RESUMEN
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 alsoknown 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.