Approximation of magnetic Schr\"odinger operators with $\delta$-interactions supported on networks

Abstract

This paper deals with the approximation of a magnetic Schr\"odinger operator with a singular $\delta$-potential that is formally given by $(i \nabla + A)^2 + Q + \alpha \delta_\Sigma$ by Schr\"odinger operators with regular potentials in the norm resolvent sense. This is done for $\Sigma$ being the finite union of $C^2$-hypersurfaces, for coefficients $A$, $Q$, and $\alpha$ under almost minimal assumptions such that the associated quadratic forms are closed and sectorial, and $Q$ and $\alpha$ are allowed to be complex-valued functions. In particular, $\Sigma$ can be a graph in $\mathbb{R}^2$ or the boundary of a piecewise $C^2$-domain. Moreover, spectral implications of the mentioned convergence result are discussed.