Two-body problem on spaces of constant curvature

TL;DR

This paper investigates the quantum and classical two-body problem on constant curvature spaces, deriving explicit Hamiltonians, spectral series, and classical reductions, advancing understanding of dynamics in curved geometries.

Contribution

It provides explicit quantum Hamiltonians, constructs self-adjoint extensions, and describes classical reductions for two-body problems on curved spaces, which are novel contributions.

Findings

Explicit quantum two-body Hamiltonian derived

Spectral series calculated for specific potentials

Classical reduced systems described on homogeneous spaces

Abstract

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.