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

Abstract We consider a two-dimensional periodic Schrodinger 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 N Γ 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.