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We study the asymptotic behaviour of eigenvalues and eigenfunctions of aboundary value problem for the Sturm-Liouville operator with general boundaryconditions and the weight function perturbed by the so-called $\delta'$-likesequence $\varepsilon^{-2}h(x/\varepsilon)$. The eigenvalue problem is realizedas a family of non-self-adjoint matrix operators acting on the same Hilbertspace and the norm resolvent convergence of this family is established. We alsoprove the Hausdorff convergence of the perturbed spectra.

On spectrum of strings with $\delta'$-like perturbations of mass density