STURM'S SOLUTIONS OF THE TWO CENTER PROBLEM IN QUANTUM MECHANICS

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.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.

math-phCross-listedarXiv:2610.01869Oct 2, 2026

ELEMENTARY SOLUTIONS OF THE QUANTUM PROBLEM OF THE MOTION OF A PARTICLE IN THE FIELD OF TWO COULOMB CENTERS.

Yu. N. Demkov

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The eigenvalues and wave functions of the two-centre problem in quantum mechanics

A. D. Nadezhin

USSR Computational Mathematics and Mathematical PhysicsJan 1, 19810 citations

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.