O(6) algebraic approach to three bound identical particles in the hyperspherical adiabatic representation

• Permutation-symmetric three-body O(6)O(6) hyper-spherical harmonics were constructed.
• Harmonics lead to simplifications in hyperspherical adiabatic representation.
• There are only four model-dependent coefficients for states with K<12K<12.
• Showing intrinsic limit of the method: the potential must be square-integrable.
• Calculated K<5K<5 shell spectra of homogeneous potentials, and in three specific cases.

We construct the three-body permutation symmetric O(6)O(6) hyperspherical harmonics and use them to solve the non-relativistic three-body Schrödinger equation in three spatial dimensions. We label the states with eigenvalues of the U(1)⊗SO(3)rot⊂U(3)⊂O(6)U(1)⊗SO(3)rot⊂U(3)⊂O(6) chain of algebras, and we present the K≤4K≤4 harmonics and tables of their matrix elements. That leads to closed algebraic form of low-K energy spectra in the adiabatic approximation for factorizable potentials with square-integrable hyper-angular parts. This includes homogeneous pairwise potentials of degree α≥−1α≥−1. More generally, a simplification is achieved in numerical calculations of non-adiabatic approximations to non-factorizable potentials by using our harmonics.