On open book analogs of quantum graphs

Abstract

Quantum graphs have become in this century a favorite playground for mathematicians, mathematical physicists, and chemists, due to their manifold applications as models of thin structures, as well as presenting sometimes simpler playground for hard higher dimensional problems. It was clear from some applications that thin surface structures (looking as stratified varieties) also arise, for instance in photonic crystals theory and dynamical systems. However, both justification and studying of these models is much harder and very little progress has been made by now. The goal of this note is to set down some basic notions and results for such structures. The name ``open book'' has been used for such geometric structures in topology and comes from an image of several smooth $n$-dimensional ``pages'' bound to an $(n-1)$- dimensional ``binding.''