Geodesic
On the localization conditions for the spectrum of a non-self-adjoint Sturm--Liouville operator with slowly growing potential
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential equations. Spectral theory, Tome 141 (2017), pp. 48-60.

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We consider the Sturm–Liouville operator $T_0$ on the semi-axis $(0,+\infty)$ with the potential $e^{i\theta}q$, where $0\theta\pi$ and $q$ is a real-valued function that can have arbitrarily slow growth at infinity. This operator does not meet any condition of the Keldysh theorem: $T_0$ is non-self-adjoint and its resolvent does not belong to the Neumann–Schatten class $\mathfrak{S}_p$ for any $p\infty$. We find conditions for $q$ and perturbations of $V$ under which the localization or the asymptotics of its spectrum is preserved.
Keywords: non-self-adjoint differential operator, Keldysh theorem, spectral stability, localization of spectrum. 
@article{INTO_2017_141_a3,
     author = {L. G. Valiullina and Kh. K. Ishkin},
     title = {On the localization conditions for the spectrum of a non-self-adjoint {Sturm--Liouville} operator with slowly growing potential},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {48--60},
     publisher = {mathdoc},
     volume = {141},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2017_141_a3/}
}
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L. G. Valiullina; Kh. K. Ishkin. On the localization conditions for the spectrum of a non-self-adjoint Sturm--Liouville operator with slowly growing potential. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential equations. Spectral theory, Tome 141 (2017), pp. 48-60. http://geodesic.mathdoc.fr/item/INTO_2017_141_a3/