Limiting absorption principle and scattering matrix for Dirac operators with <i>δ</i>-shell interactions

Behrndt, Jussi; Holzmann, Markus; Mantile, Andrea; Posilicano, Andrea

doi:10.1063/1.5123289

Cited by 4 publications

(2 citation statements)

References 43 publications

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“…The family {H τ } τ ∈R naturally arises in the context of confining δ-shell interactions. In the last decade, Dirac operators coupled with δ-shell potentials have been investigated from a mathematical perspective: their self-adjointness and spectral properties [9,10,16,17,19,20,22,39,57,68], the case of rough domains [27,26], and their approximations and other asymptotic regimes [33,50,59,58,61]; we refer to the survey [63] for further details on the state of the art of shell interactions for Dirac operators. Several of these works addressed singular perturbations of the form…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue Curves for Generalized MIT Bag Models

Arrizabalaga

Mas

Sanz-Perela

et al. 2022

Commun. Math. Phys.

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We study spectral properties of Dirac operators on bounded domains Ω ⊂ R 3 with boundary conditions of electrostatic and Lorentz scalar type and which depend on a parameter τ ∈ R; the case τ = 0 corresponds to the MIT bag model. We show that the eigenvalues are parametrized as increasing functions of τ , and we exploit this monotonicity to study the limits as τ → ±∞. We prove that if Ω is not a ball then the first positive eigenvalue is greater than the one of a ball with the same volume for all τ large enough. Moreover, we show that the first positive eigenvalue converges to the mass of the particle as τ ↓ −∞, and we also analyze its first order asymptotics.

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Section: Introductionmentioning

confidence: 99%

Eigenvalue Curves for Generalized MIT Bag Models

Arrizabalaga

Mas

Sanz-Perela

et al. 2022

Commun. Math. Phys.

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“…The family {H τ } τ ∈R naturally arises in the context of confining δ-shell interactions. In the last decade, Dirac operators coupled with singular δ-shell potentials have been investigated from a mathematical perspective: their self-adjointness and spectral properties [9,10,16,17,19,20,22,39,57,68], the case of rough domains [26,27], and their approximations and other asymptotic regimes [33,50,59,58,61]; we refer to the survey [63] for further details on the state of the art of shell interactions for Dirac operators. Several of these works addressed singular perturbations of the form…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue curves for generalized MIT bag models

Arrizabalaga¹,

Mas²,

Sanz-Perela³

et al. 2021

Preprint

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Boundary triples and Weyl functions for Dirac operators with singular interactions

Behrndt,

Holzmann,

Stelzer-Landauer

et al. 2023

Rev. Math. Phys.

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In this paper, we develop a systematic approach to treat Dirac operators [Formula: see text] with singular electrostatic, Lorentz scalar, and anomalous magnetic interactions of strengths [Formula: see text], respectively, supported on points in [Formula: see text], curves in [Formula: see text], and surfaces in [Formula: see text] that is based on boundary triples and their associated Weyl functions. First, we discuss the one-dimensional case which also serves as a motivation for the multidimensional setting. Afterwards, in the two- and three-dimensional situation we construct quasi, generalized, and ordinary boundary triples and their Weyl functions, and provide a detailed characterization of the associated Sobolev spaces, trace theorems, and the mapping properties of integral operators which play an important role in the analysis of [Formula: see text]. We make a substantial step towards more rough interaction supports [Formula: see text] and consider general compact Lipschitz hypersurfaces. We derive conditions for the interaction strengths such that the operators [Formula: see text] are self-adjoint, obtain a Krein-type resolvent formula, and characterize the essential and discrete spectrum. These conditions include purely Lorentz scalar and purely non-critical anomalous magnetic interactions as well as the confinement case, the latter having an important application in the mathematical description of graphene. Using a certain ordinary boundary triple, we also show the self-adjointness of [Formula: see text] for arbitrary critical combinations of the interaction strengths under the condition that [Formula: see text] is [Formula: see text]-smooth and derive its spectral properties. In particular, in the critical case, a loss of Sobolev regularity in the operator domain and a possible additional point of the essential spectrum are observed.

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Limiting absorption principle and scattering matrix for Dirac operators with δ-shell interactions

Cited by 4 publications

References 43 publications

Eigenvalue Curves for Generalized MIT Bag Models

Eigenvalue Curves for Generalized MIT Bag Models

Eigenvalue curves for generalized MIT bag models

Boundary triples and Weyl functions for Dirac operators with singular interactions

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