On the spectrum of the Kronig-Penney model in a constant electric field

Journal article, 2022

We are interested in the nature of the spectrum of the one-dimensional Schrödinger operator (formula presented) with F > 0 and two different choices of the coupling constants {gn}n ∈Z. In the first model, gn ≡ λ, and we prove that if F ∈ π2Q then the spectrum is R and is furthermore absolutely continuous away from an explicit discrete set of points. In the second model, the gn are independent random variables with mean zero and variance λ2. Under certain assumptions on the distribution of these random variables, we prove that almost surely the spectrum is R and it is dense pure point if F < 1/2 λ2 and purely singular continuous if F > 1/2 λ2.