| Research Abstract |
In this research, we have investigated the quantum graphs, namely, the motion of a quantum particle residing on one-dimensional lines connected at nodes to form a graph. This can be regarded as a idealized mathematical model of quantum wire, and as such, it is expected to become a model for quantum computational device. Through the author's past studies, it has been known that quantum graphs possess exotic properties which are usually absent in quantum mechanics and only found in quantum field theories. In the current study, we have successfully uncovered the fact that behind the elementary appearance of the model, there exists a parameter space structure described by U(2) group, and its non trivial topology is precisely the cause of such phenomena as "spectral anholonomy". "Berry phase", "scale anomaly", and "fermion-boson duality". We have further studied several scattering and transport phenomena in quantum graphs which are the related to these exotic features, and explored the possibility of their applications in mesoscopic physics. In the simplest setting, the quantum graph is reduced to a system of t*
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