Theory of Atom–Diatom Collisions. III. An Integral Equation Formalism for Level Widths and Shifts

An integral equation formalism for calculating level widths and shifts for Feshbach-type compound state resonances is presented. As in previous studies, the resonances are initially defined by means of the distortion approximation eigenstates of the closed channel projected Hamiltonian. The level shift then contains corrections to the real part of the resonance energy arising due to coupling of the distortion state to other closed channels as well as to the open channels. The level width contains effects due to both open and closed channels and far resonances. Because the Feshbach formalism is employed, the level width and shift are dependent on the scattering energy, in contrast to the Breit–Wigner parameterization of the resonances. The method of Sams and Kouri for obtaining noniterative solutions to integral equations is proposed as a practical method for obtaining accurate level widths and shifts beginning with the distortion approximation to the resonances. It results, in this method, that the solution of the integral equation for the level width and shift is very similar to that for the nonresonant scattering integral equation.