[Paper Review] Two-body problem on spaces of constant curvature

А. В. Щепетилов, И. Э. Степанова|ArXiv.org|Jan 7, 2005

Nonlinear Waves and Solitons24 references3 citations

TL;DR

This paper derives an explicit, geometrically natural expression for the quantum two-body Hamiltonian on spaces of constant curvature (spheres and hyperbolic spaces) using radial differential operators and generators of the isometry group. It proves self-adjointness, reduces the spectral problem to ordinary differential equations via group representation theory, and computes exact energy levels for specific potentials on $\mathbf{S}^3$, revealing quasi-exact solvability despite classical non-integrability.

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.

Motivation & Objective

• To provide a unified, geometrically natural expression for the two-body quantum Hamiltonian on constant curvature spaces $\mathbf{S}^n$ and $\mathbf{H}^n$.
• To establish the self-adjointness of the two-body Hamiltonian through a rigorous domain specification.
• To reduce the spectral problem to a system of ordinary differential equations indexed by irreducible representations of the isometry group.
• To compute explicit spectral series for specific interaction potentials on $\mathbf{S}^3$, demonstrating quasi-exact solvability.

Proposed method

• Express the quantum two-body Hamiltonian as a sum of a radial differential operator and invariant differential operators (generators of $\mathfrak{so}(n+1)$ or $\mathfrak{so}^*(1,n)$) acting on the configuration space.
• Use the method of Hamiltonian reduction via coadjoint orbits of the isometry group to derive the classical reduced Hamiltonian in terms of canonical coordinates on the cotangent bundle of a homogeneous space.
• Construct a self-adjoint extension of the quantum Hamiltonian by specifying a suitable domain using group representation theory and spectral theory.
• Reduce the spectral problem to a sequence of ordinary differential equations by decomposing the Hilbert space into irreducible representations of the isometry group.
• Solve the resulting radial equations explicitly for $\mathbf{S}^3$ under specific potentials, yielding closed-form energy levels.
• Derive the classical reduced Hamilton function from the quantum Hamiltonian structure using coadjoint orbit geometry, avoiding computer algebraic calculations.

Experimental results

Research questions

• RQ1 Can the quantum two-body Hamiltonian on constant curvature spaces be expressed in a geometrically canonical form using isometry group generators and radial operators?
• RQ2 Is the two-body Hamiltonian on $\mathbf{S}^n$ and $\mathbf{H}^n$ self-adjoint, and what is the correct domain for its self-adjoint extension?
• RQ3 Does the spectral problem for the two-body system on $\mathbf{S}^3$ admit exact solutions for certain potentials, despite classical non-integrability?
• RQ4 How does the structure of the reduced classical Hamiltonian on homogeneous spaces relate to the quantum Hamiltonian via coadjoint orbits?
• RQ5 What is the geometric and algebraic origin of the quasi-exact solvability observed in the $\mathbf{S}^3$ case?

Key findings

• The quantum two-body Hamiltonian on $\mathbf{S}^3$ is expressed as a sum of a radial differential operator and generators of $\mathfrak{so}(4)$, enabling spectral analysis via group representation theory.
• The Hamiltonian admits a self-adjoint extension, ensuring a well-defined quantum mechanical system with a discrete spectrum.
• For specific central potentials on $\mathbf{S}^3$, the spectral problem reduces to solving a finite set of ordinary differential equations, whose solutions yield explicit energy levels.
• The system exhibits quasi-exact solvability: all energy levels of certain radial equations are found in closed form, even though the full system is not classically integrable.
• The reduced classical Hamiltonian on $\mathbf{S}^3$ is derived geometrically via coadjoint orbits, yielding a natural form without reliance on computer algebra.
• The reduced phase space is diffeomorphic to $T^*I \times \mathcal{O}_\beta$, with the orbit $\mathcal{O}_\beta$ determined by the Casimir invariant $\mu$, and the Hamilton function explicitly expressed in terms of canonical coordinates on the cotangent bundle.