Two-body problem on two-point homogeneous spaces, invariant differential operators and the mass center concept

Abstract

We consider the two-body problem with central interaction on two-point homogeneous spaces from the point of view of the invariant differential operators theory. The representation of the two-particle Hamiltonian in terms of the radial differential operator and invariant operators on the symmetry group is found. The connection of different mass center definitions for these spaces to the obtained expression for Hamiltonian operator is studied.

The two-body quantum mechanical problem on spheres

“…be the corresponding Laplace-Beltrami operator, expressed through local coordinates, where γ := det g ij . We start from the description of the two-body quantum Hamiltonian on S n found in [32] and [41]. The configurations space for the two-body system on S n is…”

“…where I = (0, πR) and the factor space G/K is G-homogeneous w.r.t. left shifts [32]. The space G/K is isomorphic to the unit sphere bundle over S n [33].…”

“…Taking into account all transformations used while reducing equation (32) to the hypergeometric one we get the following expression for radial eigenfunctions (up to an arbitrary constant nonzero factor)…”

“…In [32] there was found a polynomial expression for the quantum two-body Hamiltonian H on Q through a radial differential operator and generators of the algebra Diff(Q S ). Coefficients of this polynomial depend only on the distance between particles.…”

The Volume Swept out by a Moving Planar Region

“…The first notion of center of mass for regions in symmetric spaces was developed by Cartan [2], and has been generalized to other settings (see Berger [1]; Galperin [9] gives an extrinsic definition of center of mass in S n and H n ). The notion is more subtle than in R n and, in particular, its dynamical properties do not extend (for example, the center of mass of a freely-moving rigid body does not necessarily follow a geodesic), and there are competing notions for a substitute concept in this setting [10].…”