Geodesic
An application of the fixed point theorem to the inverse Sturm--Liouville problem
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 40, Tome 370 (2009), pp. 203-218

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We consider Sturm–Liouville operators $-y''+v(x)y$ on $[0,1]$ with Dirichlet boundary conditions $y(0)=y(1)=0$. For any $1\le p\infty$, we give a short proof of the characterization theorem for the spectral data corresponding to $v\in L^p(0,1)$. Bibl. – 10 titles. 
@article{ZNSL_2009_370_a11,
     author = {D. Chelkak},
     title = {An application of the fixed point theorem to the inverse {Sturm--Liouville} problem},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {203--218},
     publisher = {mathdoc},
     volume = {370},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_370_a11/}
}
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D. Chelkak. An application of the fixed point theorem to the inverse Sturm--Liouville problem. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 40, Tome 370 (2009), pp. 203-218. http://geodesic.mathdoc.fr/item/ZNSL_2009_370_a11/