# Talk by Gregory Berkolaiko

On March 24th, 2021, Prof. Dr. Gregory Berkolaiko (Texas A&M University) gave a talk about "**Lateral variation principle of dispersion relation of periodic graphs**" as part of the research seminar Analysis of the FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.

## 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: there is a local

condition akin to second derivative test that detects if a critical point is a global (sic!) extremum. With some additional assumptions (time-reversal invariance and dimension 3 or less), we show that 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.