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 problems 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 is contained in the spectral density of the Laplace operator with the given boundary conditions in the domain, from which the short-time hyperasymptotics of the trace of the heat kernel can be obtained by the Laplace transform. However, the problem of recovering these lengths from measured values of the trace of the heat kernel (the "resurgence" problem) is unresolved. In this paper we develop a simple algorithm for extracting the lengths from the short-time hyperasymptotic expansion of the trace. We give an alternative construction of the short-time expansion of the trace by constructing a ray approximation to the heat kernel for a planar domain with Dirichlet or Neumann boundary conditions. We evaluate the trace by introducing the rays as global coordinates.