On the coexistence of absolutely continuous and singular continuous components of the spectral measure for some Sturm–Liouville operators with square summable potential

We prove that there exist some Sturm–Liouville operators with square summable potentials such that the singular continuous component of the spectral measure lies on the positive half-line. The Hausdorff dimension of the support of this singular measure can be arbitrary number from 0 to 1.