Perturbation theory for spectral gap edges of 2D periodic Schrödinger operators

We consider a two-dimensional periodic Schrödinger operator H=−Δ+W with Γ being the lattice of periods. We investigate the structure of the edges of open gaps in the spectrum of H. We show that under arbitrary small perturbation V periodic with respect to NΓ where N=N(W) is some integer, all edges of the gaps in the spectrum of H+V which are perturbation of the gaps of H become non-degenerate, i.e. are attained at finitely many points by one band function only and have non-degenerate quadratic minimum/maximum. We also discuss this problem in the discrete setting and show that changing the lattice of periods may indeed be unavoidable to achieve the non-degeneracy.