We want to consider the potential and solve on the Dirac equation:

Using the calculations of Camperosi and Higuchi, we know that in geodesic polar coordinates the eigenspinors have the factor

where is the eigenvalue number and . Let’s approximate by because we will consider the large radius . Now we are integrating over for which the volume form in polar coordinates will have a factor . If we were doing the calculation in latex then the factor of interest in the volume form would be which multiplied by produces an odd function and for we have even functions in and we have zero integral. However this vanishing does not hold for . These integrals would correspond to the inner product for eigenspinors where .

Now let us take a slightly more abstract approach. We want to calculate in order to solve the Dirac equation in eigenspinor expansions. We can essentially bring in the Dirac-squared on the right side and by dividing by and then use the self-adjointness and the Lichnerowicz formula:

where is the scalar curvature. Since the scalar curvature is constant, we have:

(*)

For the right side we can use the formula for the Laplacian in coordinates:

A small calculation gives which we can then plug in with in the integrand, use the fact that the volume form contains a . The right hand side of (*) is then the sum of and . Let’s call the right side and all of these will be finite. Then we have the expression

We can get some asymptotic approximation for by using the formula here:

and we focus on the term

Once we absorb the cosine and sine terms in the eigenspinor formula, we need only worry about and in our case. Then use the double angle formula for to get approximations for integrals of and .

Now let us return to the Dirac equation and its inner product with eigenspinor to examine the linear equation in eigenspaces. We have *either*

or the coefficient of eigenspinor in the solution expansion is zero. Since coefficients can be calculated before attempting to solve the Dirac operator (these are the effect of the multiplication by on the eigenspaces), in principle, we have a complete solution of the Dirac equation: for each , check whether $\lambda_n – \beta_{nl}$ is equal to . If so, then all the eigenfunctions in that eigenspace produce solutions to the Dirac equation; if not, then none of the eigenfunctions in the eigenspace enter into the solution to the Dirac equation.

EXPLICIT FORMULA

Gradsheyn and Ryzhik formulae in 7.391 allow us to get exact expressions for both and . For both, use the change of variables in the latter using . The general formula is (see GradshteynTableIntegrals):

where

and

We use and for integrals of both and . For the former, set and for the latter . So we have computable expressions for the right hand side of (*).

## Leave a Reply