2004/01/14 by Lachieze-Rey Marc, Marc Lachièze-Rey, Marc, Lachieze-Rey
Mathematics · Physics and Astronomy · #43A90 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.SP #msc:43A90
paper · pdf · doi:10.48550/arxiv.math/0401153
10 pages, no figure
arxiv created 2004/01/14 · openalex publication_date 2004/01/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The vector space Vk of the eigenfunctions of the Laplacian on the three sphere S3, corresponding to the same eigenvalue lambdak = -k (k +2), has dimension (k + 1)2. After recalling the standard bases for Vk, we introduce a new basis B3, constructed from the reductions to S3 of peculiar homogeneous harmonic polynomia involving null vectors. We give the transformation laws between this basis and the usual hyper-spherical harmonics. Thanks to the quaternionic representations of S3 and SO(4), we are able to write explicitely the transformation properties of B3, and thus of any eigenmode, under an arbitrary rotation of SO(4). This offers the possibility to select those functions of Vk which remain invariant under a chosen rotation of SO(4). When the rotation is an holonomy transfor- mation of a spherical space S3/Gamma, this gives a method to calculates the eigenmodes of S3/Gamma, which remains an open problem in general. We illustrate our method by (re-)deriving the eigenmodes of lens and prism space. In a forthcoming paper, we derive the eigenmodes of dodecahedral pace.