One-dimensional Schroedinger operators with delta-prime-interactions on Cantor-type sets

We introduce a novel approach for defining a ${\delta }^{\mathrm{\prime }}$-interaction on a subset of the real line of Lebesgue measure zero which is based on Sturm-Liouville differential expression with measure coefficients. This enables us to establish basic spectral properties (e.g., self-adjointness, lower semiboundedness and spectral asymptotics) of Hamiltonians with ${\delta }^{\mathrm{\prime }}$-interactions concentrated on sets of complicated structures.