The Landau-Dirac operator with shell interactions: self-adjointness and clustering
Abstract
We consider the two-dimensional Dirac operator with constant magnetic field that is perturbed by a combination of electrostatic and Lorentz-scalar delta interactions with variable coefficients supported on a smooth closed curve. Self-adjointness is studied in the so-called non-critical and critical cases. In the non-critical case the essential spectrum is unchanged-it remains to be the set of the Landau-Dirac levels, the eigenvalues of infinite multiplicity of the unperturbed operator-while in the critical case an additional interval of essential spectrum emerges in the spectral gap containing zero. Our main result concerns the discrete spectrum in the non-critical case: using the pseudodifferential properties of the involved boundary integral operators, we show that the eigenvalues accumulate at each Landau-Dirac level at a rate governed by the logarithmic capacity of the curve. A novel and surprising phenomenon is the change in the side of the accumulation depending on the position relative to the critical value. As a byproduct, clusters of eigenvalues for a family of exterior boundary value problems are obtained via confining couplings; the infinite-mass boundary condition arises as a special case.
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BibTeX
@misc{benhellal2026landaudirac,
title = {{The Landau-Dirac operator with shell interactions: self-adjointness and clustering}},
author = {Badreddine Benhellal and Vincent Bruneau and Pablo Miranda},
howpublished = {Journal of Spectral Theory},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.20665},
eprint = {2608.20665},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05785849v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Landau-Dirac operator with shell interactions: self-adjointness and clustering
AU - Badreddine Benhellal
AU - Vincent Bruneau
AU - Pablo Miranda
PY - 2026
DA - 2026/08/21
PB - Journal of Spectral Theory
DO - 10.48550/arxiv.2608.20665
UR - https://hal.science/hal-05785849v1
AB - We consider the two-dimensional Dirac operator with constant magnetic field that is perturbed by a combination of electrostatic and Lorentz-scalar delta interactions with variable coefficients supported on a smooth closed curve. Self-adjointness is studied in the so-called non-critical and critical cases. In the non-critical case the essential spectrum is unchanged-it remains to be the set of the Landau-Dirac levels, the eigenvalues of infinite multiplicity of the unperturbed operator-while in the critical case an additional interval of essential spectrum emerges in the spectral gap containing zero. Our main result concerns the discrete spectrum in the non-critical case: using the pseudodifferential properties of the involved boundary integral operators, we show that the eigenvalues accumulate at each Landau-Dirac level at a rate governed by the logarithmic capacity of the curve. A novel and surprising phenomenon is the change in the side of the accumulation depending on the position relative to the critical value. As a byproduct, clusters of eigenvalues for a family of exterior boundary value problems are obtained via confining couplings; the infinite-mass boundary condition arises as a special case.
SN - 1664-039X
SN - 1664-0403
ER -