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

We say that a discrete set $X = { x_n }_{nin N_0}$ 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 $frac{Delta x_n}{Delta x_{n-1}} o +infty$. In this paper half-line Schr dinger operators with point $delta$- and $delta'$-interactions on a sparse set are considered. Assuming that strengths of point interactions tend to $infty$ we give simple sufficient conditions for such Schr dinger operators to have non-empty singular continuous spectrum and to have purely singular continuous spectrum, which coincides with $R_+$.