Theory of Atom–Diatom Collisions. IV. On Integral Equation Formalisms for Resonance Level Widths and Positions

Abstract

Integral-equation approaches to the level width and position of compound-state resonances are examined. A new description of the resonance state in terms of the eigenvectors and eigenvalues of the Hermitian operator QHQ + QHP[P / (E − PHP)] PHQ is explored in some detail. It is shown that one may utilize the formalism of Sams and Kouri to avoid expansions of the nonlocal effective potential in terms of approximate compound-state eigenfunctions. The procedure developed thus avoids difficulties such as neglect of the continuum of the closed channel projected Hamiltonian (or effective Hamiltonian) and should permit the evaluation of accurate level widths and positions by a single solution of a set of close-coupled inhomogeneous integral equations. These coupled equations are similar in nature to the original coupled open-and closed-channel scattering equations except that the inhomogeneity in the new equations is exactly that appropriate for describing the bound complex rather than an unbound scattering state. Finally, the integral equation technique may be used to study any of the various alternative descriptions of the compound state resonance.