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Localization for the Random XXZ Spin-J Chain and Embedded Eigenvalues in Defective Periodic Quantum Graphs


This thesis is composed of two distinct projects. Although both projects are devoted to studying situations in which the eigenfunctions of certain Schrodinger operators exhibit interesting or exceptional behavior, the nature of the results as well as the tools used in each of the parts are drastically different. The first portion is dedicated to a particular spin-system model in which a type of many body localization is proved. We use the Multiscale Analysis to show that localization in higher spin systems occurs within a certain interval near the bottom of the spectrum. The second project demonstrates that a carefully constructed defect in specific kinds of periodic media, most notably multilayer quantum-graph graphene, can create an eigenvalue embedded in the continuous spectrum with a corresponding eigenfunction that has exceptional properties.

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