Perturbation theory for spectral gap edges of 2D periodic Schr\"odinger operators

We consider a two-dimensional periodic Schr"odinger operator $H=-\mathrm{\Delta }+W$ with $\mathrm{\Gamma }$ 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 $N\mathrm{\Gamma }$ 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.