On the dense point and absolutely continuous spectrum for Hamiltonians with concentric $\delta$ shells

We consider Schr\"odinger operator in dimension $d\ge 2$ with a singular interaction supported by an infinite family of concentric spheres, analogous to a system studied by Hempel and coauthors for regular potentials. The essential spectrum covers a halfline determined by the appropriate one-dimensional comparison operator; it is dense pure point in the gaps of the latter. If the interaction is radially periodic, there are absolutely continuous bands; in contrast to the regular case the measure of the p.p. segments does not vanish in the high-energy limit.