On Recovering the Shape of a Domain from the Trace of the Heat Kernel

The problem of recovering geometric properties of a domain from the trace of the heat kernel for an initial-boundary value problem arises in NMR microscopy and other applications. It is similar to the problem of ``hearing the shape of a drum'', for which a Poisson type summation formula relates geometric properties of the domain to the eigenvalues of the Dirichlet or Neumann problem for the Laplace equation. It is well known that the area, circumference, and the number of holes in a planar domain can be recovered from the short time asymptotics of the solution of the initial-boundary value problem for the heat equation. It is also known that the length spectrum of closed billiard ball trajectories in the domain can be recovered from the eigenvalues or from the solution of the wave equation. This spectrum can also be recovered from the heat kernel for a compact manifold without boundary. We show that for a planar domain with boundary, the length spectrum can be recovered directly from the short time expansion of the trace of the heat kernel. The results can be extended to higher dimensions in a straightforward manner.