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Recovering the shape of a quantum graph. (English) Zbl 1447.34026

The paper studies the inverse problem for equilateral quantum graphs with a small number of vertices. The Schrödinger operator on an equilateral metric graph with Kirchhoff’s matching conditions is considered. The questions investigated in the paper are the following: finding the shape of the graph from its spectrum and recovering the potential on the edges from the spectrum and the shape of the graph.
The authors find the eigenvalue asymptotics for simple connected graphs which are not trees. For trees they prove that the eigenvalues of the problem with a non-zero potential asymptotically approach to the eigenvalues of the problem without potential.
For graphs without potential and the number of vertices smaller or equal to 5 the authors prove that the shape of the graph is uniquely determined by their spectra. A similar result is obtained for tree graphs without potential and with the number of vertices smaller or equal to 8. Finally, the last result of the paper is a generalization of the geometric Ambarzumian’s theorem proven in [J. Boman et al., Integral Equations Oper. Theory 90, No. 3, Paper No. 40, 24 p. (2018; Zbl 1403.34030)]. The authors prove that, if the spectrum is the same as in the case without potential and the number of vertices is smaller or equal to the above values, the shape of the graph is uniquely determined and the potential is indeed zero.

MSC:

34A55 Inverse problems involving ordinary differential equations
34B45 Boundary value problems on graphs and networks for ordinary differential equations
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34B24 Sturm-Liouville theory

Citations:

Zbl 1403.34030
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References:

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