常曲率空间上的两体问题
Two-body problem on spaces of constant curvature
摘要：
考虑任意维常曲率单连通空间上具有中心相互作用的两体问题。通过径向微分算符和等距群的生成元，得到了量子两体哈密顿量的显式表达式，并构造了该哈密顿量的自伴扩张。计算了${\bf S}^3$空间中几种势函数的精确谱列。我们还描述了在李群的齐次空间上的约化经典力学系统的余伴随作用。利用这种方法给出了常曲率空间上约化的经典二体问题的描述。
英文摘要：
The two-body problem with a central interaction on simply connected constant curvature spaces of an arbitrary dimension is considered. The explicit expression for the quantum two-body Hamiltonian via a radial differential operator and generators of the isometry group is found. We construct a self-adjoint extension of this Hamiltonian. Some its exact spectral series are calculated for several potential in the space ${\bf S}^3$. We describe also the reduced classical mechanical system on a homogeneous space of a Lie group in terms of the coadjoint action of this group. Using this approach the description of the reduced classical two-body problem on constant curvature spaces is given.
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