Geodesic
Self-conjugacy of abstract differential operators of the hyperbolic type
Matematičeskie zametki, Tome 20 (1976) no. 5, pp. 703-708.

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Sufficient conditions are obtained for the self-conjugacy of certain operators generated on a semiaxis or a complete axis by a differential expression of the form $l[y]=y''+ay-q(t)y$, where $A$ is a self-conjugate operator bounded below in a separable Hilbert space $H$, and, for almost all $t$, $q(t)$ is a bounded self-conjugate operator in $H$, locally summable with the square of the norm. 
@article{MZM_1976_20_5_a8,
     author = {L. I. Vainerman},
     title = {Self-conjugacy of abstract differential operators of the hyperbolic type},
     journal = {Matemati\v{c}eskie zametki},
     pages = {703--708},
     publisher = {mathdoc},
     volume = {20},
     number = {5},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1976_20_5_a8/}
}
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L. I. Vainerman. Self-conjugacy of abstract differential operators of the hyperbolic type. Matematičeskie zametki, Tome 20 (1976) no. 5, pp. 703-708. http://geodesic.mathdoc.fr/item/MZM_1976_20_5_a8/