Algebraic Aspects of Periodic Graph Operators

Abstract: A periodic linear graph operator acts on states (functions) defined on the vertices of a graph equipped with a free translation action. Fourier transform with respect to the translation group reveals the central spectral objects, Bloch and Fermi varieties. These encode the relation between the eigenvalues of the translation group and the eigenvalues of the operator. As they are algebraic varieties, algebraic methods may be used to study the spectrum of the operator. We establish a framework in which commutative algebra directly comes to bear on the spectral theory of periodic operators, helping to distinguish their algebraic and analytic aspects. We also discuss reducibility of the Fermi variety and non-degeneracy of spectral band edges.