QGLAB: A MATLAB Package for Computations on Quantum Graphs

TL;DR

QGLAB is a MATLAB package that facilitates the analysis of quantum graphs by enabling spectrum computation, PDE solutions, and bifurcation analysis through advanced discretization techniques and user-friendly features.

Contribution

It introduces a comprehensive MATLAB toolkit for quantum graph analysis, including spectrum calculation, PDE solving, and bifurcation tracking, with innovative discretization methods for vertex conditions.

Findings

Accurately computes spectra of elliptic operators on quantum graphs.

Enables solutions to Poisson and PDEs on quantum graphs.

Supports bifurcation analysis and branch continuation.

Abstract

We describe QGLAB, a new MATLAB package for analyzing partial differential equations on quantum graphs. The software is built on the existing, object-oriented MATLAB directed-graph class, inheriting its structure and adding additional easy-to-use features. The package allows one to construct a quantum graph and accurately compute the spectrum of elliptic operators, solutions to Poisson problems, the linear and nonlinear time evolution of a variety of PDEs, the continuation of branches of steady states (including locating and switching branches at bifurcations) and more. It overcomes the major challenge of discretizing quantum graphs -- the enforcement of vertex conditions -- using non-square differentiation matrices. It uses a unified framework to implement finite-difference and Chebyshev discretizations of differential operators on a quantum graph. For simplicity, the package overloads many built-in MATLAB functions to work on the class.