Geodesic
Necessary Conditions for the Localization of the Spectrum of the Sturm--Liouville Problem on a Curve
Matematičeskie zametki, Tome 78 (2005) no. 1, pp. 72-84

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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 $\lambda_k\sim\pi\cdot k$, $k\to+\infty$. 
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     author = {Kh. K. Ishkin},
     title = {Necessary {Conditions} for the {Localization} of the {Spectrum} of the {Sturm--Liouville} {Problem} on a {Curve}},
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     url = {http://geodesic.mathdoc.fr/item/MZM_2005_78_1_a7/}
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Kh. K. Ishkin. Necessary Conditions for the Localization of the Spectrum of the Sturm--Liouville Problem on a Curve. Matematičeskie zametki, Tome 78 (2005) no. 1, pp. 72-84. http://geodesic.mathdoc.fr/item/MZM_2005_78_1_a7/