The project addresses a number of related questions concerning the spectra of differential operators acting on spaces of functions defined on metric graphs. A graph equipped with such an operator is generally referred to as a "quantum graph." The research has two central themes: investigating the correlations between the eigenvalues of quantum graphs, and understanding the morphology of their eigenfunctions. For generic families of graphs the eigenvalues appear to be correlated like eigenvalues of large random matrices. For non-generic graphs, different correlations are expected. The present project will improve the understanding of how these correlations arise both from a mathematical and physical point of view. The results obtained will be useful in understanding the appearance of correlations in the eigenvalues of other differential operators arising in mathematical physics. The second theme of the project is to understand the statistical properties of eigenfunctions of quantum graphs. For generic graphs, it is expected that the eigenfunctions should exhibit some equidistribution properties similar to those already proved to exist in other systems. A central objective of the project is to formulate an analogous theory for eigenfunctions of quantum graphs. It is also suspected that a sparse sequence of eigenfunctions may become localized (scarred) rather than equidistributed. Another part of the project will be to discover whether this can, in fact, occur, and the conditions under which it may do so.
Quantum graphs are part of a new branch of mathematics, sometimes called ``nanomathematics'', which contains within its scope areas such as transport in nanostructures, quantum information, and superconductivity. In this context, our study of quantum graphs is part of an effort to understand the theory of quantum mechanics as it relates to classical Newtonian dynamics (so-called "quantum chaology"). The wave-based theory of quantum mechanics is unavoidable on scales of the order of magnitude of present microchip manufacturing. The shapes and frequencies of waves appearing in complex quantum mechanical systems can be understood by looking at similar questions for oscillations on networks. The results of the project will be relevant to current research in many diverse areas of mesoscopic physics and engineering. In order to draw new researchers into the field, part of the project will be used as topics for an undergraduate research experience program, and an introductory graduate text will be written.
