Quantum Graphs and Their Applications

by Peter Kuchment (Editor), Robert Carlson (Editor), Stephen A. Fulling (Editor)

This volume is a collection of articles dedicated to quantum graphs, a newly emerging interdisciplinary field related to various areas of mathematics and physics. The reader can find a broad overview of the theory of quantum graphs. The articles present methods coming from different areas of number theory, combinatorics, mathematical physics, differential equations, spectral theory, global analysis, and theory of fractals. They also address various important applications, such as Anderson localization, electrical networks, quantum chaos, mesoscopic physics, superconductivity, optics, and biological modeling.

307 pages, Paperback

First published November 1, 2006

Reviews

[Kevin K. Gillette · Rating 4 out of 5]

I chose this collection of presentations from a recent conference simply out of curiosity. "What are quantum graphs, and what are they used for?" ran through my head the first time I saw the term. I didn't read all of the articles in the collection, but rather a small number that appealed to me based on the titles. I will say, however, that this sample was enough to whet my appetite for the subject, and it certainly gave me a good understanding of how quantum graphs are used, especially in physics, as discretized versions of quantum models that then take on the characteristics of the continuous quantum analogues when the size of the graphs (or some other metric of them) passed to infinity.

My favorite paper was the one on Sierpinski gaskets as electrical resistance networks, and some of the spectral results obtained therein. Not just cool math - cool pictures, too! If you don't know what Sierpinski gaskets are, check out some of the amazing illustrations of these in Mandelbrot's legendary text, "The Fractal Geometry of Nature". (Also very cool, in the same book: Peano curves!)

This conference collection is definitely not the best "first book" for this subject, but it is very effective in giving a good flavor of the subject, for newcomers and experienced practitioners alike.