Abstract

We are interested in the nature of the spectrum of the one-dimensional Schrödinger operator

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 $ℝ$ 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 $ℝ$ and it is dense pure point if $F<\frac{1}{2}{\lambda }^2$ and purely singular continuous if $F>\frac{1}{2}{\lambda }^2$.

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