Spectral edges nondegeneracy conjecture

Establish, for periodic operators (including discrete periodic graph operators), that for generic choices of edge weights and vertex potentials, each spectral edge is attained by a single band function of the real Bloch variety over the torus and that all such extrema are isolated and nondegenerate (full-rank Hessian).

Background

The real Bloch variety R_A ⊂ T^d × R encodes the spectral relation between the characters of the Z^d-action and the energy values for periodic graph operators. Spectral bands arise as images of band functions, and their endpoints (spectral edges) correspond to extrema of these band functions. Many physical and analytical properties (effective mass, Liouville property, Green’s function asymptotics, Anderson localization, homogenization) rely on these extrema being isolated and nondegenerate.

Kuchment formulated the spectral edges conjecture asserting generic nondegeneracy and uniqueness of extrema at spectral edges. Despite partial results and known counterexamples in special parameter regimes, the conjecture remains largely open, particularly for discrete periodic operators. The paper discusses this conjecture in the context of discrete periodic graph operators and provides examples, counterexamples, and computational approaches relevant to its paper.