Singular continuous spectrum of half-line Schr\"odinger 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 $\Delta x_n = x_{n+1} -x_n$ between neighbouring points satisfy the condition $\frac{\Delta x_{n}}{\Delta x_{n-1}} \rightarrow +\infty$. In this paper half-line Schr\"odinger operators with point $\delta$- and $\delta^\prime$-interactions on a sparse set are considered. Assuming that strengths of point interactions tend to $\infty$ we give simple sufficient conditions for such Schr\"odinger operators to have non-empty singular continuous spectrum and to have purely singular continuous spectrum, which coincides with $\dR_+$.