Hyperspherical harmonics with arbitrary arguments

TL;DR

This paper introduces a new derivation scheme for hyperspherical harmonics with arbitrary arguments, providing explicit formulas for dimensions 2 to 6 and applications to quantum few-body problems.

Contribution

It presents a novel method to derive hyperspherical harmonics for arbitrary arguments and explicit formulas for dimensions 2 to 6, including applications to quantum few-body problems.

Findings

Explicit expressions for HSH in 2-6 dimensions.

Derived HSH for three- and four-body quantum problems.

Analyzed particular cases in 4D and 6D spaces.

Abstract

The derivation scheme for hyperspherical harmonics (HSH) with arbitrary arguments is proposed. It is demonstrated that HSH can be presented as the product of HSH corresponding to spaces with lower dimensionality multiplied by the orthogonal (Jacobi or Gegenbauer) polynomial. The relation of HSH to quantum few-body problems is discussed. The explicit expressions for orthonormal HSH in spaces with dimensions from 2 to 6 are given. The important particular cases of four- and six-dimensional spaces are analyzed in detail and explicit expressions for HSH are given for several choices of hyperangles. In the six-dimensional space, HSH representing the kinetic energy operator corresponding to i) the three-body problem in physical space and ii) four-body planar problem are derived.