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.


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.)
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