. Since αr is Lipschitzian, we get that

››

›› ż

R³

r

αp¨q ´αrp¨ ´εzq

ε ¨ pvp¨ ´εzq∇ρ1pzqq dz

››

››

L²

ďC}v}^L2}| ¨ ||∇ρ1p¨q|}^L1,

and ›

››

› ż

R³pdivαrpyqqvpyqρ_εp¨ ´yqdy

››

››

L²

ďC}v}^L2,

so that (2.3) follows. This ends the proof of Lemma 2.1.

2.3. Proof of Corollary 1.7. Thanks to Lemma 2.1, the set DompH^‹q is included inH¹pΩq⁴. Hence, for anyψ PDompH^‹q, the trace ofψ on the setBΩ is well-defined and belongs toH^1{2pBΩq⁴. By the definition ofDompH^‹qand an integration by parts, we obtain that, for any ϕPDompHq,

0“ xψ, Hϕy^Ω´ xHψ, ϕy^Ω “ xψ,´iα¨nϕyBΩ “ xβψ, ϕyBΩ . Hence, we have, for almost any sP BΩ,

βψpsq PkerpB´14q^K “kerpB`14q, so that

ψpsq PkerpB´1⁴q, and the conclusion follows.

Appendix A. Some elementary properties Lemma A.1. For all x,yPR³, we have

pα¨xqpα¨yq “ px¨yq14`iγ5α¨ pxˆyq, βpα¨xq “ ´pα¨xqβ , βγ5 “ ´γ5β , γ5pα¨xq “ pα¨xqγ5.

Proof. We refer to [12, Appendix 1.B].

In the following lemma, we recall the proof of the symmetry of H.

Lemma A.2. pH,DompHqq is a symmetric operator.

Proof. Since the α-matrices are Hermitian, we have, thanks to the Green-Riemann formula:

(A.1) @ϕ, ψP H¹pΩ,C⁴q, xα¨Dϕ, ψy^Ω “ xϕ, α¨Dψy^Ω` xp´iα¨nqϕ, ψyBΩ . Now we consider ψ, ϕP DompHq. By usingβ² “1⁴ and the boundary condition, we get

xp´iα¨nqϕ, ψyBΩ “ xβϕ, ψyBΩ , so that, we deduce

(A.2) @ϕ, ψ PDpHq, xα¨Dϕ, ψyΩ´ xϕ, α¨DψyΩ “ xβϕ, ψyBΩ .

The left hand side of (A.2) is a skew-Hermitian expression of pϕ, ψq and the right hand side is Hermitian in pϕ, ψq since β is Hermitian. Thus both sides must be

zero.

Acknowledgments. This work was partially supported by the Henri Lebesgue Cen- ter (programme “Investissements d’avenir” – n^o ANR-11-LABX-0020-01). L. L.T.

was partially supported by the ANR project Moonrise ANR-14-CE23-0007-01. N.

A. was partially supported by ERCEA Advanced Grant 669689-HADE, MTM2014- 53145-P (MICINN, Gobierno de Espa˜na) and IT641-13 (DEUI, Gobierno Vasco).

N. A. wishes to thank the IRMAR (Universit´e de Rennes 1) for the invitation and hospitality.

References

[1] N. Arrizabalaga, L. Le Treust, and N. Raymond. On the MIT bag model in the non-relativistic limit.Communications in Mathematical Physics, 2017 (to appear).

[2] C. B¨ar and W. Ballmann. Boundary value problems for elliptic differential operators of first order. InSurveys in differential geometry. Vol. XVII, volume 17 ofSurv. Differ. Geom., pages 1–78. Int. Press, Boston, MA, 2012.

[3] C. B¨ar and W. Ballmann.Guide to Elliptic Boundary Value Problems for Dirac-Type Opera- tors, pages 43–80. Springer International Publishing, Cham, 2016.

[4] R. D. Benguria, S. Fournais, E. Stockmeyer, and H. Van Den Bosch. Spectral gaps of Dirac operators with boundary conditions relevant for graphene.Preprint: arXiv:1601.06607, 2016.

[5] B. Booß-Bavnbek, M. Lesch, and C. Zhu. The Calder´on projection: new definition and appli- cations.J. of Geom. and Phys., 59(7):784–826, 2009.

[6] H. Brezis.Functional analysis, Sobolev spaces and partial differential equations. Springer Sci- ence & Business Media, 2010.

[7] O. Hijazi, S. Montiel, and A. Rold´an. Eigenvalue boundary problems for the Dirac operator.

Comm. Math. Phys., 231(3):375–390, 2002.

[8] K. Johnson. The MIT bag model.Acta Phys. Pol., B(6):865–892, 1975.

[9] J. Neˇcas.Direct methods in the theory of elliptic equations. Springer Monographs in Mathe- matics. Springer, Heidelberg, 2012. Translated from the 1967 French original by Gerard Tronel and Alois Kufner, Editorial coordination and preface by ˇS´arka Neˇcasov´a and a contribution by Christian G. Simader.

[10] T. Ourmi`eres-Bonafos and L. Vega. A strategy for self-adjointness of Dirac operators: Appli- cations to the MIT bag model andδ-shell interactions.Preprint: arXiv:1612.07058, 2017.

[11] E. Stockmeyer and S. Vugalter. Infinite mass boundary conditions for Dirac operators.Preprint:

arXiv:1603.09657, 2016.

[12] B. Thaller. The Dirac equation. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1992.

(N. Arrizabalaga)Departamento de Matem´aticas, Universidad del Pa´ıs Vasco/Euskal Herriko Unibertsitatea (UPV/EHU), 48080 Bilbao, Spain

E-mail address: naiara.arrizabalaga@ehu.eus

(L. Le Treust) Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France

E-mail address: loic.le-treust@univ-amu.fr

(N. Raymond)IRMAR, Universit´e de Rennes 1, Campus de Beaulieu, F-35042 Rennes cedex, France

E-mail address: nicolas.raymond@univ-rennes1.fr