Searcharxiv⌕ Search

arXiv subjects

Christian Stelzer-Landauer

Publications and source records attributed to Christian Stelzer-Landauer.

4 recordsLinked to original sources
arXiv2025

Approximation of Dirac operators with $\boldsymbolδ$-shell potentials in the norm resolvent sense, II. Quantitative results

This paper is devoted to the approximation of two and three-dimensional Dirac operators $H_{\widetilde{V} δ_Σ}$ with combinations of electrostatic and Lorentz scalar $δ$-shell interactions in the norm resolvent sense. Relying on results from \cite{BHS23} an explicit smallness condition on the coupling parameters is derived so that $H_{\widetilde{V} δ_Σ}$ is the limit of Dirac operators with scaled electrostatic and Lorentz scalar potentials. Via counterexamples it is shown that this condition is sharp. The approximation of $H_{\widetilde{V} δ_Σ}$ for larger coupling constants is achieved by adding an additional scaled magnetic term.

math.SP↗
arXiv2025

Two-dimensional Schrödinger operators with non-local singular potentials

In this paper we introduce and study a family of self-adjoint realizations of the Laplacian in $L^2(\mathbb{R}^2)$ with a new type of transmission conditions along a closed bi-Lipschitz curve $Σ$. These conditions incorporate jumps in the Dirichlet traces both of the functions in the operator domains and of their Wirtinger derivatives and are non-local. Constructing a convenient generalized boundary triple, they may be parametrized by all compact hermitian operators in $L^2(Σ;\mathbb{C}^2)$. Whereas for all choices of parameters the essential spectrum is stable and equal to $[0, +\infty)$, the discrete spectrum exhibits diverse behaviour. While in many cases it is finite, we will describe also a class of parameters for which the discrete spectrum is infinite and accumulates at $-\infty$. The latter class contains a non-local version of the oblique transmission conditions. Finally, we will connect the current model to its relativistic counterpart studied recently in [L. Heriban, M. Tušek: Non-local relativistic $δ$-shell interactions].

math.SP↗
arXiv2024

Nonrelativistic Limit of Generalized MIT Bag Models and Spectral Inequalities

For a family of self-adjoint Dirac operators $-i c (α\cdot \nabla) + \frac{c^2}{2}$ subject to generalized MIT bag boundary conditions on domains in $\mathbb R^3$ it is shown that the nonrelativistic limit in the norm resolvent sense is the Dirichlet Laplacian. This allows to transfer spectral geometry results for Dirichlet Laplacians to Dirac operators for large $c$.

math.SP↗