Modified Spherical Harmonics in Several Dimensions

摘要

A modification of the classical theory of spherical harmonics is presented. The space R-d = {(x(1), ..., x(d))} is replaced by the upper half space R-+(d) = {(x(1), ..., x(d)), x(d) > 0}, and the unit sphere Sd-1 in R-d by the unit half sphere S-+(d-1) = {(x(1), ..., x(d)) : x(1)(2) + center dot center dot center dot + x(d)(2) = 1, x(d) > 0}. Instead of the Laplace equation Delta h = 0 we shall consider the Weinstein equation x(d) Delta u + (d - 2) partial derivative u/partial derivative x(d) = 0. The Euclidean scalar product for functions on Sd-1 will be replaced by a non-Euclidean one for functions on S-+(d-1). It will be shown that in this modified setting all major results from the theory of spherical harmonics stay valid. In case d = 3 and d = 4 the modified theory has already been treated.