Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909050324647936 |
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| author | Campoamor-Stursberg, Rutwig Herranz, Francisco J. Latini, Danilo Marquette, Ian Blasco, Alfonso |
| author_facet | Campoamor-Stursberg, Rutwig Herranz, Francisco J. Latini, Danilo Marquette, Ian Blasco, Alfonso |
| contents | 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 |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_02491 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum Campoamor-Stursberg, Rutwig Herranz, Francisco J. Latini, Danilo Marquette, Ian Blasco, Alfonso Quantum Physics Mathematical Physics Exactly Solvable and Integrable Systems 37J35, 22E60, 46N50 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 |
| title | Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum |
| topic | Quantum Physics Mathematical Physics Exactly Solvable and Integrable Systems 37J35, 22E60, 46N50 |
| url | https://arxiv.org/abs/2502.02491 |