The paper is devoted to the study of the similarity to self-adjoint operators of operators of the form L=−sign x|x|αp(x)d2dx2, α>−1 , in the space L2(R) with weight |x|αp(x) . As is well known, the answer to this problem in the case p(x)≡1 is positive; it was obtained by using delicate methods of the theory of Hilbert spaces with indefinite metric. The use of a general similarity criterion in combination with methods of perturbation theory for differential operators allows us to generalize this result to a much wider class of weight functions p(x) .