Can one hear the shape of a graph?

2001/05/08 by Boris Gutkin, Uzy Smilansky · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Matrix Theory and Algorithms #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #nlin.CD

paper · pdf · doi:10.1088/0305-4470/34/31/301

9 pages, 1 figure

arxiv created 2001/05/08 · openalex publication_date 2001/07/27 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

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.

Citations

• Periodic Orbit Theory and Spectral Statistics for Quantum Graphs
• Drums That Sound the Same
• Kirchhoff's rule for quantum wires

Cited by

• Nodal domains on isospectral quantum graphs: the resolution of isospectrality?
• Hearing shapes of drums: Mathematical and physical aspects of isospectrality