The shifted harmonic approximation and asymptotic SU(2) and SU(1,1) Clebsch--Gordan coefficients

TL;DR

This paper introduces the shifted harmonic approximation to derive asymptotic expressions for SU(2) and SU(1,1) Clebsch-Gordan coefficients, simplifying their calculation in large quantum number limits.

Contribution

The paper presents a novel approximation method that provides asymptotic formulas for Clebsch-Gordan coefficients of SU(2) and SU(1,1) groups, enhancing computational efficiency.

Findings

Derived asymptotic expressions for Clebsch-Gordan coefficients.

Showed eigenfunctions approach harmonic oscillator wave functions in large limits.

Provided a practical approximation method for complex group coefficients.

Abstract

Clebsch-Gordan coefficients of SU(2) and SU(1,1) are defined as eigenfunctions of a linear operator acting on the tensor product of the Hilbert spaces for two irreps of these groups. The shifted harmonic approximation is then used to solve these equations in asymptotic limits in which these eigenfunctions approach harmonic oscillator wave functions and thereby derive asymptotic expressions for these Clebsch--Gordan coefficients.