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

Abstract

The inverse eigenvalue problem for vibrating membranes (4), may also be examined in three or more dimensions. Let us suppose that λn are the eigen values of the problemwhere Ω is a closed convex region or body in En 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 λn. In analogy with the membrane problem, it is clear that the trace function may be constructed in identical fashion; thuswhere G(r, r', t) is the Green's function of the diffusion equationand 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.