Quadratic symmetry algebras and Sturm algebraization of superintegrable systems with magnetic fields

The gist: The analysis provides three complementary algebraic descriptions of the spectrum for two three-dimensional quantum superintegrable systems in nonvanishing axially symmetric magnetic fields, revealing common hidden algebraic structures after symmetry reduction.

Quadratic Algebra and Deformed-Oscillator Representation

The study begins by examining the symmetry algebra generated by the integrals of motion, which close quadratically to form an associative algebra Q(3) possessing a Casimir operator The representation problem is then converted into a deformed-oscillator problem following the construction of Daskaloyannis This method yields algebraic energy spectra where the structure function phi depends on the central elements of the polynomial algebra and therefore on the energy The resulting spectral problem is reduced to finite-dimensional representation theory by imposing conditions such as Φ(0) = 0, Φ(p + 1) = 0, and Φ(n) > 0 for n = 1,..., p. This approach is particularly useful when a direct solution of the Schrödinger equation is possible because the algebraic spectrum can then be checked against separation of variables and the corresponding wavefunctions.

Sl(2) Algebraization after Parabolic Separation

After separation in circular parabolic coordinates, each separated equation admits an sl(2) algebraization and reduces to an explicit finite tridiagonal matrix. The Sturm formulation gives a useful bridge between these viewpoints by keeping the physical energy as a parameter and promoting another parameter, like a coupling constant or a separation constant, to the spectral variable. After fixing the axial integral and gauge rotating, the full two-variable Sturm Hamiltonian and its non-central integrals belong to the universal enveloping algebra U(sl(3)). This analysis provides three complementary algebraic descriptions of the spectrum: 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.

Sl(3) Algebraization of Complete Sturm Problem

The analysis shows that for both models, fixing the axial integral reduces them to the same singular 2:1 oscillator normal form with different effective parameter embeddings, explaining the common sl(3) structure without implying equivalence of the original magnetic systems. The complete two-variable Sturm problems admit sl(3) realizations. For Model I, after fixing Z = mħ and performing a gauge rotation, the resulting two-variable operators can be expressed in U(sl(3)). The common projective sl(3) realization is a property of the reduced/Sturm normal form (7.24), not evidence that the original magnetic systems are equivalent.

Separation of Variables and Exact Solvability Checks

The spectral information obtained from the quadratic algebra is checked against direct solutions in two coordinate systems: circular parabolic coordinates, which expose sextic equations leading to biconfluent-Heun type forms, and cylindrical coordinates, which reduce the problem to a singular radial oscillator and a shifted one-dimensional oscillator. Agreement between these two separations and the algebraic spectrum provides an important consistency check. For Model I, the first polynomial termination condition fixes the finite energy sector, while the second condition determines the separation constant via an explicit sl(2) Sturm construction, which makes this second condition explicit by showing that on PN the Sturm operator becomes a tridiagonal (N + 1)×(N + 1) matrix.

Sturm Representation of Complete Hamiltonians

The inverse-square term is particularly convenient for the complete two-variable problem because after fixing the axial integral, multiplication of the Schrödinger equation by ρ squared converts the coupling u2/ρ squared into an additive spectral term. By keeping E fixed and treating u2 as a Sturm eigenvalue, and transforming the commuting integrals at the operator level, one finds that the full Sturm problem gives both the allowed values of the coupling and their wavefunctions. The fact that both models lead to this same sl(3) realization is explained by their common reduced singular-oscillator normal form after fixing Z = mħ, not by an equivalence of their original magnetic fields.

