Exact and phase-integral quantal matrix elements of a function, corresponding to a damped oscillation, between unbound states for a particle in a potential proportional to e-ax

Abstract

The Jackson-Mott formula, which gives an exact expression for the matrix elements of exp(-ax), where a>0, between unbound states for a particle moving in a real potential proportional to exp(-ax), has been generalised by Mies for states with different potentials (but with the same a). Eno and Balint-Kurti generalised the formula further to matrix elements of exp(-ax) raised to an arbitrary positive power. In the present treatment the authors make a further generalisation to matrix elements of a function corresponding to a damped oscillation exp(- gamma ax), where gamma is a complex number, between unbound states for a particle moving in a potential proportional to exp(-ax), where a>0. For the diagonal and non-diagonal matrix elements the resulting exact formula is compared to the corresponding phase-integral formula. When certain conditions are fulfilled, the respective phase-integral formula gives a very satisfactory result.