The inverse spectral problem

1 Introduction The inverse spectral problem on a Riemannian manifold (M, g), possibly with boundary, is to determine as much as possible of the geometry of (M, g) from the spectrum of its Laplacian ∆ g (with some given boundary conditions). The special inverse problem of Kac is to determine a Euclidean domain Ω ⊂ R n up to isometry from the spectrum Spec B (Ω) of its Laplacian ∆ B with Dirichlet, Neumann or more general boundary conditions B. The physical motivation is to identify physical objects from the light or sound they emit, which may be all that is observable of remote objects such as stars or atoms. The inverse spectral problem is just one among many kinds of inverse problems whose goal is to determine a metric, domain or scatterer from physically relevant invariants. A comparison with other inverse problems shows just how small a set of invariants the spectrum is. For instance, the boundary inverse problem asks to determine the metric g on a fixed bounded domain Ω ⊂ M of a Riemannian manifold (M, g) from the spectrum of the Dirichlet Laplacian on L 2 (Ω), and from the Cauchy data ∂φ λ ∂ν