Oscillator rotation branching

Shells under ordinary spatial rotations

For an isotropic scalar oscillator on $\mathbb R^d$, $d\ge3$, the creation operators transform as a spatial vector. Shell $N$ therefore has the representation $\operatorname{Sym}^N(\mathbb C^d)$. Its polynomial representation decomposes as

$$\operatorname{Sym}^N(\mathbb C^d) =\bigoplus_{k=0}^{\lfloor N/2\rfloor}r^{2k}\mathcal H_{N-2k}^{(d)}, \qquad h_d(\ell)=\dim\mathcal H_\ell^{(d)} =\binom{\ell+d-1}{d-1}-\binom{\ell+d-3}{d-1}.$$

Here $\mathcal H_\ell^{(d)}$ denotes homogeneous harmonic polynomials; binomial coefficients with top smaller than bottom are zero. To see the decomposition, subtract successive traces (Laplacians) of a homogeneous polynomial. The identity

$$\Delta(r^{2k}h_\ell)=2k(2\ell+2k+d-2)r^{2k-2}h_\ell$$

has a nonzero coefficient for $k>0$, permitting recursive removal of each trace. Dimensions then telescope to $\binom{N+d-1}{d-1}$.

For the cumulative band $0\le N<n$, the multiplicity of $\mathcal H_\ell^{(d)}$ is

$$\left\lfloor\frac{n-1-\ell}{2}\right\rfloor+1 \quad(0\le\ell<n).$$

In particular the rotational scalar occurs once in each even shell, and $\lceil n/2\rceil$ times in the cumulative band. A shell's parity condition and a cumulative band's scalar content are different counts.

Restriction to a selected three-plane

For $d>3$ choose a spatial three-plane $O$ and perpendicular complement $O^\perp$ within the same $d$ directions. Under rotations of $O$ that fix $O^\perp$, the shell representation decomposes into a sum over $N_O+N_\perp=N$. A three-dimensional oscillator shell $N_O$ contains angular momenta $\ell=N_O,N_O-2,\ldots$, each once. The complementary shell contributes scalar multiplicity $g_{d-3}(N_\perp)=\binom{N_\perp+d-4}{d-4}$. Thus

$$m_d(\ell,N)=\sum_{j=0}^{\lfloor(N-\ell)/2\rfloor} \binom{N-\ell-2j+d-4}{d-4}, \qquad \sum_{\ell=0}^N(2\ell+1)m_d(\ell,N)=\binom{N+d-1}{d-1}.$$

Summing $m_d$ over $N<n$ gives the cumulative multiplicity. At $d=6,n=13$, its values for $\ell=0,\ldots,12$ are

$$(1036,784,581,420,295,200,130,80,46,24,11,4,1),$$

and their $(2\ell+1)$-weighted sum is $18564=S(13,6)$.

This is an exact restriction of the flat oscillator representation to a specified subgroup. Matching a dimension does not identify a unique group embedding.

• Normalized oscillator modes — gives shell counts and the separated scalar functions.

• Zonal hybrid geometry — uses the resulting three-dimensional angular functions for directed overlaps.

• Resonance and profile tools — overview and verification.

resonance_profile_checks.py checks finite examples by quadrature, matrix evolution, and independent algebra.