Spectral theory of Schr\"odinger operators with potentials that are measures supported on ${\Bbb N}$

Abstract

We discuss spectral properties of the one-dimensional Schr\"odinger operator with a potential of the form $\sum V(n)\delta(x-n)$. Our main result says that the absolutely continuous spectum of such an operator covers an interval $[\alpha^2,\beta^2]$, if $V\in \ell^4$ and the Fourier series $\sum e^{2i kn}V(n)$ is a function of $k$ that is square integrable over $[\alpha,\beta]$. We prove that this result is sharp by constructing examples of potentials $V\notin\ell^2$ for which the spectrum of the Schr\"odinger operator is singular.