  Characterizations of eigenvalues and eigenfunctions of the Laplacian on a
product domain are obtained. When zero Dirichlet, Robin or Neumann boundary
conditions are specified on each factor, then the eigenfunctions on the product
domain are precisely the products of the eigenfunctions on the individual
factors. There is a related result when Steklov boundary conditions are
specified on the second factor. These results enable the characterization of
certain Hilbert-Sobolev spaces as tensor products and descriptions of some
orthogonal bases of the spaces. A different characterization of the trace space
is also found.
