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

We are interested in the nature of the spectrum of the one-dimensional Schr"odinger operator $$

\frac{d^2}{dx^2}-Fx + \sum_{n \in $Z$ }g_n \delta(x-n) \qquad $in$ L^2( $R$ ) $ $w i t h$ F>0$ and two different choices of the coupling constants ${g_{n}}_{n \in Z}$ . In the first model $g_n \equiv \lambda $a n d w e p r o v e t h a t i f$ F\in \pi^2 $Q$ $ 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 $λ^{2}$ . 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 $F < λ^{2} / 2$ and purely singular continuous if $F > λ^{2} / 2$ .

We are interested in the nature of the spectrum of the one-dimensional Schr"odinger operator $$

\frac{d^2}{dx^2}-Fx + \sum_{n \in $Z$ }g_n \delta(x-n) \qquad $in$ L^2( $R$ ) $ $w i t h$ F>0$ and two different choices of the coupling constants ${g_{n}}_{n \in Z}$ . In the first model $g_n \equiv \lambda $a n d w e p r o v e t h a t i f$ F\in \pi^2 $Q$ $ 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 $λ^{2}$ . 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 $F < λ^{2} / 2$ and purely singular continuous if $F > λ^{2} / 2$ .

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