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

Vladimir Lotoreichik

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.

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