Conclusion

The two Lie-algebraic descriptions have complementary roles: the complete two-variable Sturm problem admits an sl(3) algebraization before separation, whereas each one-variable parabolic Sturm equation admits an sl(2) algebraization after separation. The quadratic algebra Q(3) remains the symmetry algebra of the original superintegrable Hamiltonian. Taken together, these three structures connect the integrals of motion, finite-dimensional representations, separated special-function solutions and the Sturm coupling problem in a single framework. This common Lie-algebraic pattern should not be confused with equivalence of the original magnetic Hamiltonians. The two systems have the same reduced Hamiltonian and transformed quadratic-algebra templates after different effective parameter identifications, while the dependence of those parameters on the axial quantum number is different. The distinction between the classified magnetic systems is therefore retained when the fixed-m sectors are reassembled into the full three-dimensional problem. The two Lie-algebraic descriptions therefore have complementary roles. For each system, the complete two-variable Sturm problem admits an sl(3) algebraization before separation, whereas each one-variable parabolic Sturm equation admits an sl(2) algebraization after separation. The quadratic algebra Q(3) remains the symmetry algebra of the original superintegrable Hamiltonian. Taken together, these three structures connect the integrals of motion, finite-dimensional representations, separated special-function solutions and the Sturm coupling problem in a single framework. This common Lie-algebraic pattern should not be confused with equivalence of the original magnetic Hamiltonians. The two systems have the same reduced Hamiltonian and transformed quadratic-algebra templates after different effective parameter identifications, while the dependence of those parameters on the axial quantum number is different.

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Quadratic symmetry algebras and Sturm algebraization of superintegrable systems with magnetic fields Shams Araa, Md Fazlul Hoquea and Ian Marquetteb aPabna University of Science and Technology, Faculty of Science, Department of Mathematics, Pabna 6600, Bangladesh bDepartment of Mathematical and Physical Sciences, La Trobe University, Bendigo, VIC 3552, Australia Email: shams.tithi@gmail.com; fazlulmath@pust.ac.bd; i.marquette@latrobe.edu.au Abstract We study two three-dimensional quantum superintegrable systems in nonvanishing axially symmetric magnetic fields For each model the integrals of motion generate a quadratic algebra with a Casimir operator, and finite-dimensional deformed-oscillator representations yield algebraic energy spectra The same spectra are recovered independently by separation of variables In circular parabolic coordinates the separated equations are of sextic, biconfluent-Heun type and possess finite polynomial sectors We formulate these equations as Sturm eigenvalue problems in which the physical energy is kept as a parameter and the separation constant is diagonalized After a gauge transformation, each separated equation admits an sl(2) algebraization and reduces to an explicit finite tridiagonal matrix We then construct a Sturm representation of each complete Hamiltonian before separation by taking the inverse-square coupling as the Sturm eigenvalue The transformed integrals remain commuting operators and, after fixing the axial integral, changing variables and gauge rotating, the full two-variable Sturm Hamiltonian and its non-central integrals belong to the universal enveloping algebra U(sl(3)) The analysis therefore provides three complementary algebraic descriptions of the spectrum: 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 members of 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 This illustrates how distinct hidden algebraic structures, together with exact and quasi-exact solvability, can emerge at different stages of the analysis of a multiseparable superintegrable system.

1. Improved AI can perform exact spectral analysis of quantum superintegrable systems by utilizing 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. This allows for a comprehensive check between algebraic results and direct differential-equation calculations.

2. Improved AI can identify hidden algebraic structures that are shared across distinct magnetic systems by demonstrating the common sl(3) structure without implying equivalence of the original magnetic systems, which is achieved after fixing the axial integral and gauge rotating.

3. Improved AI can determine allowed physical energy branches by solving endpoint conditions of a deformed-oscillator module, as shown in Equation (3.4), and verifying that these energies are consistent with separation of variables.

4. Improved AI can explicitly calculate parabolic separation constants by analyzing the characteristic polynomial derived from the Sturm operator, which is shown to be a tridiagonal (N + 1)×(N + 1) matrix on the polynomial space PN.

5. Improved AI can derive explicit, exactly solvable wavefunctions for both circular parabolic and cylindrical coordinates by relating them to Laguerre–Hermite wavefunctions, providing an independent check of the algebraic construction against direct differential-equation solutions.

6. Improved AI can perform a third algebraic structure analysis by promoting the inverse-square coupling to a Sturm eigenvalue, which leads to a realization where the Sturm Hamiltonian and both non-central integrals belong to U(sl(3)) before separation.