Sturm–Liouville problem with a weight function being non-compact multiplier

We consider a Sturm–Liouville problem with weight generated by a self-similar $n$-link multiplier in the Sobolev space with negative smoothness index. We show that in the case of non-compact multiplier the problem is equivalent to the spectral problem for a periodic Jacobi matrix. The period of the matrix depends on the parameters of self-similarity of the weight function. We discuss the spectrum structure for the Sturm–Liouville problem with the weight.