Can one hear the shape of a graph?

Abstract. We show that the spectrum of the Schrödinger operator on a finite, metric graph determines uniquely the connectivity matrix and the bond lengths, provided that the lengths are non-commensurate and the connectivity is simple (no parallel bonds between vertices and no loops connecting a vertex to itself). That is, one can hear the shape of the graph! We also consider a related inversion problem: A compact graph can be converted into a scattering system by attaching to its vertices leads to infinity. We show that the scattering phase determines uniquely the compact part of the graph, under similar conditions as above. Can One Hear the Shape of a Graph? 2 1. Background and notations The question “Can one hear the shape of a drum?”, was posed by Marc Kac [1] as a paradigm example for a class of problems which is of fundamental importance in many physical applications: Given the quantum spectrum, can one deduce uniquely the basic interactions or the geometric constraints which specify the system? In Kac’s original paper, this inversion problem is formulated for Laplacians on compact domains with boundary conditions (billiards), and in spite of approximately fifty years of active

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