Geodesic
The inverse Sturm--Liouville problem with mixed boundary conditions
Algebra i analiz, Tome 21 (2009) no. 5, pp. 114-137.

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Let $H\psi=-\psi''+q\psi$, $\psi(0)=0$, $\psi'(1)+b\psi(1)=0$ be a selfadjoint Sturm-Liouville operator acting in $L^2(0,1)$. Let $\lambda_n(q,b)$ and $\nu_n(q,b)$ denote its eigenvalues and the so-called norming constants, respectively. A complete characterization of all spectral data $(\{\lambda_n\}_{n=0}^{+\infty};\{\nu_n\}_{n=0}^{+\infty})$ corresponding to $(q;b)\in L^2(0,1)\times\mathbb{R}$ is given, together with a similar characterization for fixed $b$ and a parametrization of isospectral manifolds. 
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E. L. Korotyaev; D. S. Chelkak. The inverse Sturm--Liouville problem with mixed boundary conditions. Algebra i analiz, Tome 21 (2009) no. 5, pp. 114-137. http://geodesic.mathdoc.fr/item/AA_2009_21_5_a5/

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