Journal of Operator Theory

Volume 93, Issue 2, Spring 2025 pp. 569-591.

On the self-adjointness of two-dimensional relativistic shell interactions

Authors: Badreddine Benhellal (1), Konstantin Pankrashkin (2), Mahdi Zreik (3)
Author institution: (1) Carl von Ossietzky Universitat Oldenburg, Fakultat V, Institut fur Mathematik, 26111 Oldenburg, Germany
(2) Carl von Ossietzky Universitat Oldenburg, Fakultat V, Institut fur Mathematik, 26111 Oldenburg, Germany
(3) Institut de Mathematiques de Toulouse, 118, route de Narbonne, 31062 Toulouse Cedex 9, France

Summary: We discuss the self-adjointness of the two-dimensional Dirac operator with a transmission condition along a closed Lipschitz curve. The new self-adjointness condition includes and extends all previous results for this class of problems. The study is particularly precise for the case of curvilinear polygons, as the angles can be taken into account in an explicit way. In particular, if the curve is a curvilinear polygon with obtuse angles, then there is a unique self-adjoint realization with domain contained in $H^{{1}/{2}}$ for the full range of non-critical coefficients in the transmission condition.

DOI: http://dx.doi.org/10.7900/jot.2023jul27.2447
Keywords: spectral analysis, Dirac operator, self-adjointness, boundary integral operators, Fredholm theory, Cauchy transform

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