Asymptotics of the Eigenvalues of the Sturm–Liouville Problem with Discrete Self-Similar Weight

We study the asymptotics of the spectrum of the boundary-value problem
$$ -y''-\lambda\rho y=0,\qquad y(0)=y(1)=0, $$
for the case in which the weight $\rho\in\mathring W_2^{-1}[0,1]$ is the generalized (in the sense of distributions) derivative of a self-similar function $P\in L_2[0,1]$ of zero spectral order.