Leaky quantum graphs: approximations by point interaction Hamiltonians

TL;DR

This paper demonstrates how certain quantum operators involving delta interactions along graphs can be approximated by point-interaction Hamiltonians, with applications to spectral analysis and scattering phenomena on complex curves.

Contribution

It provides a rigorous approximation framework for quantum graphs using point interactions and explores spectral and scattering properties in specific geometric configurations.

Findings

Operators with delta interactions along graphs can be approximated by point-interaction Hamiltonians.

Spectral properties are characterized for graphs shaped as rings or stars.

Scattering on complex curves may exhibit resonances due to tunneling and reflections.

Abstract

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