Can one hear the shape of a graph

By Boris S. Gutkin, Uzy Smilansky. 2001

232 citations recorded by the source corpus.

We show that the spectrum of the Schroperator 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.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ddf94d56bcceee528a867092655c71857ba037af

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Graph Laplacians and topology (2008, methodology): “Note that the approach in [18] was developed further in [15] (and later in [19]).”
Schrödinger operators on graphs and geometry I: Essentially bounded potentials (2008, methodology): “On the other hand, very few papers are devoted to operators on arbitrary graphs (with cycles) and here we would like to mention J. von Below [5], R. Carlson [10], L. Friedlander [14], B. Gutkin and U. Smilansky [ 18 ], V. Kostrykin and R. Schrader [22], J. Boman, P. Kurasov, F. Stenberg and M. Nowac”
Inverse spectral problem for quantum graphs (2006, methodology): “Our presentation follows essentially [21, 24], but we were able to correct a few minor mistakes making the presentation mathematically rigorous.”
Inverse problem for the Sturm–Liouville equation on a star‐shaped graph (2007, background): “The paper [31] is entitled “Can one hear the shape of a network?” and the paper [14] is entitled “Can one hear the shape of a graph?” with reminiscence to the famous M.”
Homogeneous trees of second order Sturm-Liouville equations: A general theory and program (2008, background): “However, the beautiful construction of [115] shows that “one can hear the shape of a quantum graph” if one assumes rational independence of edges’ lengths.”