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Inverse problem for the Sturm–Liouville equation on a star-shaped graph

✍ Scribed by V. Pivovarchik

Publisher
John Wiley and Sons
Year
2007
Tongue
English
Weight
268 KB
Volume
280
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

The problem of small vibrations of a graph consisting of n smooth inhomogeneous stretched strings joined at the vertex with the pendant ends fixed is reduced to the Sturm–Liouville boundary problem on a star‐shaped graph. The obtained problem occurs also in quantum mechanics. The spectrum of such a problem which consists of normal eigenvalues accumulating at infinity is investigated in comparison with the union of spectra of the Dirichlet–Dirichlet problems on the edges of the graph. It is shown that the eigenvalues of the spectra interlace in certain sense, thus an analogue of Sturm theorem is established. If the (n + 1) spectra (the spectrum of the boundary problem on the graph and the n spectra of the mentioned Dirichlet–Dirichlet problems) do not intersect the inverse problem of recovering the potentials on the edges from the (n + 1) spectra is uniquely solvable. The procedure of recovering of the potentials is presented. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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