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

We discuss spectral properties of the one-dimensional Schrödinger 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 $V\in {\mathrm{\ell }}^4$ and the Fourier series $\sum e^{2ikn}V(n)$ is a function of $k$ that is square integrable over $[\alpha ,\beta ]$. We prove that this result is sharp by constructing examples of potentials $V\mathrm{\notin }{\mathrm{\ell }}^2$ for which the spectrum of the Schrödinger operator is singular.