Quadratic symmetry algebras and Sturm algebraization of superintegrable systems with magnetic fields
Abstract
We study two three-dimensional quantum superintegrable systems in nonvanishing axially symmetric magnetic fields. For each model the integrals generate a quadratic algebra with a Casimir, and finite-dimensional deformed-oscillator representations yield algebraic spectra. The same spectra are recovered by separation of variables. In circular parabolic coordinates the separated equations are sextic biconfluent-Heun equations with finite polynomial sectors. We formulate them as Sturm eigenvalue problems with the physical energy as a parameter and the separation constant diagonalized. After gauge transformation, each equation admits an sl(2) algebraization and reduces to a finite tridiagonal matrix. We also construct a pre-separation Sturm representation of each complete Hamiltonian, taking the inverse-square coupling as the Sturm eigenvalue. The transformed integrals remain commuting and, after fixing the axial integral, changing variables and gauge rotating, the two-variable Sturm Hamiltonian and its non-central integrals belong to the universal enveloping algebra U(sl(3)). We thus obtain three complementary algebraic descriptions: the quadratic-algebra/deformed-oscillator representation of the original Hamiltonian, the sl(2) algebraization after parabolic separation, and the sl(3) algebraization of the complete Sturm problem before separation. Although the two magnetic Hamiltonians are distinct in the original classification, fixing the axial integral reduces them to the same singular 2:1 oscillator normal form with different effective parameter embeddings. This explains the common sl(3) structure without implying equivalence of the original magnetic systems. These results show how distinct hidden algebraic structures and exact or quasi-exact solvability can emerge at different stages of a multiseparable superintegrable system.
Related
- Symmetry and Invariant Theory of Quadratic Dynamics over Eight-Dimensional Composition Algebras
- Equivalence of superintegrable Poisson projection chains: magnetic cotangent bundles and spin Calogero-Moser models
- Superintegrable systems in curved spaces via coalgebras and deformed Racah algebras
- An elementary proof of the Artin--Springer theorem for generalized quadratic forms over quaternion algebras
- Varieties of associative algebras with quadratic codimension growth
BibTeX
@misc{ara2026quadratic,
title = {{Quadratic symmetry algebras and Sturm algebraization of superintegrable systems with magnetic fields}},
author = {Shams Ara and Md Fazlul Hoque and Ian Marquette},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01869},
eprint = {2610.01869},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01869},
}
RIS
TY - UNPB
TI - Quadratic symmetry algebras and Sturm algebraization of superintegrable systems with magnetic fields
AU - Shams Ara
AU - Md Fazlul Hoque
AU - Ian Marquette
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01869
UR - https://arxiv.org/abs/2610.01869
AB - We study two three-dimensional quantum superintegrable systems in nonvanishing axially symmetric magnetic fields. For each model the integrals generate a quadratic algebra with a Casimir, and finite-dimensional deformed-oscillator representations yield algebraic spectra. The same spectra are recovered by separation of variables. In circular parabolic coordinates the separated equations are sextic biconfluent-Heun equations with finite polynomial sectors. We formulate them as Sturm eigenvalue problems with the physical energy as a parameter and the separation constant diagonalized. After gauge transformation, each equation admits an sl(2) algebraization and reduces to a finite tridiagonal matrix. We also construct a pre-separation Sturm representation of each complete Hamiltonian, taking the inverse-square coupling as the Sturm eigenvalue. The transformed integrals remain commuting and, after fixing the axial integral, changing variables and gauge rotating, the two-variable Sturm Hamiltonian and its non-central integrals belong to the universal enveloping algebra U(sl(3)). We thus obtain three complementary algebraic descriptions: the quadratic-algebra/deformed-oscillator representation of the original Hamiltonian, the sl(2) algebraization after parabolic separation, and the sl(3) algebraization of the complete Sturm problem before separation. Although the two magnetic Hamiltonians are distinct in the original classification, fixing the axial integral reduces them to the same singular 2:1 oscillator normal form with different effective parameter embeddings. This explains the common sl(3) structure without implying equivalence of the original magnetic systems. These results show how distinct hidden algebraic structures and exact or quasi-exact solvability can emerge at different stages of a multiseparable superintegrable system.
ER -