An overview of periodic elliptic operators

TL;DR

This paper reviews the spectral theory of periodic elliptic operators, focusing on analytic properties of Bloch and Fermi varieties, and discusses various techniques applicable to a broad class of such operators in mathematical physics.

Contribution

It provides a comprehensive overview of techniques and results in the spectral theory of periodic elliptic operators, highlighting their applicability to diverse mathematical models.

Findings

Analysis of Bloch and Fermi varieties influences spectral properties

Applicable to Schrödinger operators with periodic potentials

Extends to elliptic equations, systems, and operators on graphs

Abstract

The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which influence significantly most properties of such operators. The approaches described are applicable not only to the standard model example of Schr\"odinger operator with periodic electric potential $-\Delta+V(x)$, but to a wide variety of elliptic periodic equations and systems, equations on graphs, $\overline{\partial}$-operator, and other operators on abelian coverings of compact bases. Many important applications are mentioned. However, due to the size restrictions, they are not dealt with in details.