Leaky Quantum Graphs: A Review

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 $-\mathrm{\Delta }-\alpha \delta (x-\mathrm{\Gamma })$ 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 $\mathrm{\Gamma }$. Furthermore, we are going to discuss their properties, in particular, the way in which the geometry of $\mathrm{\Gamma }$ 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.