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

65 pages

References in corpus (2)

• Uniform spectral properties of one-dimensional quasicrystals, III. $\alpha$-continuity
• Stark-Wannier type operators with purely singular spectrum