Recovering the Shape of a Quantum Graph

Abstract

Sturm–Liouville problems on simple connected equilateral graphs of $$\le 5$$ vertices and trees of $$\le 8$$ vertices are considered with Kirchhoff’s and continuity conditions at the interior vertices and Neumann conditions at the pendant vertices and the same potential on the edges. It is proved that if the spectrum of such problem is unperturbed (such as in case of zero potential) then this spectrum uniquely determines the shape of the graph and the zero potential. This is a generalization of the ’geometric’ Ambarzumian’s theorem of Boman et al. (Integral Equ. Oper. Theory 90:40, 2018. https://doi.org/10.1007/s00020-018-2467-1 ).