Quantizing graphs, one way or two?

TL;DR

This paper reviews different quantum graph models, comparing their mathematical foundations and exploring conditions under which they are equivalent, including the role of self-adjoint operators and scattering matrices.

Contribution

It clarifies the relationship between self-adjoint Hamiltonian models and scattering matrix models, and introduces a Dirac operator approach for graph spectra.

Findings

Self-adjoint Hamiltonians can correspond to certain scattering matrices.

Dirac operators with specific conditions produce matching secular equations.

The models are related under zero-mass particle assumptions.

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.