Particles, Astrophysics, and Nuclear Physics Seminar

The Coulomb problem of two fixed charge centers and an electron (ppe) is considered. It is integrable and admits a variable separation in prolate spheroidal coordinates. The problem has hidden algebraic structure sl₂ ⊕ sl₂.

Combining the WKB expansion at large distances (leading and sub-leading terms) and perturbation theory at small distances, a compact uniform approximation for eigenfunctions is constructed. For the two lowest eigenstates, 1s_g and 2p_u, this approximation provides the relative accuracy ≤ 10-5 (not less than 5 s.d.) for any real x in eigenfunctions and for the total energy E(R), it gives 10-11 significant digits (s.d.) for internuclear distances R in the range [0-50]. Corrections to the proposed approximations are evaluated perturbatively. Separation constants and the oscillator strength for the transition 1s_g↔2p_u are calculated and compared with existing numerical data.