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Published online by Cambridge University Press: 30 May 2025
In this survey we review positive inverse spectral and inverse resonant results for the following kinds of problems: Laplacians on bounded domains, Laplace-Beltrami operators on compact manifolds, Schrödinger operators, Laplacians on exterior domains, and Laplacians on manifolds which are hyperbolic near infinity.
Marc Kac [1966], in a famous paper, raised the following question: Let Ωℝ be a bounded domain and let be the eigenvalues of the nonnegative Euclidean Laplacian ΔΩ with either Dirichlet or Neumann boundary conditions. Is Ω determined up to isometries from the sequence λ0, λ1, . . .? We can ask the same question about bounded domains in Rn, and below we will discuss other generalizations as well. Physically, one motivation for this problem is identifying distant physical objects, such as stars or atoms, from the light or sound they emit. These inverse spectral problems, as some engineers have recently proposed in [Reuter 2007; Reuter et al. 2007; 2009; Peinecke et al. 2007], may also have interesting applications in shape-matching, copyright and medical shape analysis.
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