Indefinite Sturm--Liouville problem for some classes of self-similar singular weights

We study the Sturm-Liouville problem $-y^{\mathrm{\prime }\mathrm{\prime }}-\rho y=0$, $y(0)=y(1)=0$. $\rho$ is a generalized derivative of function $P\in L_2[0,1]$. For self-similar $P$ asymptotic formulas for eigenvalues are obtained. In this paper we consider two cases of self-similarity of $P$: 1) arithmetic degenerate, and 2) non-arithmetic.