On the Davies Formula for the Distribution of Eigenvalues of a Non-Self-Adjoint Differential Operator

In this paper, we analyze conditions under which the spectrum of the Sturm–Liouville operator on a certain smooth curve is localized near a countable number of rays. In the case where the potential is piecewise analytical, an asymptotic of eigenvalues is found for each series localized near the corresponding ray. The result obtained allows one to generalize the well-known formula for the asymptotic of the distribution function of the spectrum stated by Davies in the case of a finite number of localization rays.