  Spectrum of the Laplacian on spherical domains is analyzed from the point of
view of the heat equation on the cone. The series solution to the heat equation
on the cone is known to lead to a study of the Laplacian eigenvalue problem on
domains on the sphere in higher dimensions. It is found that the solution leads
naturally to a spectral function, a `generating function' for the eigenvalues
and multiplicities of the Laplacian, expressible in closed form for certain
domains on the sphere. Analytical properties of the spectral function suggest a
simple scaling procedure for estimating the eigenvalues. Comparison of the
first eigenvalue estimate with the available theoretical and numerical results
for some specific domains shows remarkable agreement.
