  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.
