Two-dimensional Dirac operators with singular interactions supported on closed curves

Behrndt, Jussi; Holzmann, Markus; Ourmières-Bonafos, Thomas; Pankrashkin, Konstantin

doi:10.1016/j.jfa.2020.108700

Cited by 30 publications

(67 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We expect that in the critical case ϑ(x) 2 = 1 for some x ∈ ∂Ω the spectral properties of A ϑ may significantly differ from the non-critical case, e.g., essential spectrum may arise also for bounded domains or in the gap (−m, m). Similar difficulties and effects were observed in the 2D situation in [22,61] and also in the analysis of Dirac operators with singular interactions in [4,10,15,16,56].…”

Section: Introductionsupporting

confidence: 67%

“…Assume first that Ω is an unbounded C 2domain with compact boundary. The proof of (ii) is based on the same argument as the proof of [16,Proposition 3.8].…”

Section: Self-adjointness and Spectral Properties Of A ϑmentioning

confidence: 99%

“…is compactly supported in B(0, R)\B(0, r) due to the construction of g 1 and g 2 . Thus, B 2 has only finitely many eigenvalues below m 2 ; for a proof see, e.g., [16,Proof of Proposition 3.8].…”

Section: Self-adjointness and Spectral Properties Of A ϑmentioning

confidence: 99%

“…Moreover, Dirac operators with boundary conditions are also closely related to Dirac operators with singular δ-shell interactions supported on surfaces for special choices of the interaction strengths in the so-called confinement case, i.e., when the δ-potential is impenetrable for the particle; cf. [5,12,16,37].…”

Section: Introductionmentioning

confidence: 99%

“…However, this approach does not allow directly a systematic spectral analysis of these operators. Finally, we mention that in recent years the self-adjointness and the spectral and scattering properties of the closely related Dirac operators with singular δ-shell interactions were studied comprehensively in [4][5][6]10,12,15,16,42,[49][50][51]55,56].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Behrndt

Holzmann

Blesa

2020

Ann. Henri Poincaré

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionsupporting

confidence: 67%

“…Assume first that Ω is an unbounded C 2domain with compact boundary. The proof of (ii) is based on the same argument as the proof of [16,Proposition 3.8].…”

Section: Self-adjointness and Spectral Properties Of A ϑmentioning

confidence: 99%

Section: Self-adjointness and Spectral Properties Of A ϑmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Behrndt

Holzmann

Blesa

2020

Ann. Henri Poincaré

Self Cite

View full text Add to dashboard Cite

show abstract

On the calculation of the discrete spectra of one‐dimensional Dirac operators

Barrera-Figueroa

Rabinovich

Loredo‐Ramírez

2022

Math Methods in App Sciences

View full text Add to dashboard Cite

In this work, we consider the one‐dimensional Dirac operator where is Pauli's matrix, is a ‐matrix representing a regular potential that includes the electrostatic and scalar interactions as well as the anomalous magnetic momentum, is a singular potential consisting of delta distributions ( ) are ‐matrices representing the strengths of Dirac deltas, and is a two‐spinor. We associate to the operator an unbounded in symmetric operator denoted by , where is the support of singular potential . The operator includes only the regular potential together with certain interaction conditions at each point . The paper presents a method for determining the discrete spectrum of the operator for arbitrary potential whose entries are given by ‐functions. The eigenvalues of the operator are the zeros of a dispersion equation , where the characteristic function is determined explicitly in terms of power series involving the spectral parameter . The construction of the characteristic function from a set of monodromy matrices and the interaction conditions is presented in the paper. Moreover, its power series representation leads to an efficient numerical method for calculating the eigenvalues of the Dirac operator from the zeros of certain approximate function which is obtained by truncating the series up to a finite number of terms. Several examples show the applicability and accuracy of the numerical method.

show abstract

Interaction Problems for n-Dimensional Dirac Operators with Singular Potentials

Rabinovich

2024

Trends in Mathematics

View full text Add to dashboard Cite

Two-dimensional Dirac operators with singular interactions supported on closed curves

Cited by 30 publications

References 25 publications

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

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

On the calculation of the discrete spectra of one‐dimensional Dirac operators

Interaction Problems for n-Dimensional Dirac Operators with Singular Potentials

Contact Info

Product

Resources

About