  We study the asymptotic behaviour of eigenvalues and eigenfunctions of a
boundary value problem for the Sturm-Liouville operator with general boundary
conditions and the weight function perturbed by the so-called $\delta'$-like
sequence $\varepsilon^{-2}h(x/\varepsilon)$. The eigenvalue problem is realized
as a family of non-self-adjoint matrix operators acting on the same Hilbert
space and the norm resolvent convergence of this family is established. We also
prove the Hausdorff convergence of the perturbed spectra.
