Graphs in spectral analysis of complex systems

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The project focuses on several inter-related questions of spectral analysis of graphs and the use of graphs in spectral analysis of more complicated system. Most questions draw inspiration from the use of quantum graphs as models for quantum chaology, a branch of mathematical physics concerned with the properties of quantum systems that in the classical limit exhibit chaotic dynamics. The project addresses several open problems on quantum graphs and related systems that can be tackled within a 3-year time frame. Using graphs as models has already resulted in some notable successes, in particular in studying the universality of spectral statistics and in studying the nodal statistics of graph eigenfunctions. There remain, however, many important unanswered questions and some extremely promising direction that are addressed by the project. The first is a foundation-type question of whether the spectrum of a generic graph is simple. Further, recent results on the number of nodal domains of graph eigenfunctions hint at the existence of a trace formula, connecting this number with the properties of the periodic orbits of the graph. The PI studies the dynamics of zeros on an open graph (a graph with infinite leads attached) when the spectral parameter is varied. The aim is to quantify the events in which the number of zeros changes. The PI also studies the correlations within the set of long periodic orbits (longer than the graph size) using a special type of graphs as a model. Finally the PI is looking at ways to extend the diagrammatic summation schemes to new applied questions and also searches for algebraic structures unifying several existing diagrammatic schemes. Possible direct applications to questions in combinatorics and computational molecular biology are considered.The project addresses several questions of spectral properties of graphs where our current understanding is insufficient. It also uses graphs as a model to explore questions for which the experimental answer is known but mathematical proofs are lacking and, furthermore, to explore questions for which even experimental answers are yet unknown. Namely, the PI works on a highly promising idea of establishing a "trace formula" for the nodal count on graphs. If successful, this would be a break-through development, a first formula to tie together the structure of the quantum state (microscopic structure) with the structure of the classical closed trajectories on the graph (macroscopic structure). So far, similar formulae were only available for the quantum energies of a system, and any information about the fine structure of the corresponding quantum state of the system is a great step forward. The PI also builds on his experience in a wide variety of research areas (mathematical physics, graphs and combinatorics) to search for the unifying algebraic structures behind semiclassical evaluation of the quantities related to the quantum transport. Algebraic structures can help to formalize calculations in future questions and for different physical quantities; right now the calculations are performed in each case starting from the first principles. The PI uses graphs to achieve deeper mathematical understanding in questions that are pertinent to other, more complicated systems, like quantum cavities. Thus the questions studied in the project have direct relevance to mesoscopic physics and engineering. Answering them holds the key to the quantum effects which happen on the scales that are within reach of today's chip manufacturers. Applications to combinatorics and computational molecular biology are also considered in the project. Quantum graphs also serve as a perfect educational medium for new researchers and students. The PI is writing an introductory text on quantum graphs and uses easier tasks within the project as graduate and undergraduate research projects.