Calculation of Energies and Widths of Resonances in Inelastic Scattering: Stabilization Method

In previous work, the stabilization method of calculating resonance parameters was applied to potential scattering and to elastic scattering from a target. The method is here extended to compound-state resonances in inelastic scattering and its application to a model problem for a target with three bound states is examined. The eigenfunctions associated with eigenvalues ${\epsilon }_j$ obtained from the diagonalization of the exact Hamiltonian in appropriately chosen sets of square-integrable basis functions are good approximations, in the inner region, to particular linear combinations of the degenerate exact scattering solutions at $E={\epsilon }_j$ (above inelastic threshold). The partial widths are calculated from a Fermi's-"Golden-Rule"-like formula involving the matrix elements of the exact Hamiltonian between the square-integrable eigenfunctions representing the resonance state and potential-scattering solutions at the same energy. The slowly varying (as a function of $E$) potential-scattering $S$ matrix, knowledge of which is required in the calculation of the decay widths, is determined using the criterion that several good approximations to the resonance state yield exactly the same widths. For the exactly soluble model problem studied here, the resonance parameters obtained with the stabilization method compare well with the exact values, especially for narrow resonances. The theoretical limitations of the method are discussed.