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

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_+.Comment: 14 pages, submitte

Asymptotics of resonances induced by point interactions

We consider the resonances of the self-adjoint three-dimensional Schr\"odinger operator with point interactions of constant strength supported on the set $X = \{ x_n \}_{n=1}^N$. The size of $X$ is defined by $V_X = \max_{\pi\in\Pi_N} \sum_{n=1}^N |x_n - x_{\pi(n)}|$, where $\Pi_N$ is the family of all the permutations of the set $\{1,2,\dots,N\}$. We prove that the number of resonances counted with multiplicities and lying inside the disc of radius $R$ behaves asymptotically linear $\frac{W_X}{\pi} R + \mathcal{O}(1)$ as $R \to \infty$, where the constant $W_X \in [0,V_X]$ can be seen as the effective size of $X$. Moreover, we show that there exist configurations of any number of points such that $W_X = V_X$. Finally, we construct an example for $N = 4$ with $W_X < V_X$, which can be viewed as an analogue of a quantum graph with non-Weyl asymptotics of resonances.Comment: 14 pages, 1 figure, submission to the proceedings of the 8th Workshop on Quantum Chaos and Localisation Phenomena, Warsaw, May 201