Spherical Harmonics - Higher Dimensions

Higher Dimensions

The classical spherical harmonics are defined as functions on the unit sphere S2 inside three-dimensional Euclidean space. Spherical harmonics can be generalized to higher dimensional Euclidean space Rn as follows. Let Pℓ denote the space of homogeneous polynomials of degree ℓ in n variables. That is, a polynomial P is in Pℓ provided that

Let Aℓ denote the subspace of Pℓ consisting of all harmonic polynomials; these are the solid spherical harmonics. Let Hℓ denote the space of functions on the unit sphere

obtained by restriction from Aℓ.

The following properties hold:

  • The spaces Hℓ are dense in the set of continuous functions on Sn−1 with respect to the uniform topology, by the Stone-Weierstrass theorem. As a result, they are also dense in the space L2(Sn−1) of square-integrable functions on the sphere.
  • For all ƒ ∈ Hℓ, one has
where ΔSn−1 is the Laplace–Beltrami operator on Sn−1. This operator is the analog of the angular part of the Laplacian in three dimensions; to wit, the Laplacian in n dimensions decomposes as
  • It follows from the Stokes theorem and the preceding property that the spaces Hℓ are orthogonal with respect to the inner product from L2(Sn−1). That is to say,
for ƒ ∈ Hℓ and g ∈ Hk for k ≠ ℓ.
  • Conversely, the spaces Hℓ are precisely the eigenspaces of ΔSn−1. In particular, an application of the spectral theorem to the Riesz potential gives another proof that the spaces Hℓ are pairwise orthogonal and complete in L2(Sn−1).
  • Every homogeneous polynomial P ∈ Pℓ can be uniquely written in the form
P(x) = P_\ell(x) + |x|^2P_{\ell-2} + \cdots + \begin{cases} |x|^\ell P_0 & \ell \rm{\ even}\\ |x|^{\ell-1} P_1(x) & \ell\rm{\ odd} \end{cases}
where Pj ∈ Aj. In particular,

An orthogonal basis of spherical harmonics in higher dimensions can be constructed inductively by the method of separation of variables, by solving the Sturm-Liouville problem for the spherical Laplacian

where φ is the axial coordinate in a spherical coordinate system on Sn−1. The end result of such a procedure is

Y_{l_1, \dots l_{n-1}} (\theta_1, \dots \theta_{n-1}) = \frac{1}{\sqrt{2\pi}} e^{i l_1 \theta_1} \prod_{j = 2}^{n-1} {}_j \bar{P}^{l_{n-1} - 1}_{l_j} (\theta_j)

where the indices satisfy |ℓ1| ≤ ℓ2 ≤ ... ≤ ℓn−1 and the eigenvalue is −ℓn−1(ℓn−1 + n−2). The functions in the product are defined in terms of the Legendre function

{}_j \bar{P}^l_{L} (\theta) = \sqrt{\frac{2L+j-1}{2} \frac{(L+l+j-2)!}{(L-l)!}} \sin^{\frac{2-j}{2}} (\theta) P^{-(l + \frac{j-2}{2})}_{L+\frac{j-2}{2}} (\cos \theta)

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