Approximation of Dirac operators with $\boldsymbol{\delta}$-shell potentials in the norm resolvent sense, I. Qualitative results

In this paper the approximation of Dirac operators with general $\delta$-shell potentials supported on $C^2$-curves in $\mathbb{R}^2$ or $C^2$-surfaces in $\mathbb{R}^3$, which may be bounded or unbounded, is studied. It is shown under suitable conditions on the weight of the $\delta$-interaction that a family of Dirac operators with regular, squeezed potentials converges in the norm resolvent sense to the Dirac operator with the $\delta$-shell interaction.