Self-Adjointness of Dirac Operators with Infinite Mass Boundary Conditions in Sectors

Treust, Loïc Le; Ourmières-Bonafos, Thomas

doi:10.1007/s00023-018-0661-y

Cited by 23 publications

(16 citation statements)

References 27 publications

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“…Many considerations in [ 22 , 23 , 61 ] are based on complex analysis techniques, which are not available in the 3D situation. We also refer to [ 28 , 47 , 48 , 57 ] for self-adjointness and spectral problems of 2D Dirac operators on different types of domains with special boundary conditions. In contrast to the 2D setting, was not directly investigated for general boundary parameters in 3D, as far as we know only the particular MIT bag operator is well studied.…”

Section: Introductionmentioning

confidence: 99%

Self-Adjoint Dirac Operators on Domains in $$\mathbb {R}^3$$

Behrndt

Holzmann

Blesa

2020

Ann. Henri Poincaré

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In this paper, the spectral and scattering properties of a family of self-adjoint Dirac operators in $$L^2(\Omega ; \mathbb {C}^4)$$ L 2 ( Ω ; C 4 ) , where $$\Omega \subset \mathbb {R}^3$$ Ω ⊂ R 3 is either a bounded or an unbounded domain with a compact $$C^2$$ C 2 -smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with boundary conditions as of Robin type. Our approach is based on abstract boundary triple techniques from extension theory of symmetric operators and a thorough study of certain classes of (boundary) integral operators, that appear in a Krein-type resolvent formula. The analysis of the perturbation term in this formula leads to a description of the spectrum and a Birman–Schwinger principle, a qualitative understanding of the scattering properties in the case that $$\Omega $$ Ω is an exterior domain, and corresponding trace formulas.

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Section: Introductionmentioning

confidence: 99%

Self-Adjoint Dirac Operators on Domains in $$\mathbb {R}^3$$

Behrndt

Holzmann

Blesa

2020

Ann. Henri Poincaré

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“…Eventually, in [27], a detailed study of the spectral properties of A 0,τ for purely scalar potentials was provided; in particular, it was shown that the discrete eigenvalues in the large mass limit are characterized by an effective operator on the surface . Furthermore, there is a great interest recently in the study of self-adjoint Dirac operators on domains with boundary conditions, see, e.g., [2,3,10,11,25,30,31,35,38].…”

mentioning

confidence: 99%

On Dirac operators in $$\mathbb {R}^3$$ R 3 with electrostatic and Lorentz scalar $$\delta $$

Behrndt

Exner

Holzmann

et al. 2019

Quantum Stud.: Math. Found.

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In this article, Dirac operators A η,τ coupled with combinations of electrostatic and Lorentz scalar δshell interactions of constant strength η and τ , respectively, supported on compact surfaces ⊂ R 3 are studied. In the rigorous definition of these operators, the δ-potentials are modeled by coupling conditions at . In the proof of the self-adjointness of A η,τ , a Krein-type resolvent formula and a Birman-Schwinger principle are obtained. With their help, a detailed study of the qualitative spectral properties of A η,τ is possible. In particular, the essential spectrum of A η,τ is determined, it is shown that at most finitely many discrete eigenvalues can appear, and several symmetry relations in the point spectrum are obtained. Moreover, the nonrelativistic limit of A η,τ is computed and it is discussed that for some special interaction strengths, A η,τ is decoupled to two operators acting in the domains with the common boundary .

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“…We follow the same strategy as in the proof of Lemma 3.2. By Proposition 3.4 and (19) we have, for ε −1 1 + c,…”

Section: Resolvent Estimatesmentioning

confidence: 75%

“…For further recent study of H Ω from the mathematical point of view see e.g. [7,19,10]. Note that in dimension 3, this infinite mass limit is strongly related to the MIT bag model for quarks confinement.…”

Section: Setting and Main Resultmentioning

confidence: 95%

Resolvent Convergence to Dirac Operators on Planar Domains

Barbaroux

Cornean

Treust

et al. 2019

Ann. Henri Poincaré

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Consider a Dirac operator defined on the whole plane with a mass term of size m supported outside a domain Ω. We give a simple proof for the norm resolvent convergence, as m goes to infinity, of this operator to a Dirac operator defined on Ω with infinite mass boundary conditions. The result is valid for bounded and unbounded domains and gives estimates on the speed of convergence. Moreover, the method easily extends when adding external matrix-valued potentials.

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Self-Adjointness of Dirac Operators with Infinite Mass Boundary Conditions in Sectors

Cited by 23 publications

References 27 publications

Self-Adjoint Dirac Operators on Domains in $$\mathbb {R}^3$$

Self-Adjoint Dirac Operators on Domains in $$\mathbb {R}^3$$

On Dirac operators in $$\mathbb {R}^3$$ R 3 with electrostatic and Lorentz scalar $$\delta $$

Resolvent Convergence to Dirac Operators on Planar Domains

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