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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 λk with large numbers are localized near a single ray, then this ray is the positive real semiaxis. Moreover, if the eigenvalues λk are numbered with algebraic multiplicities taken into account, then λk ∼ π · k as k → +∞.
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