Singular continuous spectrum of half-line Schrödinger operators with point interactions on a sparse set

Abstract

We say that a discrete set $X =\{x_n\}_{n\in\dN_0}$ on the half-line $$0=x_0 < x_1 <x_2 <x_3<... <x_n<... <+\infty$$ is sparse if the distances $Δx_n = x_{n+1} -x_n$ between neighbouring points satisfy the condition $\frac{Δx_{n}}{Δx_{n-1}} \rightarrow +\infty$. In this paper half-line Schrödinger operators with point $δ$- and $δ^\prime$-interactions on a sparse set are considered. Assuming that strengths of point interactions tend to $\infty$ we give simple sufficient conditions for such Schrödinger operators to have non-empty singular continuous spectrum and to have purely singular continuous spectrum, which coincides with $\dR_+$.