Generalized quantum Zernike superintegrability and polynomial symmetry conjecture

Let $N$ be a positive integer and let $\gamma_k$ be coefficients satisfying $1\leq k\leq N$. Let $\hat{\mathcal{H}}_N$ be the quantum Hamiltonian, let $\hat{\mathcal{C}}$ be the quantum angular momentum operator, and let $\hat I_N$ and $\hat I'_N$ be the two additional operators. Let $\hat{\mathcal{K}}$, $\hat{\mathcal{K}}_+$, and $\hat{\mathcal{K}}_-$ denote the number and ladder operators, and let $\operatorname{Id}$ be the identity operator. Generalized quantum Zernike conjecture. For every $N$ and every choice of the coefficients $\gamma_k$, $\hat{\mathcal{H}}_N$ admits the three quantum symmetries $\hat{\mathcal{C}}$, $\hat I_N$, and $\hat I'_N$, where $\hat I_N$ and $\hat I'_N$ are quadratic in the momenta for $N=1,2$ and of $N^{\rm th}$ order for $N>2$. The sets $\{\hat{\mathcal{H}}_N,\hat{\mathcal{C}},\hat I_N\}$ and $\{\hat{\mathcal{H}}_N,\hat{\mathcal{C}},\hat I'_N\}$ consist of three algebraically independent operators, so that $\hat{\mathcal{H}}_N$ determines a superintegrable system. Moreover, the operators generated by $\{\hat{\mathcal{C}},\hat I_N,\hat I'_N\}$ yield $(\hat{\mathcal{K}}_1,\hat{\mathcal{K}}_2,\hat{\mathcal{K}}_3)$ and hence $(\hat{\mathcal{K}},\hat{\mathcal{K}}_+,\hat{\mathcal{K}}_-)$ satisfying the stated commutation relations with structure function $\Phi=\Phi_1\Phi_2$, where

$$\Phi_1(\hat{\mathcal{H}}_N,\hat{\mathcal{K}})=\frac14\left(\hat{\mathcal{H}}_N-\sum_{k=1}^N(2\mathrm{i})^k\gamma_k\hat{\mathcal{K}}^k\right),$$

and

$$\Phi_2(\hat{\mathcal{H}}_N,\hat{\mathcal{K}})=\hat{\mathcal{H}}_N-\sum_{k=1}^N(-2\mathrm{i})^k\gamma_k(\hat{\mathcal{K}}-\operatorname{Id})^k.$$

If the finite-dimensional representation is introduced with representation parameters $B$, $E$, and $u$, then the resulting structure function is obtained by substituting $\hat{\mathcal{H}}_N\mapsto E$ and $\hat{\mathcal{K}}\mapsto B+u$ in these factors, and the representation equations yield two solutions, Type I and Type II, for every $n\in\{1,2,\dots\}$ and arbitrary coefficients $\gamma_k$, with the displayed energies and structure functions. This conjectural extrapolation extends the explicitly solved cases $N<5$ and, in particular, the explicitly solved case $N=5$; its validity for arbitrary $N$ is obstructed by exponentially increasing computational complexity. If valid, the operators $(\hat{\mathcal{K}},\hat{\mathcal{K}}_+,\hat{\mathcal{K}}_-)$ should span a $(2N-1)^{\rm th}$-order polynomial symmetry algebra of Higgs type, denoted $\mathfrak{sl}^{(2N-1)}(2,\mathbb R)$.