On the string equation with a singular weight belonging to the space of multipliers in Sobolev spaces with negative index of smoothness

We study spectral properties of the boundary-value problem
$$\begin{gather*} -y''-\lambda\rho y=0, \\ y(0)=y(1)=0, \end{gather*}$$
in the case when the weight $\rho$ belongs to the space $\mathcal M$ of multipliers from the space $\overset{\circ}{W}{}_2^1[0,1]$ to the dual space $\bigl(\overset{\circ}{W}{}_2^1[0,1]\bigr)'$. We obtain a criterion for the generalized derivative (in the sense of distributions) of a piecewise-constant affinely self-similar function to lie in $\mathcal M$. For general weights in this class we show that the spectrum of the problem is discrete and the eigenvalues grow exponentially. The nature of this growth is determined by the parameters of self-similarity. When the parameters of self-similarity reach the boundary of the set where $\rho\in\mathcal M$, the problem exhibits continuous spectrum.