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**The two-body quantum mechanical problem on spheres.**
*(English)*
Zbl 1092.81019

Summary: The quantum mechanical two-body problem with a central interaction on the sphere \(\mathbb{S}^{n}\) is considered. Using recent results in representation theory, an ordinary differential equation for some energy levels is found. For several interactive potentials these energy levels are calculated in explicit form.

Reviewer’s addendum: The quantum mechanical problem of two-body central interactions on spaces of constant curvature as considered by the author [in I. M. Mladenov (ed.) et al., Proceedings of the international conference on geometry, integrability and quantization, Varna, Bulgaria, September 1–10, 1999. Sofia: Coral Press Scientific Publishing. 229–240 (2000; Zbl 1014.70009); Theor. Math. Phys. 118, No. 2, 197–208 (1999; Zbl 0966.81013); Rep. Math. Phys. 44, No.1–2, 191–198 (1999; Zbl 0964.81024)] and others [A. A. Bogush and V. S. Otchik, J. Phys. A, Math. Gen. 30, No. 2, 559–571 (1997; Zbl 0960.81075)], namely on the sphere \(\mathbb{S}^n\), is studied. In beginning, the basic information concerning differential operators on homogeneous spaces and regular representations of compact Lie group is presented. A description of the quantum two-body Hamiltonian on the sphere \(\mathbb{S}^n\) through radial differential operators and generators of the algebra \(\text{Diff}(\mathbb{S}_S^n)\) [the author, J. Phys. A, Math. Gen. 36, No. 26, 7361–7396 (2003; Zbl 1042.43002)] is given. For the cases of Coulomb and oscillator potentials, a separate ordinary second-order differential equation for the radial part of eigenfunctions is derived and reduced to the hypergeometric equation. As a result, an explicit form of some infinite energy level series is obtained and calculated.

In the appendices, orthogonal complex Lie algebras and their representations, self-adjointness of Schrödinger operators on Riemannian spaces and some Fuchsian differential equations are briefly described.

Reviewer’s addendum: The quantum mechanical problem of two-body central interactions on spaces of constant curvature as considered by the author [in I. M. Mladenov (ed.) et al., Proceedings of the international conference on geometry, integrability and quantization, Varna, Bulgaria, September 1–10, 1999. Sofia: Coral Press Scientific Publishing. 229–240 (2000; Zbl 1014.70009); Theor. Math. Phys. 118, No. 2, 197–208 (1999; Zbl 0966.81013); Rep. Math. Phys. 44, No.1–2, 191–198 (1999; Zbl 0964.81024)] and others [A. A. Bogush and V. S. Otchik, J. Phys. A, Math. Gen. 30, No. 2, 559–571 (1997; Zbl 0960.81075)], namely on the sphere \(\mathbb{S}^n\), is studied. In beginning, the basic information concerning differential operators on homogeneous spaces and regular representations of compact Lie group is presented. A description of the quantum two-body Hamiltonian on the sphere \(\mathbb{S}^n\) through radial differential operators and generators of the algebra \(\text{Diff}(\mathbb{S}_S^n)\) [the author, J. Phys. A, Math. Gen. 36, No. 26, 7361–7396 (2003; Zbl 1042.43002)] is given. For the cases of Coulomb and oscillator potentials, a separate ordinary second-order differential equation for the radial part of eigenfunctions is derived and reduced to the hypergeometric equation. As a result, an explicit form of some infinite energy level series is obtained and calculated.

In the appendices, orthogonal complex Lie algebras and their representations, self-adjointness of Schrödinger operators on Riemannian spaces and some Fuchsian differential equations are briefly described.

Reviewer: A. A. Bogush (Minsk)

### MSC:

81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |

81V99 | Applications of quantum theory to specific physical systems |

70G65 | Symmetries, Lie group and Lie algebra methods for problems in mechanics |

22E70 | Applications of Lie groups to the sciences; explicit representations |

57S25 | Groups acting on specific manifolds |

43A85 | Harmonic analysis on homogeneous spaces |

34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |