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On open book analogs of quantum graphs

Setenay Akduman, Peter Kuchment

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.''

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 -dimensional ``pages'' bound to an - dimensional ``binding.''

Paper Structure

This paper contains 6 sections, 4 theorems, 12 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Open book structures
  3. Operators on open books. "Quantum Open Books"
  4. Binding (junction) conditions
  5. Selfadjointness
  6. Comments and acknowledgments

Key Result

Lemma 3.1

The operator $\mathcal{H}$ is bounded as an operator from the space $\widetilde{H}^2(M)$ to $L_2(M)$.

Figures (3)

  • Figure 1: The local structure, explaining the name "open book.
  • Figure 2: Three spherical "pages" (in fact, six pieces of them) connected at a circular "binding."
  • Figure 3: A "dumbbell" open book structure with two circular bindings, one cylindrical, and two spherical cap pages.

Theorems & Definitions (15)

  • Definition 2.1
  • Remark 2.1
  • Definition 3.1
  • Lemma 3.1
  • Definition 3.2
  • Remark 3.1
  • Definition 3.3
  • Definition 3.4
  • Theorem \oldthetheorem
  • proof
  • ...and 5 more