  We analyze the spectrum of the generalized Schrodinger operator in
$L^2(R^\nu) \nu \geq 2$, with a general local, rotationally invariant singular
interaction supported by an infinite family of concentric, equidistantly spaced
spheres. It is shown that the essential spectrum consists of interlaced
segments of the dense point and absolutely continuous character, and that the
relation of their lengths at high energies depends on the choice of the
interaction parameters; generically the p.p. component is asymptotically
dominant. We also show that for $\nu=2$ there is an infinite family of
eigenvalues below the lowest band.
