On one condition for the discreteness of the spectrum and the compactness of the resolvent of a nonsectorial Sturm–Liouville operator on the semiaxis

The spectral properties of the Sturm–Liouville operator on the semi-axis with the complex-valued potential with the range exceeding the half-plane, has been little studied. The operator in this case can be non-sectorial, the numerical range can coincide with the entire complex plane. In this situation we propose the conditions ensuring the discreteness of the spectrum and the compactness of the resolvent.