Three-dimensional Laplace spherical-harmonic addition theorem

Let ${\mathcal{P}}_{\mathrm{\ell }}$ be the homogeneous degree-$\mathrm{\ell }$ polynomials in three variables; it has $\left(\genfrac{}{}{0px}{}{\mathrm{\ell }+2}{2}\right)$ monomials. For a homogeneous polynomial $g$ of degree $d$, direct differentiation and Euler's identity give $\mathrm{\Delta }(\mathrm{\mid }x{\mathrm{\mid }}^{2s}g)=2s(2d+2s+1)\mathrm{\mid }x{\mathrm{\mid }}^{2s-2}g+\mathrm{\mid }x{\mathrm{\mid }}^{2s}\mathrm{\Delta }g$. If $\mathrm{\ell }\ge 2$, put $n=\mathrm{\ell }-2$, $J=\lfloor n\mathrm{/}2\rfloor$, $A_j=2(j+1)(2n-2j+3)$, $c_0=1\mathrm{/}{A}_0$, and $c_j=-c_{j-1}\mathrm{/}{A}_j$ for $1\le j\le J$. For $f\in {\mathcal{P}}_n$, the degree-$\mathrm{\ell }$ polynomial $q={\sum }_{j=0}^J{c}_j\mathrm{\mid }x{\mathrm{\mid }}^{2(j+1)}{\mathrm{\Delta }}^{j}f$ satisfies $\mathrm{\Delta }q=f$: the coefficients of $\mathrm{\mid }x{\mathrm{\mid }}^{2j}{\mathrm{\Delta }}^{j}f$ telescope by $c_0{A}_0=1$ and $c_j{A}_j+c_{j-1}=0$, and ${\mathrm{\Delta }}^{J+1}f=0$. Hence $\mathrm{\Delta }:{\mathcal{P}}_{\mathrm{\ell }}\to {\mathcal{P}}_{\mathrm{\ell }-2}$ is onto; for $\mathrm{\ell }=0,1$ the target is zero.

Restriction of a homogeneous polynomial to ${\mathbb{S}}^2$ is injective, because homogeneity recovers its value at every nonzero $x$ from its value at $x\mathrm{/}\mathrm{\mid }x\mathrm{\mid }$. Rank-nullity therefore gives $\dim {\mathcal{H}}_{\mathrm{\ell }}=\left(\genfrac{}{}{0px}{}{\mathrm{\ell }+2}{2}\right)-\left(\genfrac{}{}{0px}{}{\mathrm{\ell }}{2}\right)=2\mathrm{\ell }+1$ for $\mathrm{\ell }\ge 2$, and direct counting gives $1$ and $3$ for $\mathrm{\ell }=0,1$. Gram-Schmidt on any finite spanning list gives an orthonormal basis; no choice axiom is used.

For an orthonormal basis define $K_{\mathrm{\ell }}(\omega ,{\omega }^{\mathrm{\prime }})={\sum }_m{Y}_{\mathrm{\ell }m}(\omega )\overline{Y_{\mathrm{\ell }m}({\omega }^{\mathrm{\prime }})}$. A unitary basis change leaves this sum unchanged, and rotations carry one orthonormal basis of ${\mathcal{H}}_{\mathrm{\ell }}$ to another, so $K_{\mathrm{\ell }}$ is rotation-invariant. Thus $K_{\mathrm{\ell }}(\omega ,\omega )$ is constant in $\omega$; integrating it over the sphere gives $2\mathrm{\ell }+1$, so this constant is $(2\mathrm{\ell }+1)\mathrm{/}(4\pi )$.

