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Curvature contribution to the essential spectrum of Dirac operators with critical shell interactions

Badreddine Benhellal and Konstantin Pankrashkin

Vol. 6 (2024), No. 1, 237–252
DOI: 10.2140/paa.2024.6.237

We discuss the spectral properties of three-dimensional Dirac operators with critical combinations of electrostatic and Lorentz scalar shell interactions supported by a compact smooth surface. It turns out that the criticality of the interaction may result in a new interval of essential spectrum. The position and the length of the interval are explicitly controlled by the coupling constants and the principal curvatures of the surface. This effect is completely new compared to lower-dimensional critical situations or special geometries considered up to now, in which only a single new point in the essential spectrum was observed.

Dirac operator, pseudodifferential operators, essential spectrum, principal curvature, transmission condition, boundary integral operator
Mathematical Subject Classification
Primary: 35Q40, 47A10, 58J40, 53A05
Received: 28 February 2023
Revised: 8 August 2023
Accepted: 18 September 2023
Published: 22 February 2024
Badreddine Benhellal
Carl von Ossietzky Universität Oldenburg
Konstantin Pankrashkin
Carl von Ossietzky Universität Oldenburg