# the spectrum and determinant of the Laplacian on $S^3$

I came across the following statement in a paper:

On $S^3$, the eigenvalues of the vector Laplacian on divergenceless vector fields is $(\ell + 1)^2$ with degeneracy $2\ell(\ell+2)$ with $\ell \in \mathbb{ Z}$.

Is it possible to prove the spectrum and degeneracy using the representation theory of $SO(4)$? Perhaps there is a general result for the n-sphere.

The paper then proceeds to make the non-sense statement (RHS is divergent):

$$\det \big(-\Delta + a\big) = \prod_{\ell=1}^\infty \big((\ell + 1)^2 + a \big)^{2\ell(\ell+2)}$$

How do we make sense of the determinant of the Laplacian on the space of divergenceless vector fields?

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My question is how the spectra are calculated in the first place - using harmonic analysis - it was the 1st of several spectra in the paper. Then there is a separate question about regularizing the infinite product. Hardy published a book on divergent series. – cactus314 Aug 13 '13 at 15:30
Yes, the repn theory of $SO(4)$ is useful to determine the spectrum. For example, the functions on the 3-sphere are functions that descend to $SO(4)/SO(3)$. From the regular repn of $L^2(SO(4))$, functions on $S^3$ decompose as the sum of $\pi^{SO(3)}\otimes \check{\pi}$ where $\pi$ runs over irreducibles. This gives multiplicities in terms of those dimensions. The eigenvalues are eigenvalues of Casimir. – paul garrett Aug 13 '13 at 16:13
@paulgarrett OK. Then I have decompose (divergenceless) vector fields on $S^3$ - which I guess is not $L^2(S^3)\oplus L^2(S^3)$ - into eigenspaces. – cactus314 Aug 13 '13 at 16:25