Fix ${\omega }^{\mathrm{\prime }}$. After rotating it to the third coordinate axis, use $w=x_1+i{x}_2$ and $\bar{w}=x_1-i{x}_2$. A rotation sends these to $e^{i\theta }w$ and $e^{-i\theta }\bar{w}$, so an invariant monomial $w^a{\bar{w}}^b{x}_3^c$ must have $a=b$. The invariant homogeneous polynomials are therefore sums of $(x_1^2+x_2^2{)}^j{x}_3^{\mathrm{\ell }-2j}$, whose restrictions to the sphere are polynomials $f(t)$ of degree at most $\mathrm{\ell }$, with $t=x_3$. The polar-coordinate Laplacian is ${\mathrm{\partial }}_{rr}+2r^{-1}{\mathrm{\partial }}_r+r^{-2}{\mathrm{\Delta }}_{{\mathbb{S}}^2}$; applying it to $r^{\mathrm{\ell }}f(\cos \theta )$ gives ${\mathrm{\Delta }}_{{\mathbb{S}}^2}f=-\mathrm{\ell }(\mathrm{\ell }+1)f$. For a zonal $f$, the angular term is $(\sin \theta {)}^{-1}{\mathrm{\partial }}_{\theta }(\sin \theta {\mathrm{\partial }}_{\theta }f)=(1-t^2)f^{\mathrm{\prime }\mathrm{\prime }}-2t{f}^{\mathrm{\prime }}$ by $t=\cos \theta$, so $(1-t^2)f^{\mathrm{\prime }\mathrm{\prime }}-2t{f}^{\mathrm{\prime }}+\mathrm{\ell }(\mathrm{\ell }+1)f=0$. Writing $f(t)={\sum }_{k=0}^{\mathrm{\ell }}{a}_k{t}^k$ yields $(k+2)(k+1)a_{k+2}+(\mathrm{\ell }(\mathrm{\ell }+1)-k(k+1))a_k=0$. For $\mathrm{\ell }\ge 1$, the equation at $k=\mathrm{\ell }-1$, using $a_{\mathrm{\ell }+1}=0$, gives $2\mathrm{\ell }{a}_{\mathrm{\ell }-1}=0$; descending through $k=\mathrm{\ell }-3,\mathrm{\ell }-5,\dots$ forces every coefficient of parity opposite to $\mathrm{\ell }$ to vanish. For $k=\mathrm{\ell }-2,\mathrm{\ell }-4,\dots$, the coefficient $\mathrm{\ell }(\mathrm{\ell }+1)-k(k+1)$ is positive, so each lower coefficient of parity $\mathrm{\ell }$ is determined by the next one. Thus the zonal solution space has dimension at most one (for $\mathrm{\ell }=0$ it is the constants). The function $K_{\mathrm{\ell }}(\cdot ,{\omega }^{\mathrm{\prime }})$ is a zonal harmonic, and $K_{\mathrm{\ell }}({\omega }^{\mathrm{\prime }},{\omega }^{\mathrm{\prime }})=(2\mathrm{\ell }+1)\mathrm{/}(4\pi )>0$, so a nonzero such solution exists. Define $P_{\mathrm{\ell }}(t)=K_{\mathrm{\ell }}(\omega ,{\omega }^{\mathrm{\prime }})\mathrm{/}{K}_{\mathrm{\ell }}({\omega }^{\mathrm{\prime }},{\omega }^{\mathrm{\prime }})$ when $t=\omega \cdot {\omega }^{\mathrm{\prime }}$. Rotation invariance makes this depend only on $t$, and $P_{\mathrm{\ell }}(1)=1$. The diagonal value in step 3.1 now gives the displayed addition theorem.

Cauchy-Schwarz applied to the finite basis sum gives $\mathrm{\mid }{K}_{\mathrm{\ell }}(\omega ,{\omega }^{\mathrm{\prime }})\mathrm{\mid }\le \sqrt{K_{\mathrm{\ell }}(\omega ,\omega )K_{\mathrm{\ell }}({\omega }^{\mathrm{\prime }},{\omega }^{\mathrm{\prime }})}=(2\mathrm{\ell }+1)\mathrm{/}(4\pi )$. Dividing by the diagonal constant proves $\mathrm{\mid }{P}_{\mathrm{\ell }}(t)\mathrm{\mid }\le 1$.

Fix ${\omega }^{\mathrm{\prime }}\in {\mathbb{S}}^2$ and let $u(x)=\mathrm{\mid }{\omega }^{\mathrm{\prime }}-x{\mathrm{\mid }}^{-1}$ for $\mathrm{\mid }x\mathrm{\mid }<1$. The radial formula $\mathrm{\Delta }\mathrm{\mid }x-{\omega }^{\mathrm{\prime }}{\mathrm{\mid }}^{-1}=f^{\mathrm{\prime }\mathrm{\prime }}(r)+2f^{\mathrm{\prime }}(r)\mathrm{/}r=0$ away from ${\omega }^{\mathrm{\prime }}$ shows $u$ is harmonic in the unit ball. Its homogeneous Taylor term of degree $\mathrm{\ell }$ is therefore harmonic, since the homogeneous terms in $\mathrm{\Delta }u=0$ of each degree vanish separately; it is also invariant under rotations fixing ${\omega }^{\mathrm{\prime }}$. By step 4.1 it has the form $b_{\mathrm{\ell }}{r}^{\mathrm{\ell }}{P}_{\mathrm{\ell }}(\omega \cdot {\omega }^{\mathrm{\prime }})$. On the ray $x=r{\omega }^{\mathrm{\prime }}$, $u(r{\omega }^{\mathrm{\prime }})=(1-r{)}^{-1}={\sum }_{\mathrm{\ell }\ge 0}{r}^{\mathrm{\ell }}$, so coefficient comparison gives $b_{\mathrm{\ell }}=1$. Finally $u(r\omega )=(1-2r(\omega \cdot {\omega }^{\mathrm{\prime }})+r^2{)}^{-1\mathrm{/}2}$, whose Taylor series in $r$ has radius at least $1$: its possible denominator zeros are $t\pm i\sqrt{1-t^2}$, both of modulus one. This proves the generating identity for $0\le r<1$.