Pavel Exner
Leaky Quantum Graphs: A Review
(575K, LaTeX, amsproc.sty)
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.