标题： 广义量子Zernike哈密顿量：多项式Higgs型代数和谱的代数推导 显示英文标题

标题： Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum

摘要： 我们考虑由哈密顿量给出的广义泽尼克系统的量子模拟：$$ \hat{\mathcal{H}} _N =\hat{p}_1^2+\hat{p}_2^2+\sum_{k=1}^N \gamma_k (\hat{q}_1 \hat{p}_1+\hat{q}_2 \hat{p}_2)^k , $$，其中包含规范算子$\hat{q}_i,\, \hat{p}_i$和任意系数$\gamma_k$。 这个二维量子模型除了角动量守恒外，在海森堡代数的包络代数中还表现出高阶运动积分$\mathfrak h_2$。 通过构造这些积分的适当组合，我们发现了一种多项式Higgs型对称代数，通过适当的基变换，该代数产生一种变形振子代数。 相关的结构函数$\Phi$被证明可以分解为两个对易分量$\Phi=\Phi_1 \Phi_2$。 此框架使得可以代数地确定模型在情况$N=2,3,4$的可能能量谱，情况$N=1$与简谐振子是规范等价的。 基于这些发现，我们提出了两个猜想，这些猜想推广了所有$N\ge 2$和系数$\gamma_k$的任意值的结果，它们对于$N=5$已经被明确证明。 此外，所有这些结果都可以解释为对应于$N=2$的原始量子泽尼克系统的超可积扰动，这些结果也被分析并应用于球面、双曲和欧几里得空间中的各向同性振子。 显示英文摘要

摘要： We consider the quantum analog of the generalized Zernike systems given by the Hamiltonian: $$ \hat{\mathcal{H}} _N =\hat{p}_1^2+\hat{p}_2^2+\sum_{k=1}^N \gamma_k (\hat{q}_1 \hat{p}_1+\hat{q}_2 \hat{p}_2)^k , $$ with canonical operators $\hat{q}_i,\, \hat{p}_i$ and arbitrary coefficients $\gamma_k$. 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 $\mathfrak h_2$. 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 $\Phi$ is shown to factorize into two commuting components $\Phi=\Phi_1 \Phi_2$. This framework enables an algebraic determination of the possible energy spectra of the model for the cases $N=2,3,4$, 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 $N\ge 2$ and any value of the coefficients $\gamma_k$, that they are explicitly proven for $N=5$. In addition, all of these results can be interpreted as 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.