Opuscula Math. 31, no. 4 (), 615-628
http://dx.doi.org/10.7494/OpMath.2011.31.4.615
Opuscula Mathematica

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

Abstract. We say that a discrete set $$X = \{ x_n \}_{n\in \mathbb{N}_0}$$ on the half-line $0 = x_0 \lt x_1 \lt x_2 \lt x_3 \lt ... \lt x_n \lt ... \lt +\infty$ 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 $$\delta'$$-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 $$\mathbb{R}_+$$.
Keywords: half-line Schrödinger operators, $$\delta$$-interactions, $$\delta '$$-interactions, singular continuous spectrum.
Mathematics Subject Classification: 34L05, 34L40, 47E05.