An example of embedded singular continuous spectrum for one-dimensional Schrödinger operators (Q2571663)

One-dimensional Schrödinger operators on \(L_2[0,\infty)\), given by \[ (H_\alpha u)(x)=-u''(x)+V(x)u(x), \quad u(0)\cos\alpha+u'(0)\sin\alpha=0, \quad \alpha\in[0,\pi), \] are considered. There is a decomposition of the spectrum of these operators corresponding to the decomposition of a measure on \(\mathbb{R}\) into pure point, absolutely continuous (with respect to the Lebesgue measure) and singular continuous part. The author presents a new example of a potential such that the corresponding Schrödinger operator has a singular continuous spectrum embedded in the absolutely continuous spectrum. This example generalizes a result of \textit{C. Remling} [Trans. Am. Math. Soc. 351, 2479--2497 (1999; Zbl 0918.34074)].