In this pedagogical talk, we will discuss a simple setting in which to explore the methods we saw in Evan Wickenden's talk earlier this semester: discrete Schrodinger operators (1d periodic). We will compare the two methods that were introduced: the Floquet discriminant and the Bloch transform. In the 1d periodic case, there are some patterns predicted by Hill's theorem which can be disturbed by a suitably strong perturbation of the operator. The latter observation is related to some ideas of spectral engineering: resonant gap opening and flat bands.
We will not present any research results, although some numerical simulation we will see may lead to a research project.
Explorations of periodic Schrodinger operator