ABSTRACT
Quantum graphs of connected one-dimensional wires were introduced almost 80 years ago by Pauling [1].The idea of quantum graphs was used later by Kuhn [2] in order to describe organic molecules with free electron models. Quantum graphs are often used as idealizations of physical networks in the limit where the widths of the wires are much smaller than their lengths, assuming that the propagating waves remain in a single transversal mode. Quantum graphs were successfully applied to model a variety of physical problems, e.g., electromagnetic optical waveguides [3,4], quantum wires [5,6], mesoscopic systems [7,8], excitation of fractons in fractal structures [9,10], and isoscattering systems [11] (see also Reference 12 and references cited therein). Quantum graphs can also be realized experimentally. Recent developments in various epitaxy techniques also allowed for the fabrication and design of quantum nanowire networks[13,14]. The statistical properties of spectra of quantum graphs were studied in the series of theoretical papers by Kottos and Smilansky [15–17]. They have shown that quantum graphs are excellent paradigms of quantumchaos. Their findings have been confirmed in numerous theoretical investigations of this topic [18–27] and in the experiments with microwave networks simulating quantum graphs [28–32].
