Leaky Quantum Graphs: A Review

TL;DR

This review discusses the mathematical modeling, spectral properties, and scattering behavior of leaky quantum graphs, emphasizing the influence of geometry and exploring open problems in the field.

Contribution

It provides a comprehensive overview of the theory of leaky quantum graphs, including their definitions, properties, and open research challenges.

Findings

Geometry of graphs affects spectral properties.

Analysis of scattering and asymptotic behavior.

Identification of open problems in leaky quantum graph theory.

Abstract

The aim of this review is to provide an overview of a recent work concerning ``leaky'' quantum graphs described by Hamiltonians given formally by the expression $-\Delta -\alpha \delta (x-\Gamma)$ with a singular attractive interaction supported by a graph-like set in $\mathbb{R}^\nu,\: \nu=2,3$. We will explain how such singular Schr\"odinger operators can be properly defined for different codimensions of $\Gamma$. Furthermore, we are going to discuss their properties, in particular, the way in which the geometry of $\Gamma$ influences their spectra and the scattering, strong-coupling asymptotic behavior, and a discrete counterpart to leaky-graph Hamiltonians using point interactions. The subject cannot be regarded as closed at present, and we will add a list of open problems hoping that the reader will take some of them as a challenge.