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

Abstract

The paper is devoted to the study of the similarity to self-adjoint operators of operators of the form $L = - \frac{{{\text{sign }}x}}{{|x|^\alpha p(x)}}\frac{{d^2 }}{{dx^2 }},{\text{ }}\alpha >- 1$, 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 $p(x)$.