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

TL;DR

This paper investigates the spectral properties of a one-dimensional Schrödinger operator with a periodic delta potential and a constant electric field, revealing conditions for absolutely continuous, pure point, and singular continuous spectra.

Contribution

It provides a detailed analysis of the spectrum of the Kronig-Penney model under a constant electric field for both periodic and random coupling constants, identifying spectral types based on parameters.

Findings

Spectrum is real and absolutely continuous away from discrete points when F is rational multiple of pi^2.

Almost sure spectrum is real for random couplings with certain distribution assumptions.

Spectral type depends on the relation between F and the variance of the random couplings, being pure point or singular continuous.

Abstract

We are interested in the nature of the spectrum of the one-dimensional Schr\"odinger operator $$ - \frac{d^2}{dx^2}-Fx + \sum_{n \in \mathbb{Z}}g_n \delta(x-n) \qquad\text{in } L^2(\mathbb{R}) $$ 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 $F> \lambda^2/2$.