On the Similarity of Some Differential Operators to Self-Adjoint Ones

The paper is devoted to the study of the similarity to self-adjoint operators of operators of the form $L=-\frac {\operatorname {sign}x}{|x|^\alpha p(x)}\,\frac {d^2}{dx^2}$, $\alpha >-1$, in the space $L_2(\mathbb R)$ with weight $|x|^\alpha p(x)$. As is well known, the answer to this problem in the case $p(x)\equiv 1$ 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 $p(x)$.