Quantum Ergodicity on Circulant Graphs

Quantum ergodicity is a fundamental property of the quantum mechanics of systems where the corresponding classical dynamics is chaotic. These systems are described as exhibiting quantum chaos. In quantum mechanics, wave functions express the probability of finding a particle in a region. Quantum ergodicity refers to a situation where typical wave functions that are energy eigenfunctions (wave functions with a fixed value of energy) become evenly distributed as the energy of the wave function increases. Some quantized chaotic systems exhibit a stronger form of this behavior where all energy eigenfunctions become evenly distributed as the energy increases, rather than this being a feature of most of the eigenfunctions, a property known as quantum unique ergodicity. Recent work by Magee, Thomas, and Zhao introduced a notion of quantum unique ergodicity on graphs (networks). In this thesis, we describe quantum ergodicity and quantum unique ergodicity, explain this recent definition of a discrete quantum unique ergodicity for networks, and prove new results relating to quantum unique ergodicity for families of circulant graphs (networks with a rotational symmetry). In particular, we find families of circulant graphs that do not display discrete quantum unique ergodicity. However, when the definition of discrete quantum ergodicity is relaxed to allow complex bases of eigenfunctions such circulant graph families display discrete quantum unique ergodicity. This was not observed by Magee, Thomas, and Zhao, where discrete quantum unique ergodicity was proved for real or complex orthonormal bases of eigenfunctions for Cayley graphs of quasirandom groups.