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

TL;DR

This paper establishes a rigorous approximation of magnetic Schr"odinger operators with $oldsymbol{ ext{delta}}$-interactions supported on networks by regular potential operators, analyzing spectral properties and convergence in the norm resolvent sense.

Contribution

It provides the first comprehensive approximation framework for magnetic Schr"odinger operators with $oldsymbol{ ext{delta}}$-interactions on complex networks, including spectral analysis under minimal assumptions.

Findings

Convergence of regularized operators to the singular magnetic Schr"odinger operator in the norm resolvent sense.

Spectral properties are preserved under the approximation, with implications for quantum graph models.

Applicable to networks with complex coefficients and geometries such as graphs and boundary surfaces.

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.