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

Abstract

We discuss spectral properties of the one-dimensional Schrödinger operator with a potential of the form $\sum V(n)δ(x-n)$. Our main result says that the absolutely continuous spectum of such an operator covers an interval $[α^2,β^2]$, 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 $[α,β]$. We prove that this result is sharp by constructing examples of potentials $V\notin\ell^2$ for which the spectrum of the Schrödinger operator is singular.