Non-accretive Sturm–Liouville operator with discrete spectrum

Abstract

For the first time, the Sturm-Liouville equations with a complex potential were studied by M.A. Naimark. M.A. Naimark managed to find sufficient conditions for a complex potential when the corresponding Sturm-Liouville operator on the semi-axis has a discrete spectrum. later, the result of M.A. Naimark was strengthened in the works of V.B. Lidskii. the conditions for the complex potential given by V.B. Lidskii guarantee the accretivity of the studied Sturm-Liouville operators. the question of the existence of non-discrete Sturm-Liouville operators with a discrete spectrum remained relevant. The proposed article provides an answer to this question. For the Sturm - Liouville equation with a complex potential, we have constructed a special solution that decreases at infinity and, for each fixed value of the independent variable, is an entire function of the spectral parameter. Using this solution, a generalization of the well-known theorem of V.B. Lidskii on the conditions on the potential under which the spectrum of the corresponding Sturm – Liouville operator is discrete and the system of root vectors is complete and minimal. In contrast to Lidskii’s work, instead of bounded below the real part or semi-boundedness of the imaginary part of the potential, it is only required that the region of potential values lie outside a certain angle of an arbitrary opening with a bisector along the negative real semiaxis.