On the similarity of some singular differential operators to selfadjoint ones

The singular differential operator $Lf(x)=-\operatorname{sign}x\frac{d^2f(x)}{dx^2}+p(x)f(x)$ is studied. It is proved that if the second moment of $p$ is finite and $L$ has no nonreal eigenvalues, then $L$ is similar to a selfadjoint operator. The proof is based upon an integral resolvent criterion for the similarity applied to a wide class of functions $p(x)$.