Mathematical Physics and Harmonic Analysis Seminar

Date: April 10, 2020

Time: 1:50PM - 2:50PM

Location: Zoom seminar

Speaker: Gregory Berkolaiko, Texas A&M University

Title: Zoom Seminar: Global extrema of dispersion relation of tight-binding models

Abstract: Tight-binding approximation is frequently used in physics to analyze wave propagation through periodic medium. Its Floquet–Bloch transform is a compact graph with a parameter-dependent operator defined on it. The graph of the eigenvalues as functions of parameters is called the dispersion relation. Extrema (minima and maxima) of the n-th eigenvalue give rise to band edges: endpoints of intervals supporting continuous spectrum and therefore allowing wave propagation. Locating the extrema can be difficult in general; there are examples where extrema occur away from the set of parameters with special symmetry. In this talk we will show that a large family of tight-binding models have a curious property: there is a local condition akin to the second derivative test that detects if a critical point is a global (sic!) extremum. Under some additional assumptions (time-reversal 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 a joint project with Yaiza Canzani, Graham Cox, Jeremy Marzuola.