Geodesic
On the similarity of some singular differential operators to selfadjoint ones
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 28, Tome 270 (2000), pp. 336-349

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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 $p(x)$. 
@article{ZNSL_2000_270_a17,
     author = {M. M. Faddeev and R. G. Shterenberg},
     title = {On the similarity of some singular differential operators to selfadjoint ones},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {336--349},
     publisher = {mathdoc},
     volume = {270},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2000_270_a17/}
}
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M. M. Faddeev; R. G. Shterenberg. On the similarity of some singular differential operators to selfadjoint ones. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 28, Tome 270 (2000), pp. 336-349. http://geodesic.mathdoc.fr/item/ZNSL_2000_270_a17/