On the localization conditions for the spectrum of a non-self-adjoint Sturm–Liouville operator with slowly growing potential

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.