Harmonic polynomial dimension

[An explicit Laplacian right inverse with the correct degree] Fix $l\ge 2$ and $q\in {\mathcal{P}}_{l-2}$. Let $J=\lfloor (l-2)\mathrm{/}2\rfloor$ and $A_j=2(j+1)(2l-2j-1)$ for $0\le j\le J$. Every $A_j>0$. Define $c_0=(4l-2{)}^{-1}$ and $c_j=-c_{j-1}\mathrm{/}{A}_j$ for $1\le j\le J$, and put $S_{l}q={\sum }_{j=0}^J{c}_j{r}^{2j+2}{\mathrm{\Delta }}^{j}q$. Every term has degree $2j+2+(l-2-2j)=l$, and the whole polynomial lies in $r^2{\mathcal{P}}_{l-2}$. More generally [F1] gives $\mathrm{\Delta }(r^{2t}p)=2t(2d+2t+1)r^{2t-2}p+r^{2t}\mathrm{\Delta }p$ for $p\in {\mathcal{P}}_d$: compute $\mathrm{\nabla }{r}^{2t}=2t{r}^{2t-2}x$ and $\mathrm{\Delta }{r}^{2t}=2t(2t+1)r^{2t-2}$ and substitute Euler's identity. For $t=j+1$, $p={\mathrm{\Delta }}^{j}q$ this is $\mathrm{\Delta }(r^{2j+2}{\mathrm{\Delta }}^{j}q)=A_j{r}^{2j}{\mathrm{\Delta }}^{j}q+r^{2j+2}{\mathrm{\Delta }}^{j+1}q$. In $\mathrm{\Delta }{S}_{l}q$, the coefficient of $q$ is $c_0{A}_0=1$, each coefficient of $r^{2j}{\mathrm{\Delta }}^{j}q$ for $j\ge 1$ is $c_j{A}_j+c_{j-1}=0$, and the last leftover vanishes since ${\mathrm{\Delta }}^{J+1}q=0$. Hence $\mathrm{\Delta }{S}_{l}q=q$.

[Legendre parity makes the solid expression a polynomial] Rodrigues' formula in [F3] gives $P_l(-t)=(-1{)}^l{P}_l(t)$ by differentiating an even polynomial $l$ times. After $m$ more derivatives, $P_l^{(m)}(-t)=(-1{)}^{l-m}{P}_l^{(m)}(t)$, and its degree is $l-m$ with nonzero leading coefficient for $0\le m\le l$. Thus if $P_l^{(m)}(t)={\sum }_k{d}_k{t}^k$, every nonzero term has $0\le k\le l-m$ and $k\equiv l-m(\mathrm{m}\mathrm{o}\mathrm{d}2)$. Consequently $r^{l-m}{P}_l^{(m)}(x_3\mathrm{/}r)={\sum }_k{d}_k{x}_3^k(r^2{)}^{(l-m-k)\mathrm{/}2}$ is a polynomial: every exponent of $r^2$ is a nonnegative integer. Multiplication by $(-1{)}^m{b}_{lm}(x_1+i{x}_2{)}^m\mathrm{/}\sqrt{2\pi }$ makes the displayed $h_{lm}$ homogeneous of degree $l$. This polynomial determines the unique extension from $x\mathrm{\ne }0$, taking value zero for $l>0$ and the nonzero constant $b_{00}\mathrm{/}\sqrt{2\pi }$ for $l=0$; since $P_0=1$ and ${\int }_{-1}^1{P}_0^2=2$, this constant is $1\mathrm{/}\sqrt{4\pi }$. In angle coordinates $x_1+i{x}_2=r\sin \theta e^{i\phi }$, so the factor $(-1{)}^m$ gives exactly $h_{lm}=r^l{Y}_{lm}$, including the required convention.

[Polynomial Gaussian core and the top-degree isomorphism] Write $G(x)=e^{-r^2\mathrm{/}(2{\mathrm{\ell }}^2)}$. The leading terms in [F4] show by induction on degree that every complex polynomial $P$ of degree at most $D$ is a finite linear combination of the $Q_N$ with $\mathrm{\mid }N\mathrm{\mid }\le D$: subtract the matching nonzero multiples of all leading monomials of degree $D$ and apply the induction to the polynomial of lower degree. The same argument uses real coefficients when $P$ is real. Hence every $PG$ belongs to the oscillator's finite product core. An element of $E_M$ is uniquely $PG$ with degree at most $M$, because $G$ never vanishes; equality of the continuous functions as $L^2$ classes implies equality everywhere and thus polynomial uniqueness. On the basis $({\psi }_N{)}_{\mathrm{\mid }N\mathrm{\mid }=M}$, the degree-$M$ map ${\mathrm{\Phi }}_M$ takes each vector to a nonzero real multiple of its distinct monomial $x^N$. These monomials are a basis of ${\mathcal{P}}_M^{\mathbb{C}}$, so ${\mathrm{\Phi }}_M$ is directly an isomorphism, and its restriction to real spans is the stated real isomorphism. This avoids the incorrect assertion that different multi-indices necessarily have different energies.

