Quantum graphs: scattering, randomisation, and devices modelling

Abstract

This work has as its main topic the study of scattering in quantum graphs.

Throughout it, the Green’s function method for quantum graphs is ap- plied, which is obtained by the adjacency matrix of the graph and provides

the scattering amplitudes in these structures. An informational entropy of scattering is proposed, relating the probability of reflection or transmission for each of the scattering channels with an informational content about the distribution of the scattering along these channels. Next, applications of quantum graphs to model the formation of a filament between electrodes and to study memristors are proposed. The transport in randomised graph structures and the approximation by taking samples of their subgraphs are also studied. Finally, a quantum state model of scattering is also proposed, where the scattering in a graph can act as a control of the structure of a second quantum graph, and the cases in which this system can result in entanglement of the states are studied.