Necessary Conditions for the Localization of the Spectrum of the Sturm–Liouville Problem on a Curve

Abstract We consider the Sturm–Liouville operator on a convex smooth curve lying in the complex plane and connecting the points 0 and 1. We prove that if the eigenvalues $\lambda_k$ with large numbers are localized near a single ray, then this ray is the positive real semiaxis. Moreover, if the eigenvalues $\lambda_k$ are numbered with algebraic multiplicities taken into account, then $\lambda_k\sim\pi\cdot k$, $k\to+\infty$.