[Direct sum and all dimensions] Multiplication by $r^2$ is injective on polynomials, so $r^2{\mathcal{P}}_{l-2}$ and ${\mathcal{P}}_{l-2}$ have the same finite dimension. Step 1.1 proves that $\mathrm{\Delta }:r^2{\mathcal{P}}_{l-2}\to {\mathcal{P}}_{l-2}$ is onto, hence bijective by [F2]. For $p\in {\mathcal{P}}_l$, $p-S_l(\mathrm{\Delta }p)\in {\mathcal{H}}_l$ and $S_l(\mathrm{\Delta }p)\in r^2{\mathcal{P}}_{l-2}$, proving the sum. If a vector in the intersection is $r^{2}q$ and harmonic, bijectivity makes $q=0$, proving directness. Thus $\dim {\mathcal{H}}_l=\left(\genfrac{}{}{0px}{}{l+2}{2}\right)-\left(\genfrac{}{}{0px}{}{l}{2}\right)=2l+1$ for $l\ge 2$; for $l=0,1$, every polynomial is harmonic and the dimensions are $1,3$. Applying the real decomposition separately to real and imaginary parts proves the complex decomposition. A real basis remains a complex basis of its explicit complexification, since taking real and imaginary parts verifies both spanning and independence, so the complex dimensions agree.

[Rotations preserve oscillator levels before intertwining] Let $f=PG\in E_M$ and $R\in SO(3)$. Since $\mathrm{\mid }{R}^{-1}x\mathrm{\mid }=\mathrm{\mid }x\mathrm{\mid }$, its pullback is $(P\circ R^{-1})G$, a polynomial Gaussian of degree at most $M$; step 1.3 places it in the finite product core, hence in $D(H)$. The chain rule and $R^{-1}(R^{-1}{)}^{\mathsf{T}}=I$ give $\mathrm{\Delta }(f\circ R^{-1})=(\mathrm{\Delta }f)\circ R^{-1}$, and multiplication by $r^2$ also commutes with the pullback. The exact differential expression of [F4] on the core therefore gives $H\pi (R)f=\pi (R)Hf={\lambda }_M\pi (R)f$. Expand the rotated vector in its finite product basis. Since the energies are ${\lambda }_K=\mathrm{\hslash }\omega (K+3\mathrm{/}2)$ for total degree $K$ and are distinct when $K\mathrm{\ne }M$, the eigen-equation kills every coefficient outside total degree $M$. Thus $\pi (R)f\in E_M$, and applying $R^{-1}$ proves equality of the preserved space. Only after establishing this invariance define $\pi (R)$ on $E_M$. An invertible linear substitution preserves each homogeneous degree, so the degree-$M$ part of $P\circ R^{-1}$ is $({\mathrm{\Phi }}_{M}f)\circ R^{-1}$. Hence ${\mathrm{\Phi }}_M\pi (R)=\rho (R){\mathrm{\Phi }}_M$, also on the real subspaces.

