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

We are interested in the nature of spectrum one-dimensional Schr\"odinger operator $$ - \frac{d^2}{dx^2}-Fx + \sum_{n \in \mathbb{Z}}g_n \delta(x-n) \qquad\text{in } L^2(\mathbb{R}) with $F>0$ and two different choices coupling constants $\{g_n\}_{n\in \mathbb{Z}}$. In first model $g_n \equiv \lambda$ we prove that if $F\in \pi^2 \mathbb{Q}$ then is $\mathbb{R}$ furthermore absolutely continuous away from an explicit discrete set points. second $g_n$ independent random variables mean zero variance $\lambda^2$. Under certain assumptions on distribution these almost surely it dense pure point $F < \lambda^2/2$ purely singular $F> \lambda^2/2$.