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.
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.