[Harmonicity, complex basis and its real counterpart] On the open polar-coordinate set $r>0$, $0<\theta <\pi$, [F1] gives that the inverse coordinates are $C^1$ and that their derivative is the inverse Jacobian of $T$. Differentiating the coordinate map with the sine and cosine derivatives in [F1], the Jacobian of $T$ has columns ${\mathrm{\partial }}_{r}T=(\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta )$, ${\mathrm{\partial }}_{\theta }T=(r\cos \theta \cos \phi ,r\cos \theta \sin \phi ,-r\sin \theta )$, and ${\mathrm{\partial }}_{\phi }T=(-r\sin \theta \sin \phi ,r\sin \theta \cos \phi ,0)$; its determinant is $r^2\sin \theta \mathrm{\ne }0$. Matrix inversion gives the inverse Jacobian rows, hence ${\mathrm{\partial }}_{x_1}=\sin \theta \cos \phi {\mathrm{\partial }}_r+\frac{\cos \theta \cos \phi }{r}{\mathrm{\partial }}_{\theta }-\frac{\sin \phi }{r\sin \theta }{\mathrm{\partial }}_{\phi }$, ${\mathrm{\partial }}_{x_2}=\sin \theta \sin \phi {\mathrm{\partial }}_r+\frac{\cos \theta \sin \phi }{r}{\mathrm{\partial }}_{\theta }+\frac{\cos \phi }{r\sin \theta }{\mathrm{\partial }}_{\phi }$, and ${\mathrm{\partial }}_{x_3}=\cos \theta {\mathrm{\partial }}_r-\frac{\sin \theta }{r}{\mathrm{\partial }}_{\theta }$. Squaring these operators and adding with the product rule, the mixed terms cancel, the second-derivative coefficients are $1$, $r^{-2}$, and $(r^2{\sin }^2\theta {)}^{-1}$, and the first-derivative coefficients are $2r^{-1}$, $r^{-2}\cot \theta$, and $0$. Thus $\mathrm{\Delta }={\mathrm{\partial }}_r^2+\frac{2}{r}{\mathrm{\partial }}_r+\frac{1}{r^2}({\mathrm{\partial }}_{\theta }^2+\cot \theta {\mathrm{\partial }}_{\theta }+\frac{1}{{\sin }^2\theta }{\mathrm{\partial }}_{\phi }^2)$. Applying this identity to $h_{lm}(x)=r^l{Y}_{lm}(\theta ,\phi )$ gives $\mathrm{\Delta }{h}_{lm}=r^{l-2}(l(l+1)Y_{lm}-\mathcal{S}{Y}_{lm})=0$ by [F3], where $\mathcal{S}$ is the angular differential expression from Sphere Laplacian diagonal domain, Sphere integrals in angle coordinates, The derivatives of sine and cosine are cosine and minus sine, The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$, The Euclidean inverse function theorem and $\mathcal{S}{Y}_{lm}=l(l+1)Y_{lm}$. The polynomial $\mathrm{\Delta }{h}_{lm}$ is continuous and vanishes on this dense set, hence vanishes everywhere. Its real and imaginary parts are therefore real harmonic polynomials. Negative indices follow by complex conjugation, and nonvanishing follows from the norm-one restriction $Y_{lm}$. Restriction is injective: if a homogeneous polynomial vanishes on $S^2$, then $p(x)=\mathrm{\mid }x{\mathrm{\mid }}^{l}p(x\mathrm{/}\mathrm{\mid }x\mathrm{\mid })=0$ for $x\mathrm{\ne }0$, and continuity also gives its value at zero. Thus the $2l+1$ orthonormal restrictions are linearly independent in a complex space of dimension $2l+1$ by step 2.1 and hence form a basis. The invertible pairwise transformation from $h_{lm},h_{l,-m}$ to $\sqrt{2}\mathrm{Re}{h}_{lm},\sqrt{2}\mathrm{Im}{h}_{lm}$, together with real $h_{l0}$, produces $2l+1$ real harmonic polynomials which are complex-linearly independent, hence real-linearly independent and a real basis.

[Full oscillator-level decomposition and multiplicities] Iterating the proved direct decomposition of step 2.1 yields ${\mathcal{P}}_M={⨁}_{j=0}^{\lfloor M\mathrm{/}2\rfloor }{r}^{2j}{\mathcal{H}}_{M-2j}$: at each substitution directness is preserved because multiplication by $r^2$ is injective. Complexifying gives the same result for ${\mathcal{P}}_M^{\mathbb{C}}$. A rotation commutes with the Laplacian, preserves degree, and preserves $r^2$, so each displayed harmonic-degree summand is invariant. Transfer it through the intertwining isomorphism of steps 1.3 and 2.2 to obtain exactly the stated decomposition of $E_M$, with one copy of each degree $M,M-2,\dots$. All listed degrees occur because their dimensions $2(M-2j)+1$ are positive; no other degree is part of the exhaustive direct decomposition. Their sum is $\left(\genfrac{}{}{0px}{}{M+2}{2}\right)$ by the polynomial-space dimension, and equals the number of oscillator basis vectors with $\mathrm{\mid }N\mathrm{\mid }=M$ by [F2] and [F4]. The same transfer over $\mathbb{R}$ gives the claimed real decomposition.

[Conclusion] Steps 1.1 and 2.1 prove (a) and (b), steps 1.2 and 3.1 prove (c), and steps 1.3, 2.2 and 3.2 prove (d1)–(d3) with the full real and complex spaces explicit. Countable Choice is retained for the invoked sphere and realised-oscillator suppliers; the finite polynomial recursions, finite basis changes and fixed top-degree maps require no additional choice principle.