Oi. Tolstikhin et M. Matsuzawa, Hyperspherical elliptic harmonics and their relation to the Heun equation - art. no. 032510, PHYS REV A, 6303(3), 2001, pp. 2510
Hyperspherical elliptic (HSE) harmonics are the eigenfunctions of the gener
alized angular momentum operator obtained by separating variables in HSE co
ordinates. These functions depend on accessory parameters characterizing th
e HSE coordinate system and present a more flexible basis on a hypersphere
as compared with more familiar hyperspherical polar harmonics. We discuss a
special set of HSE harmonics arising in hyperspherical treatments of the t
hree-body problem in the HSE coordinate system introduced in an earlier pap
er [Tolstikhin et al., Phys. Rev. Lett. 74, 3573 (1995)]. The separation of
variables in these coordinates leads to the Heun equation, which is a gene
ralization of the Gauss hypergeometric equation. We develop an efficient me
thod to solve the corresponding one-dimensional eigenvalue problem and thus
construct the HSE harmonics, which opens a way for their application in th
e studies of various atomic, molecular, and nuclear three-body systems.