Spectral Statistics Of Quantum Graphs
The spectral geometry concerns the relation between the geometric structure of a mathematical object and the spectra (eigenvalues) of differential operators acting on that object. The idea that interactions of systems of entities can be represented in the form of a network has been long-established and researched. One of the most conventional practices in graph theory is to deduce the characteristics and structure from graph’s spectrum. In discrete networks, the spectrum plays a central role in understanding numerous physical and social phenomena. Although these discrete network models can successfully describe complex systems, the geometry of the connections between the nodes is neglected. This results in one fundamental limitation: each edge in the network is identified as a pair of vertices it is connecting, and the length of each edge plays no role. It is, therefore, more realistic to use metric graphs with edges having meaningful and precise lengths. The corresponding, continuous as opposed to discrete, dynamics is described by differential operators. A metric graph together with a differential operator is generally called a quantum graph. In this talk I will discuss the spectral dependence of quantum graphs on its structural parameters. I will also discuss some basic techniques based on variational principle for estimating the discrete eigenvalues.
Meeting ID: 949 7997 8351