Wigner–von Neumann type perturbations of the periodic one-dimensional Schrödinger operator are considered. The asymptotics of the solution to the generalized eigenfunction equation is investigated. It is proven that a subordinated solution and therefore an embedded eigenvalue may occur at the points of the absolutely continuous spectrum satisfying a certain resonance (quantization) condition between the frequencies of the perturbation, the frequency of the background potential and the corresponding quasimomentum.

Let A be a Banach algebra, and let {xλ:λ ∈ Λ} be a collection of central elements of A. We show that the ideal is closed whenever it is prime. In particular, this shows that prime ideals generated by finitely many central elements of a Banach algebra are automatically closed.