On hearing the shape of a drum: an extension to higher dimensions

The inverse eigenvalue problem for vibrating membranes (4), may also be examined in three or more dimensions. Let us suppose that λ_n are the eigen values of the problem

where Ω is a closed convex region or body in E_n and S is the bounding surface of Ω. The basic problem is to determine the precise shape of Ω on being given the spectrum of eigenvalues λ_n. In analogy with the membrane problem, it is clear that the trace function may be constructed in identical fashion; thus

where G(r, r', t) is the Green's function of the diffusion equation

and satisfies the Dirichiet condition G(r, r', t) = 0, r∈S, and the initial condition G(r, r', t) → δ(r–r') as t → 0.