Extrema of spectral band functions of two dimensional discrete periodic Schr\"odinger operators

TL;DR

This paper employs algebraic geometry tools like Bézout's theorem to analyze the extrema of spectral band functions in 2D discrete periodic Schrödinger operators, advancing previous spectral analysis methods.

Contribution

It introduces novel algebraic geometric techniques to study spectral band extrema, improving upon earlier analytical approaches.

Findings

Enhanced understanding of spectral band extrema structure.

Application of algebraic geometry to spectral theory.

Improved bounds on spectral extrema.

Abstract

We use B\'{e}zout's theorem and Bernstein-Khovanskii-Kushnirenko theorem to analyze the level sets of the extrema of the spectral band functions of discrete periodic Schr\"odinger operators on $\mathbb{Z}^2$. These approaches improve upon previous results of Liu and Filonov-Kachkovskiy.