Opuscula Math. 40, no. 6 (2020), 737-750

Opuscula Mathematica

Echoes and glimpses of a distant drum

John Wm. Turner

Abstract. To what extent does the spectrum of the Laplacian operator on a domain \(D\) with prescribed boundary conditions determine its shape? This paper first retraces the history of this problem, then Kac's approach in terms of a diffusion process with absorbing boundary conditions. It is shown how the restriction to a polygonal boundary for \(D\) in this method, which required taking the limit of an infinite number of sides to obtain a smooth one, can be avoided by using the Duhamel method.

Keywords: Kac drum problem, inverse methods, diffusion process.

Mathematics Subject Classification: 01A70, 35B30, 35J05, 35P20.

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  1. M. Berger, P. Gauduchon, E. Mazet, Le spectre d'une variété riemanienne, Lecture Notes in Mathematics, vol. 194, Springer-Verlag, Berlin-New York, 1971 [in French].
  2. O. Giraud, K. Thas, Hearing shapes of drums: Mathematical and physical aspects of isospectrality, Rev. Mod. Phys. 82 (2010), 2213-2255.
  3. C. Gordon, D.L. Webb, S. Wolpert, One cannot hear the shape of a drum, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 134-138.
  4. F. John, Partial Differential Equations, 4th edition, Applied Mathematical Sciences, Springer-Verlag, New York, 1982.
  5. M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), 1-23.
  6. H.P. McKean Jr., I.M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 (1967), 43-69.
  7. S. Minakshisundaram, Å. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canad. J. Math. 1 (1949), 242-256.
  8. F.W.J. Olver, The asymptotic expansion of Bessel functions of large order, Philos. Trans. Roy. Soc. London Ser. A 247 (1954), 328-368.
  9. Å. Pleijel, A study of certain Green's functions with applications in the theory of vibrating membranes, Ark. Mat. 2 (1954), 553-569.
  10. L. Smith, The asymptotics of the heat equation for a boundary value problem, Invent. Math. 63 (1981), 467-493.
  11. M. van den Berg, S. Srisatkunarajah, Heat equation for a region in \(\mathbb{R}^2\) with a polygonal boundary, J. London Math. Soc. (2) 37 (1988), 119-127.
  12. S. Zelditch, Inverse spectral problem for analytic domains. II, \(\mathbb{Z}_2\)-symmetric domains, Ann. of Math. (2) 170 (2009), 205-269.
  • Communicated by Jean Mawhin.
  • Received: 2020-03-03.
  • Revised: 2020-10-14.
  • Accepted: 2020-10-16.
  • Published online: 2020-12-01.
Opuscula Mathematica - cover

Cite this article as:
John Wm. Turner, Echoes and glimpses of a distant drum, Opuscula Math. 40, no. 6 (2020), 737-750, https://doi.org/10.7494/OpMath.2020.40.6.737

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