On the Localization Conditions for the Spectrum of a Non-Self-Adjoint Sturm–Liouville Operator with Slowly Growing Potential

Abstract

We consider the Sturm–Liouville operator T₀ on the semi-axis (0,+∞) with the potential e^iθq, where 0 < θ < π and q is a real-valued function that may have arbitrarily slow growth at infinity. This operator does not meet any condition of the Keldysh theorem: T₀ is non-self-adjoint and its resolvent does not belong to the Neumann–Schatten class for any p < ∞. We find conditions for q and perturbations of V under which the localization or the asymptotics of its spectrum is preserved.