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## Eigenspinors of Dirac on S4 and Jacobi polynomials

The explicit calculation of eigenspinors of the Dirac operator on spheres given by Camporesi and Higuchi show that they are products of Jacobi polynomials with eigenspinors of lower dimensional spheres.  We can hope to glean some information about localization using products of Jacobi polynomials.  For $x \in [-1,1]$ the asymptotics are given by the Darboux formula
$P_n^{(a,b)}(\cos \theta) = n^{-1/2} k(\theta) \cos( N\theta + \gamma) + O(n^{-3/2})$

where

$k(\theta) = \pi^{-1/2} \sin^{-a-1/2}(\theta/2)\cos^{-b-1/2}(\theta/2)$

$N = n + \frac{a+b+1}{2}$

$\gamma = -(a+1/2)\pi/2$

The eigenspinors of the 4-sphere can be reduced to eigenspinors of the 3-sphere multiplied by functions of form, for $l \le n$

$\phi_{nl}(\theta) = (\cos \frac{\theta}{2})^{l+1}(\sin \frac{\theta}{2})^l P_{n-l}^{(2+l-1,2+l)}(\cos \theta)$

The case of $l=0$ shows that the $\cos(N\theta + \gamma)$ term disallows concentration or space localization.  This lack of localization is analogous to lack of localization of the Fourier series of the delta function.