Title: Lateral variation principle and extrema of dispersion relation of periodic graphs
Abstract: The first step in the proofs of several spectral geometry theorems is perturbing the operator ``along'' a given eigenfunction $f$, i.e. adding a perturbation $P$ that vanishes on $f$ and therefore leaves the corresponding eigenvalue $\lambda_0$ in its place. But such perturbation may still affect the sequential number of $\lambda_0$ in the spectrum, creating a spectral shift. We will discuss a general theorem that recovers the value of the spectral shift by looking at the stability of $\lambda_0$ with respect to small variations of the perturbation $P$. As an application of this result, we show that a large family of tight-binding models have a curious property: any local extremum of a given sheet of the dispersion relation is in fact the global extremum. Based on joint work with Y.Canzani, G.Cox, P.Kuchment and J.Marzuola.