Recovering quantum graphs from their Bloch spectrum

Rueckriemen, Ralf

Annales de l'Institut Fourier, Tome 63 (2013) no. 3, pp. 1149-1176.

Nous définissons le spectre de Bloch d’un graphe quantique comme la fonction qui assigne à chaque élément de la cohomologie de deRham le spectre d’un opérateur de Schrödinger magnétique associé. On montre que le spectre de Bloch détermine le tore d’Albanese, la structure de bloc et la planarité du graphe. Il détermine un dual géometrique d’un graphe planaire. Cela nous permet de montrer que le spectre de Bloch identifie et détermine complètement les graphes quantiques planaires $3$-connexes.

We define the Bloch spectrum of a quantum graph to be the map that assigns to each element in the deRham cohomology the spectrum of an associated magnetic Schrödinger operator. We show that the Bloch spectrum determines the Albanese torus, the block structure and the planarity of the graph. It determines a geometric dual of a planar graph. This enables us to show that the Bloch spectrum indentifies and completely determines planar $3$-connected quantum graphs.

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