On the Neumann problem for Sturm-Liouville equation with self-similar Cantor type weight

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).