Explicit asymptotics of coupling matrix elements for central potentials in the hyperspherical harmonics expansion method

TL;DR

This paper derives explicit asymptotic laws for coupling matrix elements in hyperspherical harmonics expansion, revealing how different potentials influence channel coupling decay and convergence in three-body quantum systems.

Contribution

It provides new explicit asymptotic formulas for coupling elements for various potentials, clarifying their decay behavior and impact on hyperspherical expansion convergence.

Findings

Short-range potentials decay algebraically as ρ^{-(2l+3)}

Coulomb potential coupling decays as 1/ρ, indicating persistent interactions

Results aid in truncating hyperradial domains for practical calculations

Abstract

The analytic structure and asymptotic behavior of channel-coupling potentials in three-body systems are investigated within the framework of the hyperspherical harmonics expansion method. The coupling between different Jacobi partitions is expressed using Raynal--Revai transformation coefficients and a reduced hyperangular integral that contains the two-body interaction. For central potentials, this integral is factorised into geometric and dynamical components. Explicit asymptotic scaling laws are derived for the hyperradial coupling strength in the limit of large hyperradius $\rho \to \infty$ for representative nuclear potentials: Gaussian, Yukawa, and Woods--Saxon potentials (short-range), and Coulomb potential (long-range). These short-range potentials are found to exhibit an algebraic decay $\propto \rho^{-(2\ell_{\eta_i}+3)}$, where $\ell_{\eta_i}$ is the orbital angular momentum of the interacting pair. This decay is shown to lead to efficient asymptotic decoupling of hyperspherical channels. In contrast, the Coulomb interaction yields couplings that decay only as $1/\rho$, indicating persistent channel coupling at large distances and explaining the slow convergence of hyperspherical expansions for charged systems. These results provide a quantitative basis for truncating the hyperradial domain, for example, in choosing a matching radius for scattering calculations or an upper integration limit in bound-state problems.