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 
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/
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     author = {D. Chelkak},
     title = {An application of the fixed point theorem to the inverse {Sturm{\textendash}Liouville} problem},
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     year = {2009},
     volume = {370},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_370_a11/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.

[1] G. Borg, “Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte”, Acta Math. (German), 78 (1946), 1–96 | DOI | MR | Zbl

[2] D. Chelkak, P. Kargaev, E. Korotyaev, “Inverse problem for harmonic oscillator perturbed by potential, characterization”, Comm. Math. Phys., 249:1 (2004), 133–196 | DOI | MR | Zbl

[3] D. Chelkak, E. Korotyaev, “Weyl–Titchmarsh functions of vector-valued Sturm–Liouville operators on the unit interval”, J. Funct. Anal., 257 (2009), 1546–1588 | DOI | MR | Zbl

[4] R. E. Edwards, Fourier Series. A Modern Introduction, vol. 2, Springer-Verlag, New York, 1982 | MR | Zbl

[5] B. M. Levitan, Inverse Sturm–Liouville problems, Translated from the Russian by O. Efimov, VSP, Zeist, 1987, 240 pp. | MR

[6] V. A. Marčenko, “Concerning the theory of a differential operator of the second order”, Dokl. Akad. Nauk SSSR, 72 (1950), 457–460 | MR | Zbl

[7] V. A. Marchenko, Sturm–Liouville operators and applications, Operator Theory: Adv. Appl., 22, Birkhauser Verlag, Basel, 1986, 367 pp. | DOI | MR | Zbl

[8] J. Pöschel, E. Trubowitz, Inverse spectral theory, Pure Appl. Math., 130, Academic Press, Inc., Boston, MA, 1987, 192 pp. | MR | Zbl

[9] M. Reed, B. Simon, Methods of Modern Mathematical Physics. Vol. I. Functional Analysis, Academic Press, New York, 1980 | MR | Zbl

[10] A. M. Savchuk, A. A. Shkalikov, “On the properties of maps connected with inverse Sturm–Liouville problems”, Proceedings of the Steklov Institute of Mathematics, 260, 2008, 218–237 | DOI | MR | Zbl