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

Abstract

We are interested in the nature of the spectrum of the one-dimensional Schrödinger operator −d²/dx² − Fx + ∑_(n∈ℤ) g_nδ(x−n)in L²(ℝ) with F > 0 and two different choices of the coupling constants {g_n}n ∈ ℤ. In the first model g² ≡ 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 g_n 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 < λ²/2 and purely singular continuous if F > λ²/2.