Perturbation theory for spectral gap edges of 2D periodic Schrödinger operators (Q526860)

The paper is focused on the properties of the spectral edges of open gaps in the spectrum of the two-dimensional Schrödinger operator \(H=-\Delta+W\) in \(\mathbb{R}^d\), \(d\geq 2\), where \(W=W(x)\) is a smooth periodic potential, with \(\Gamma\) its lattice of periods. The authors prove that all spectral edges of \(H\) can be made non-degenerate by perturbing it with arbitrarily small periodic potential \(V\) with a smaller lattice of periods \(\widetilde\Gamma\subset \Gamma\). Several examples of discrete periodic Schrödinger operators for which some properties are not satisfied are also presented, and then the authors show that changing the lattice of periods may be unavoidable to achieve the non-degeneracy.