ON HEARING THE SHAPE OF RECTILINEAR REGIONS

From complete knowledge of the eigenvalues of the negative Laplacian o n a bounded domain, one may extract information on the geometry and th e boundary conditions by analyzing the asymptotic expansion of a spect ral function. Explicit calculations are performed for an equilateral t riangular domain with Dirichlet or Neumann boundary conditions, yieldi ng in particular the comer angle terms. In three dimensions, some appl ications to eigenvalue problems for an equilateral triangular prism ar e dealt with, including the solid vertex terms.