Hearing shapes of drums - mathematical and physical aspects of isospectrality

TL;DR

This paper reviews the mathematical and physical aspects of isospectrality, exploring the question of whether different shapes can produce identical spectral properties, a problem originating from Kac's famous question.

Contribution

It provides a comprehensive overview of the development and understanding of isospectrality, including the construction of noncongruent shapes with identical spectra.

Findings

Existence of nonisometric isospectral pairs confirmed in 1992

Mathematical techniques for constructing isospectral shapes explained

Physical implications of isospectrality discussed

Abstract

In a celebrated paper '"Can one hear the shape of a drum?"' M. Kac [Amer. 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.