Approximation of Dirac operators with δ-shell potentials supported on a straight line

Christian Stelzer (TU Graz)

Nov 09. 2022, 16:00 — 16:30

In this talk we study the approximation of two-dimensional Dirac operators with electrostatic and Lorentz scalar δ-shell potentials supported on a straight line by Dirac operators with squeezed potentials. These singular potentials are used as idealized replacements for potentials which are strongly localized in a neighbourhood of a straight line. In order to justify these replacements, we show that Dirac operators with δ-shell potentials can indeed be approximated by Dirac operators with squeezed potentials. In contrast to existing literature where convergence in the strong sense was proven for similar problems, we show convergence in norm resolvent sense. Our results hold for interaction strengths satisfying d = η^2 − τ^2 ∈ (−4,4), where η and τ denote the electrostatic and Lorentz-scalar interaction strengths of the δ-shell potential, respectively. These parameters depend non-linearly on the interaction strengths of the squeezed potentials. If       d ∈ R \ [−4,4], one can use a unitary transformation in order to approximate Dirac operators with δ-shell potentials.

This talk is based on joint work with Jussi Behrndt and Markus Holzmann.

Further Information
ESI Boltzmann Lecture Hall
Associated Event:
Spectral Theory of Differential Operators in Quantum Theory (Workshop)
Jussi Behrndt (TU Graz)
Fritz Gesztesy (Baylor U, Waco)
Ari Laptev (Imperial College, London)
Christiane Tretter (U Bern)