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

TL;DR

This paper investigates the spectral properties of half-line Schrödinger operators with point interactions on a sparse set, establishing conditions under which these operators exhibit purely singular continuous spectra covering the positive real line.

Contribution

It provides new sufficient conditions for Schrödinger operators with point interactions on sparse sets to have purely singular continuous spectra.

Findings

Operators can have non-empty singular continuous spectrum under certain conditions.

Purely singular continuous spectrum can coincide with the entire positive real line.

Conditions relate to the sparsity of the interaction points and the strength of interactions.

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 $\dR_+$.