Geodesic
The eigenvalue function of a family of Sturm--Liouville operators
Izvestiya. Mathematics , Tome 74 (2010) no. 3, pp. 439-459.

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We define a function $\mu^-(\gamma)$ in such a way that its value at every point $\gamma\in(-\infty,\pi)$, $\gamma=\beta-\pi n$, $\beta\in[0,\pi)$, $n=0,1,2,\dots$, coincides with an eigenvalue $\mu_n(\alpha,\beta)$ of the Sturm–Liouville problem $-y''+q(x)y=\mu y$, $y(0)\cos\alpha+y'(0)\sin\alpha=0$, $y(\pi)\cos\beta+y'(\pi)\sin\beta=0$ (for some $\alpha\,{\in}\,(0,\pi]$). We find necessary and sufficient conditions for a function to have this property for a real $q\in L^1[0,\pi]$.
Keywords: eigenvalue function, inverse problem.
Mots-clés : Sturm–Liouville problem 
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T. N. Harutyunyan. The eigenvalue function of a family of Sturm--Liouville operators. Izvestiya. Mathematics , Tome 74 (2010) no. 3, pp. 439-459. http://geodesic.mathdoc.fr/item/IM2_2010_74_3_a0/

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