Leaky quantum graphs: approximations by point interaction Hamiltonians

Abstract

We prove an approximation result showing how operators of the type $-Δ-γδ(x-Γ)$ in $L^2(\mathbb{R}^2)$, where $Γ$ is a graph, can be modeled in the strong resolvent sense by point-interaction Hamiltonians with an appropriate arrangement of the $δ$ potentials. The result is illustrated on finding the spectral properties in cases when $Γ$ is a ring or a star. Furthermore, we use this method to indicate that scattering on an infinite curve $Γ$ which is locally close to a loop shape or has multiple bends may exhibit resonances due to quantum tunneling or repeated reflections.