Sturm–Liouville problems on simple connected equilateral graphs of $$\le 5$$ ≤ 5 vertices and trees of $$\le 8$$ ≤ 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 ).

中文翻译：

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恢复量子图的形状

$$\le 5$$ ≤ 5 个顶点的简单连通等边图和 $$\le 8$$ ≤ 8 个顶点的树上的 Sturm-Liouville 问题在内部顶点处使用 Kirchhoff 和连续性条件，在悬垂处使用 Neumann 条件顶点和边上的相同势。证明如果此类问题的频谱不受干扰（例如在零电位的情况下），则该频谱唯一地确定了图形的形状和零电位。这是 Boman 等人的“几何”Ambarzumian 定理的推广。（Integral Equ. Oper. Theory 90:40, 2018. https://doi.org/10.1007/s00020-018-2467-1 ）。