The spectral edge conjecture for periodic Schrödinger operators

Let $-elta+V$ be a Schrödinger operator on $\mathbb{R}^n$ with periodic potential, and let $\lambda_j(k)$ denote its band functions, with $k$ the quasimomentum. Spectral edges are endpoints of spectral gaps and correspond to extremal values of band functions. Spectral edge conjecture. Generically, with respect to the potentials and other free parameters of the operator, such as the metric in the Laplace–Beltrami operator, the extrema of the band functions satisfy: each extremal value is attained by a single band $\lambda_j(k)$; the loci of extrema are isolated; and the extrema are non-degenerate, meaning that the corresponding band functions have non-degenerate Hessians at them. This conjecture predicts that, generically, the dispersion relation near a spectral edge has a parabolic shape, resembling the dispersion relation at the bottom of the spectrum of the free operator $-\Delta$.