Spectral Edges Conjecture for Discrete Periodic Operators

Determine whether, for every connected Zd-periodic graph Γ and for generic choices of real-valued periodic functions V and E defining a discrete periodic operator H on Γ, all extrema of the dispersion relation are isolated, nondegenerate, and each occurs on a single spectral band function (the Spectral Edges Conjecture).

Background

The Spectral Edges Conjecture concerns the structure of extrema of spectral band functions arising from Floquet theory for discrete periodic operators on graphs. While it is known to hold in dimension 1, counterexamples exist for discrete periodic Schrödinger operators in higher dimensions, demonstrating that the conjecture is false in that restricted setting.

In contrast, for general discrete periodic operators (where both the potential V and edge weights E vary), the conjecture is not settled. The paper provides infinite families of periodic graphs for which the conjecture holds, but the general case across all discrete periodic operators remains unresolved.