  Sturm-Liouville problem with generalized derivative of self-similar Cantor
type function as a weight is considered. Under Neumann and mixed boundary
conditions the oscillating properties of the eigenfunctions are studied. The
spectral asymptotics are made more precise then in previous papers. Namely, it
is shown that for known asymptotics $N(\lambda)=\lambda^D\cdot
[s(\ln\lambda)+o(1)]$ the function $s$ is a product of decreasing exponent and
nondecreasing purely singular function (and hence it is not constant).
