Dirac Operators on Hypersurfaces as Large Mass Limits

Moroianu, Andrei; Ourmières-Bonafos, Thomas; Pankrashkin, Konstantin

doi:10.1007/s00220-019-03642-x

Cited by 18 publications

(33 citation statements)

References 19 publications

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“…In contrast to the 2D setting, A ϑ was not directly investigated for general boundary parameters in 3D, as far as we know only the particular MIT bag operator is well studied. We emphasize the recent papers [2,56] for the analysis of general properties of the MIT bag operator and [3,9,24,54,62], where it is shown that the MIT bag boundary conditions and their 2D analogues can be interpreted as infinite mass boundary conditions (i.e., Ω is surrounded by a medium with infinite mass). The strategy developed in [56] employing Calderón projections can also be used to study the self-adjointness of Dirac operators of the form (1.1).…”

Section: Introductionmentioning

confidence: 96%

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: 96%

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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“…Various mathematical studies have been undertaken, starting with a rigorous mathematical derivation of such Hamiltonians, see e.g. [19] for the effective Hamiltonian derivation or [3,8,30,38] for the justification of the so-called infinite mass boundary conditions. Many properties of such operators have been investigated as their self-adjointness in bounded domains with specified boundary conditions or coupled with the so-called δ-interactions, see [9,11].…”

Section: Introductionmentioning

confidence: 99%

A Variational Formulation for Dirac Operators in Bounded Domains. Applications to Spectral Geometric Inequalities

Antunes

Benguria

Lotoreichik

et al. 2021

Commun. Math. Phys.

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We investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of R 2 . Motivated by spectral geometric inequalities, we prove a non-linear variational formulation to characterize its principal eigenvalue. This characterization turns out to be very robust and allows for a simple proof of a Szegő type inequality as well as a new reformulation of a Faber-Krahn type inequality for this operator. The paper is complemented with strong numerical evidences 1 supporting the existence of a Faber-Krahn type inequality.

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Dirac Operators on Hypersurfaces as Large Mass Limits

Cited by 18 publications

References 19 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 $$\delta $$

A Variational Formulation for Dirac Operators in Bounded Domains. Applications to Spectral Geometric Inequalities

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