Spectral Properties of the Non-Sectorial Sturm–Liouville Operator on the Semiaxis

The paper deals with some spectral properties of the Sturm–Liouville operator on the semiaxis $\mathbb{R}_+$ with a complex potential growing at infinity. Instead of the well-known V. B. Lidskii conditions concerning the boundedness from below of the real part or the semiboundedness of the imaginary part of the potential, it is assumed that the range of the potential is disjoint from some small sector containing the negative real semiaxis. Under some additional conditions on the potential, of the type of smoothness and regularity of the growth at infinity, it is shown that the numerical range of the operator fills the entire complex plane, the spectrum is discrete, there is a sector which is free from the spectrum, and any ray in this sector is a ray of the best decay of the resolvent. These facts are used to establish the basis property of the system of root vectors for the summation by the Abel–Lidskii method.