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Inverse spectral problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions. (English) Zbl 0960.34010

Theorems analogous to those of V. A. Marchenko [Dokl. Akad. Nauk SSSR, n. Ser. 72, 457-460 (1950; Zbl 0040.34301)] and H. Hochstadt [J. Differ. Equations 17, 220-235 (1975; Zbl 0266.34027)] are proved for the boundary value problem \[ -(p(x)y')'+q(x)y=\lambda r(x)y,\quad 0<x<1, \]
\[ y(0)\cos\alpha+y'(0)\sin\alpha=0,\quad (a\lambda+b)y(1)=(c\lambda+d)y'(1), \] where \(r(x), p(x)\in C^1\) are positive with absolutely continuous derivatives, \(q(x)\in L^1, \alpha, a, b, c, d\) are real, \(c\neq 0,\) and \(ad-bc>0.\) The case of two eigenparameter-dependent boundary conditions is also considered.

MSC:

34B07 Linear boundary value problems for ordinary differential equations with nonlinear dependence on the spectral parameter
34A55 Inverse problems involving ordinary differential equations
34L05 General spectral theory of ordinary differential operators
34B24 Sturm-Liouville theory
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