广义量子泽尼克哈密顿量：多项式Higgs型代数和代数谱的推导
Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum
AI总结：
本文研究了广义量子泽尼克系统的量子类比，通过构造合适的积分组合，揭示了多项式Higgs型对称代数，并推导了模型可能的能量谱，同时提出了两个猜想以推广结果。
摘要：
我们考虑由哈密顿量给出的广义泽尼克系统的量子类比：$$\hat{\mathcal{H}}_N =\hat{p}_1^2+\hat{p}_2^2+\sum_{k=1}^N γ_k (\hat{q}_1 \hat{p}_1+\hat{q}_2 \hat{p}_2)^k ,$$ 其中具有标准算符$\hat{q}_i,\, \hat{p}_i$和任意系数$γ_k$。这个二维量子模型除了角动量守恒外，还表现出在二维海森堡代数 enveloping 代数中的高阶积分。通过构造合适的积分组合，我们揭示了一个多项式Higgs型对称代数，通过适当的基变换，得到变形的振子代数。相关的结构函数$Φ$被证明可以分解为两个可交换的组成部分$Φ=Φ_1 Φ_2$。该框架使得能够代数地确定模型在$1\le N \le 5$情况下的可能能量谱，其中$N=1$情况与谐振子模型 canonically 等价。基于这些发现，我们提出两个猜想，以推广结果适用于所有$N\ge 1$和任意系数$γ_k$。此外，所有这些结果都可以被解释为原始量子泽尼克系统对应于$N=2$的更高阶超可积扰动，这些也被分析并应用于球面、双曲空间和欧几里得空间上的各向同性振子。
英文摘要：
We consider the quantum analog of the generalized Zernike systems given by the Hamiltonian: $$\hat{\mathcal{H}}_N =\hat{p}_1^2+\hat{p}_2^2+\sum_{k=1}^N γ_k (\hat{q}_1 \hat{p}_1+\hat{q}_2 \hat{p}_2)^k ,$$ with canonical operators $\hat{q}_i,\, \hat{p}_i$ and arbitrary coefficients $γ_k$. This two-dimensional quantum model, besides the conservation of the angular momentum, exhibits higher-order integrals of motion within the enveloping algebra of the Heisenberg algebra in two dimensions. By constructing suitable combinations of these integrals, we uncover a polynomial Higgs-type symmetry algebra that, through an appropriate change of basis, gives rise to a deformed oscillator algebra. The associated structure function $Φ$ is shown to factorize into two commuting components $Φ=Φ_1 Φ_2$. This framework enables an algebraic determination of the possible energy spectra of the model for the cases $1\le N \le 5$, the case $N=1$ being canonically equivalent to the harmonic oscillator. Based on these findings, we propose two conjectures which generalize the results for all $N\ge 1$ and any value of the coefficients $γ_k$. In addition, all of these results can be interpreted as higher-order superintegrable perturbations of the original quantum Zernike system corresponding to $N=2$, which are also analyzed and applied to the isotropic oscillator on the sphere, hyperbolic and Euclidean spaces
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