Quantum Graphs/Introduction

Studying operators of Schrödinger type on metric graphs is a growing subfield of mathematical physics which is motivated both by direct applications of the graph models to physical phenomena and by use of graphs as a simpler setting in which to study complex phenomena of quantum mechanics, such as Anderson localization, universality of spectral statistics, nodal statistics, scattering and resonances, to name but a few.

The name "quantum graphs" is most likely a shortening of the title of the article "Quantum Chaos on Graphs" by Kottos and Smilansky \cite{KotSmi_prl97}. The model itself has been studied well before the name appeared, for example in \cite{Pau_jcp36,RueSch_jcp53,Rot_crasp83,Bel_laa85,Nic_incol85}.

Several reviews and monographs cover various directions within the quantum graphs research \cite{GnuSmi_ap06,Post_book12,Mugnolo_book}. However, when starting a research project with students, both (post-) graduate and undergraduate, the author felt that a more elementary introduction would be helpful. The present manuscript grew out of the same preparatory lecture repeated, at different points of time, to several students. It is basically a collection of minimal examples of quantum graphs which already exhibit behavior typical to larger graphs. We supply the examples with pointers to the more general facts and theorems. Only in the last sections we explore a research topic (the nodal statistics on graphs) in some depth.

For obvious reasons the pointers often lead to the monograph \cite{BerKuc_graphs}.