Algebraic solution of the problem of two Coulomb centres—the continuous spectrum

The two-Coulomb-centre problem for the continuous spectrum is treated in prolate spheroidal coordinates. Solutions of the one-dimensional equations that are obtained after separation of spatial variables in the Schrödinger equation are found for large distances $R$ between the Coulomb centres. The solutions are obtained in a closed algebraic form convenient for their further application. The obtained solutions present the expansions of the exact eigenvalues and eigenfunctions of the quasiradial and quasiangular equations in inverse powers of $R.$ The derived wavefunctions allow us to investigate completely the cosmological recombination problem, namely, to include in the calculation a quasimolecular mechanism of formation of atomic hydrogen in the early universe. (paper)