Geodesic
On the Uniqueness Criterion for Solutions of the Sturm--Liouville Equation
Matematičeskie zametki, Tome 84 (2008) no. 4, pp. 552-566.

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We consider the Sturm–Liouville equation $$ -y''+qy=\lambda^2y $$ in an annular domain $K$ from $\mathbb C$ and obtain necessary and sufficient conditions on the potential $q$ under which all solutions of the equation $-y''(z)+q(z)y(z)=\lambda^2y(z)$, $z\in\gamma$, where $\gamma$ is a certain curve, are unique in the domain $K$ for all values of the parameter $\lambda\in\mathbb C$.
Keywords: spectral problem, holomorphic function, uniqueness problem, Bessel function, Rouché theorem, meromorphic function
Mots-clés : Sturm–Liouville equation, simple pole. 
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Kh. K. Ishkin. On the Uniqueness Criterion for Solutions of the Sturm--Liouville Equation. Matematičeskie zametki, Tome 84 (2008) no. 4, pp. 552-566. http://geodesic.mathdoc.fr/item/MZM_2008_84_4_a6/

[1] X. K. Ishkin, “O neobkhodimykh usloviyakh lokalizatsii spektra zadachi Shturma–Liuvillya na krivoi”, Matem. zametki, 78:1 (2005), 72–84 | MR | Zbl

[2] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, t. 2: Funktsii Besselya, funktsii parabolicheskogo tsilindra, ortogonalnye mnogochleny, Nauka, M., 1974 | MR | Zbl

[3] F. Khartman, Obyknovennye differentsialnye uravneniya, Mir, M., 1970 | MR | Zbl

[4] B. Riman, Sochineniya, GITTL, M., 1948

[5] V. I. Arnold, Yu. S. Ilyashenko, “Obyknovennye differentsialnye uravneniya”, Itogi nauki i tekhniki. Sovrem. problemy matem. Fundam. napr., 1, VINITI, M., 1985, 7–146 | MR | Zbl

[6] Problemy Gilberta, ed. P. S. Aleksandrov, Nauka, M., 1969 | MR

[7] H. von Koch, “Sur les déterminants infinis et les équations différentielles linéares”, Acta Math., 16:1 (1892), 217–295 | DOI | MR | Zbl

[8] A. I. Markushevich, Kratkii kurs teorii analiticheskikh funktsii, Nauka, M., 1978 | MR

[9] B. A. Marchenko, Operatory Shturma–Liuvillya i ikh prilozheniya, Naukova dumka, Kiev, 1977 | MR | Zbl

[10] V. E. Lyantse, “Analog obratnoi zadachi teorii rasseyaniya dlya nesamosopryazhennogo operatora”, Matem. sb., 72:4 (1967), 537–557 | MR | Zbl

[11] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR | Zbl

[12] M. Rid, B. Saimon, Metody sovremennoi matematicheskoi fiziki, t. 1: Funktsionalnyi analiz, Mir, M., 1977 | MR | Zbl