Geodesic
Ambarzumian's Theorem for a Sturm--Liouville Boundary Value Problem on a Star-Shaped Graph
Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 2, pp. 78-81

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Ambarzumian's theorem describes the exceptional case in which the spectrum of a single Sturm–Liouville problem on a finite interval uniquely determines the potential. In this paper, an analog of Ambarzumian's theorem is proved for the case of a Sturm–Liouville problem on a compact star-shaped graph. This case is also exceptional and corresponds to the Neumann boundary conditions at the pendant vertices and zero potentials on the edges.
Keywords: inverse problem, Neumann boundary conditions, normal eigenvalue, multiplicity of an eigenvalue, least eigenvalue, minimax principle. 
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     author = {V. N. Pyvovarchyk},
     title = {Ambarzumian's {Theorem} for a {Sturm--Liouville} {Boundary} {Value} {Problem} on a {Star-Shaped} {Graph}},
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V. N. Pyvovarchyk. Ambarzumian's Theorem for a Sturm--Liouville Boundary Value Problem on a Star-Shaped Graph. Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 2, pp. 78-81. http://geodesic.mathdoc.fr/item/FAA_2005_39_2_a8/