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 $\Delta x_n=x_{n+1}-x_n$ between neighbouring points satisfy the condition $\frac{\Delta x_n}{\Delta x_{n-1}}\to +\infty$. In this paper half-line Schrödinger 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 Schrödinger operators to have non-empty singular continuous spectrum and to have purely singular continuous spectrum, which coincides with $\dR_+$.

14 pages, submitted

References in corpus (1)

• Singular Continuous Spectrum for the Laplacian on Certain Sparse Trees