Permutation-symmetric three-particle hyper-spherical harmonics based on the S3⊗SO(3)rot⊂O(2)⊗SO(3)rot⊂U(3)⋊S2⊂O(6) subgroup chain

• Main Authors: Igor Salom, V. Dmitrašinović
• Format: Article
• Language: English
• Published: Elsevier 2017-07-01
• Series: Nuclear Physics B
• Online Access: http://www.sciencedirect.com/science/article/pii/S0550321317301542
• ISSN: 0550-3213; 1873-1562

We construct the three-body permutation symmetric hyperspherical harmonics to be used in the non-relativistic three-body Schrödinger equation in three spatial dimensions (3D). We label the state vectors according to the S3⊗SO(3)rot⊂O(2)⊗SO(3)rot⊂U(3)⋊S2⊂O(6) subgroup chain, where S3 is the three-body permutation group and S2 is its two element subgroup containing transposition of first two particles, O(2) is the “democracy transformation”, or “kinematic rotation” group for three particles; SO(3)rot is the 3D rotation group, and U(3),O(6) are the usual Lie groups. We discuss the good quantum numbers implied by the above chain of algebras, as well as their relation to the S3 permutation properties of the harmonics, particularly in view of the SO(3)rot⊂SU(3) degeneracy. We provide a definite, practically implementable algorithm for the calculation of harmonics with arbitrary finite integer values of the hyper angular momentum K, and show an explicit example of this construction in a specific case with degeneracy, as well as tables of K≤6 harmonics. All harmonics are expressed as homogeneous polynomials in the Jacobi vectors (λ,ρ) with coefficients given as algebraic numbers unless the “operator method” is chosen for the lifting of the SO(3)rot⊂SU(3) multiplicity and the dimension of the degenerate subspace is greater than four – in which case one must resort to numerical diagonalization; the latter condition is not met by any K≤15 harmonic, or by any L≤7 harmonic with arbitrary K. We also calculate a certain type of matrix elements (the Gaunt integrals of products of three harmonics) in two ways: 1) by explicit evaluation of integrals and 2) by reduction to known SU(3) Clebsch–Gordan coefficients. In this way we complete the calculation of the ingredients sufficient for the solution to the quantum-mechanical three-body bound state problem.