Hearing shapes of drums: Mathematical and physical aspects of isospectrality

2010/08/06 by Olivier Giraud, O. Giraud, Koen Thas +1 · 3 citations
Engineering · Mathematics · Medicine · Physics and Astronomy · Psychology · #Algebra over a field #Calculus (dental) #Cognitive science #Computer graphics (images) #Computer science #Drum #Engineering #Isospectral #Mathematical Dynamics and Fractals #Mathematical physics #Mathematics #Mechanical engineering #Medicine #Physics #Planar #Psychology #Pure mathematics #Quantum chaos and dynamical systems #Quantum mechanics #Scientific Research and Discoveries #Spectrum (functional analysis) #Theoretical physics #math-ph #math.MP #nlin.CD #quant-ph

paper · pdf · doi:10.1103/revmodphys.82.2213

published as Rev. Mod. Phys. 82, 2213 (2010) · 42 pages, 60 figures

openalex publication_date 2010/08/06 · arxiv created 2011/01/06 · arxiv updated 2015/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In a celebrated paper ``Can one hear the shape of a drum?'' M. Kac [Am. Math. Monthly 73, 1 (1966)] asked his famous question about the existence of nonisometric billiards having the same spectrum of the Laplacian. This question was eventually answered positively in 1992 by the construction of noncongruent planar isospectral pairs. This review highlights mathematical and physical aspects of isospectrality.

Citations

• Buildings of Spherical Type and Finite BN-Pairs
• Random Matrix Theories in Quantum Physics: Common Concepts
• Eine Identität zwischen Modulformen zweiten Grades
• Drums That Sound the Same
• Riemannian Coverings and Isospectral Manifolds
• A brief history of the classification of the finite simple groups
• Can one hear the shape of a graph?
• EIGENVALUES OF THE LAPLACE OPERATOR ON CERTAIN MANIFOLDS
• Spectral determination of analytic bi-axisymmetric plane domains
• Constructing Isospectral Manifolds
• Numerical investigation of isospectral cavities built from triangles
• The Geometries of 3-Manifolds
• Isospectral plane domains and surfaces via Riemannian orbifolds
• Quantum Phase Extraction in Isospectral Electronic Nanostructures
• The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics
• Level clustering in the regular spectrum
• Geometrical Theory of Diffraction*
• Which graphs are determined by their spectrum?
• Varietes Riemanniennes Isospectrales et non Isometriques
• Some planar isospectral domains
• Drums That Sound the Same
• Can One Hear the Shape of a Drum?

Cited by

• Relativistic Quantum Chaos in Neutrino Billiards
• A convexity result in the spectral geometry of conformally equivalent metrics on surfaces
• Hearing the shape of a drum [wikipedia]

Related

• We can't hear the shape of drum: revisited in 3D case
• Analytical aspects of isospectral drums
• Some planar isospectral domains
• Isospectrality For Orbifold Lens Spaces
• Designing the sound of a cut-off drum