M Lachièze-Rey 2004 J. Phys. A: Math. Gen. 37 5625 doi:10.1088/0305-4470/37/21/008
Laplacian eigenmodes for the three-sphere
M Lachièze-Rey
Show affiliationsThe vector space 
of the eigenfunctions of the Laplacian on the three-sphere S3, corresponding to the same eigenvalue λk = −k(k + 2), has dimension (k + 1)2. After recalling the standard bases for 
, we introduce a new basis B3, constructed from the reductions to S3 of a peculiar homogeneous harmonic polynomial 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 explicitly the transformation properties of B3, and thus of any eigenmode, under an arbitrary rotation of SO(4). This offers the possibility of selecting those functions of 
which remain invariant under a chosen rotation of SO(4). When the rotation is a holonomy transformation of a spherical space S3/Γ, this gives a method for calculating the eigenmodes of S3/Γ, which remains an open problem in general. We illustrate our method by (re-)deriving the eigenmodes of lens and prism space. In a companion paper, we present the derivation for dodecahedral space.
- PACS
- MSC
-
15A18 Eigenvalues, singular values, and eigenvectors
- Subjects
- Dates
-
Issue 21 (28 May 2004)
Received 14 January 2004
Published 12 May 2004
-
Laplacian eigenmodes for the three-sphere
M Lachièze-Rey 2004 J. Phys. A: Math. Gen. 37 5625
-
Self-dual Yang–Mills fields in pseudo-Euclidean spaces
E K Loginov 2004 J. Phys. A: Math. Gen. 37 6599
-
Performance measurements of a depth-encoding PET detector module based on position-sensitive avalanche photodiode read-out
P A Dokhale et al 2004 Phys. Med. Biol. 49 4293
-
Electron angular distributions after above-threshold multiphoton detachment of
by 1064 nm radiation
Dmitry Telnov and Shih-I Chu 1996 J. Phys. B: At. Mol. Opt. Phys. 29 4401
-
Real-time prediction of respiratory motion based on local regression methods
D Ruan et al 2007 Phys. Med. Biol. 52 7137
-
Aligned InN nanofingers prepared by the ion-beam assisted filtered cathodic vacuum arc technique
X H Ji et al 2005 Nanotechnology 16 3069
-
Moment of inertia of doubly connecting bonds in two-dimensional bond percolation
J D Miller 1990 J. Phys. A: Math. Gen. 23 L551
-
The cyclicity of period annulus of a quadratic reversible Lotka–Volterra system
Chengzhi Li and Jaume Llibre 2009 Nonlinearity 22 2971
-
From Poincaré to affine invariance: how does the Dirac equation generalize?
Ingo Kirsch and Djordje Sijacki 2002 Class. Quantum Grav. 19 3157
-
The absence of finite-temperature phase transitions in low-dimensional many-body models: a survey and new results
Axel Gelfert and Wolfgang Nolting 2001 J. Phys.: Condens. Matter 13 R505
Related review articles
What's this?- Generalized Bernoulli-Hurwitz numbers and the universal Bernoulli numbers
- Universality of Wigner random matrices: a survey of recent results
- A property of the set of prime numbers