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Quantizing graphs, one way or two?

Jon Harrison

Abstract

Quantum graphs were introduced to model free electrons in organic molecules using a self-adjoint Hamiltonian on a network of intervals. A second graph quantization describes wave propagation on a graph by specifying scattering matrices at the vertices. A question that is frequently raised is the extent to which these models are the same or complementary. In particular, are all energy independent unitary vertex scattering matrices associated with a self-adjoint Hamiltonian? Here we review results related to this issue. In addition, we observe that a self-adjoint Dirac operator with four component spinors produces a secular equation for the graph spectrum that matches the secular equation associated with wave propagation on the graph when the Dirac operator describes particles with zero mass and the vertex conditions do not allow spin rotation at the vertices.

Quantizing graphs, one way or two?

Abstract

Quantum graphs were introduced to model free electrons in organic molecules using a self-adjoint Hamiltonian on a network of intervals. A second graph quantization describes wave propagation on a graph by specifying scattering matrices at the vertices. A question that is frequently raised is the extent to which these models are the same or complementary. In particular, are all energy independent unitary vertex scattering matrices associated with a self-adjoint Hamiltonian? Here we review results related to this issue. In addition, we observe that a self-adjoint Dirac operator with four component spinors produces a secular equation for the graph spectrum that matches the secular equation associated with wave propagation on the graph when the Dirac operator describes particles with zero mass and the vertex conditions do not allow spin rotation at the vertices.
Paper Structure (6 sections, 1 theorem, 28 equations, 1 figure)

This paper contains 6 sections, 1 theorem, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Graph Quantizations
  3. A self-adjoint Hamiltonian on a metric graph
  4. Wave scattering quantum graphs
  5. Relations Between the Quantizations
  6. Wave Scattering of the Dirac Operator on a Graph

Key Result

Proposition 3.1

At a vertex $v$ the following are equivalent.

Figures (1)

  • Figure 1: A simple graph with $8$ vertices and $18$ edges.

Theorems & Definitions (1)

  • Proposition 3.1