Quantum Graph Package Readme

This package contains an object-oriented framework for working with quantum graphs in MATLAB. The current version defines a quantum graph object and allows the following operations

• Define a quantum graph with arbitrary topology, edge lengths, and vertex conditions.
• Compute the spectral determinant symbolically (Neumann/Kirchhoff vertex conditions only, Dirichlet & Robin coming soon.)
• Define a Laplacian operator on the quantum graph, using second-order centered differences on the edges and second order vertex conditions implemented using ghost points.
• Solve cubic nonlinear Schrödinger equations, and continue branches of solutions. Switch branches.
• Beautiful and easy plots of functions defined over metric graphs, and of bifurcation diagrams.

This code is written as a MATLAB Project. In order to run it, first type >> openProject QGObject.prj at the MATLAB prompt or else click on the file QGObject.prj in the file listing on the left. This will open up a project window. This does three things that allow you to run the codes.

1. Set the paths.
2. Change some plotting defaults. The users defaults are saved in a tmp directory and are restored when the user closes the project by closing the project window.
3. Verify that certain MATLAB toolboxes are installed that are necessary for certain functioning of the package.

There's some documentation:

Basic instructions are in a matlab live script quantumGraphRoutines.mlx, which has also been saved as an html file.

Some additional examples are in live scripts in directories source/examples, and documentation.

A wide variety of graphs are implemented in source/templates. These are demonstrated in \source\templates\templateGallery.mlx.

This package has been used in the following papers

• Goodman, R. H., NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph. Discrete & Continuous Dynamical Systems - A, 39(4), 2203–2232, (2019). http://doi.org/10.3934/dcds.2019093
• Kairzhan, A., Pelinovsky, D. E., & Goodman, R. H. Drift of Spectrally Stable Shifted States on Star Graphs. SIAM Journal on Applied Dynamical Systems, 18(4), 1723–1755, (2019). http://doi.org/10.1137/19M1246146 (Much of this is reproduced in source/examples/shiftedStates.)
• Berkolaiko, G., Marzuola, J. L., & Pelinovsky, D.M., Edge-localized states on quantum graphs in the limit of large mass. arXiv:1910.03449v2 (2020)
• Beck, T., Bors, I., Conte, G., Cox, G. & Marzuola, J.L., Limiting eigenfunctions of Sturm–Liouville operators subject to a spectral flow. Ann. Math. Québec (2020). https://doi.org/10.1007/s40316-020-00142-6 (Gracie Conte's spectrally-accurate modifcation.)

Coming Soon

• Secular determinants for a wider variety of vertex conditions
• Spectral (Chebyshev) implementation of Laplace operator
• Happy to consider adding features, or merge in contributions.