A Remark on Boundary Value Problems for the Dirac Operator

Schmidt, Karl Michael

doi:10.1093/qmath/46.4.509

Cited by 28 publications

(45 citation statements)

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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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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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“…The following description of the adjoint of the Dirac operator with Dirichlet boundary conditions may be seen as a consequence of the principle of elliptic regularity [41, Chapter 1, Theorem 7.2]. It is also possible to give an elementary proof following [38]. Theorem 6.3.…”

Section: Pre-spectral Triple For the Model Examplementioning

confidence: 99%

Noncommutative geometry for symmetric non-self-adjoint operators

Connes

Levitina

McDonald

et al. 2019

Journal of Functional Analysis

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We introduce the notion of a pre-spectral triple, which is a generalisation of a spectral triple (A, H, D) where D is no longer required to be selfadjoint, but closed and symmetric. Despite having weaker assumptions, pre-spectral triples allow us to introduce noncompact noncommutative geometry with boundary. In particular, we derive the Hochschild character theorem in this setting. We give a detailed study of Dirac operators with Dirichlet boundary conditions on domains in R d , d ≥ 2.

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“…Thus, these boundary conditions do not mix the two valleys. We obtain two copies of D π/2 , which is not self-adjoint on H 1 (Ω, C 2 ) and has zero as an eigenvalue of infinite multiplicity [21]. Infinite Mass boundary conditions.…”

Section: Application To the Two-valley Description Of Graphenementioning

confidence: 99%

Spectral Gaps of Dirac Operators Describing Graphene Quantum Dots

Benguria

Fournais

Stockmeyer

et al. 2017

Math Phys Anal Geom

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The two-dimensional Dirac operator describes low-energy excitations in graphene. Different choices for the boundary conditions give rise to qualitative differences in the spectrum of the resulting operator. For a family of boundary conditions, we find a lower bound to the spectral gap around zero, proportional to |Ω| −1/2 , where Ω ⊂ R 2 is the bounded region where the Dirac operator acts. This family contains the so-called infinite mass and armchair cases used in the physics literature for the description of graphene quantum dots.

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A Remark on Boundary Value Problems for the Dirac Operator

Cited by 28 publications

References 0 publications

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

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

Noncommutative geometry for symmetric non-self-adjoint operators

Spectral Gaps of Dirac Operators Describing Graphene Quantum Dots

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