Wave Functions of Continuous Spectrum of the Coulomb Two-center Problem 1

The effective algorithm of the calculation of the wave functions of the continuous spectrum of the Coulomb two-center problem is proposed. The problem is solved by applying the finite-difference scheme of 4th-order and the continuous analog of the Newton method. The wave functions of the continuous spectrum of the electron of a positive hydrogen molecular ion together with the phase shifts are calculated and the corresponding pictures are presented. The absolute accuracy of the calculated phase shift is of the order of 10⁻⁶ for the electron momentum k ≥ 1 and the order of 10⁻⁴ for k ∼ 0.1. The typical matrix elements of operators of the third component of the momentum and kinetic energy of an electron between the states of the continuous and discrete spectrum are also calculated.