What is the Fourier transform of spherical harmonics?

What is the definition (or some sources) of the Fourier transform of spherical harmonics?

Solution 1:

This problem can be done simply at least formally: use the plane wave expansion into spherical harmonics and spherical Bessel functions,

$$e^{i\vec{k}\cdot\vec{r}} = (4\pi)\sum_{lm} i^l j_l(kr) Y_{lm}(\hat{k})Y_{lm}^*(\hat{r})$$

We then have

$$FT\{Y_{l'm'}(\hat{r})\}(\hat{k})=\sum_{lm}Y_{lm}(\hat{k})i^l\int d^3\vec{r} j_l(kr) Y_{lm}^*(\hat{r})Y_{l'm'}(\hat{r}).$$

The angular integral can be done by orthogonality of the spherical harmonics, setting $lm = l'm'$, so we find

$$FT\{Y_{l'm'}(\hat{r})\}(\hat{k})=Y_{lm}(\hat{k})4\pi i^l \int r^2 dr j_l(kr)$$.

The integral of $j_l$ against $r^2 dr$ is divergent because as $x\to \infty$ all the spherical Bessel functions behave as $1/x$. Nonetheless this is a useful representation because one can then resolve the $j_l$ integral into a Dirac delta function or derivatives of it. For instance, using the identity

$$\int j_l(ax) j_l(bx) x^2 dx =\frac{1}{2\pi b^2}\delta_D(a-b)$$

with $l=0$ and $a=0$, the first spherical Bessel function goes to unity and we have the desired integral for $l=0$. One should be able to use recursion relations to write the higher order $j_l$ in terms of lower order $j_l$ and their derivatives, and also Rayleigh's formula to write integrals using parametric differentiation where needed.

For instance,

\begin{align} \int x^2 dx j_2(kx) &= \int x^2 dx j_0(kx) +\frac{3}{k} \int x dx j_1(kx)\\ &=\frac{\pi}{2k^2}\delta_D(k)+\frac{3\pi}{2k^3}. \end{align}

We used the relation $j_{n+1}(x)=(2n+1)j_n(x)/x-j_{n-1}(x)$ to split the integral and then for the second integral that $\int x j_1(kx) dx=-\partial/\partial k \int j_0(kx)dx$, an application of Rayleigh's formula.