On open book analogs of quantum graphs

TL;DR

This paper introduces the concept of open book analogs of quantum graphs, exploring their basic properties and potential applications in modeling thin structures and stratified varieties in physics and mathematics.

Contribution

It proposes a new framework for open book structures in quantum graph theory, extending the classical models to more complex stratified geometries.

Findings

Defined basic notions of open book quantum structures

Established initial results on their mathematical properties

Suggested potential applications in physics and dynamical systems

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.''