Decomposition of K and T matrices for inelastic scattering using variational principles

A family of related approximate equations for K and T matrices applicable to inelastic scattering are derived from a variational principle using operator decompositions and projection operators. These approximations utilize the computational simplicity of first and second Born integrals but have the character of a resumjned series such as to extend the usefulness of perturbation theory to stronger interactions and to provide information on multiquanta transitions. The approximations can be viewed as a reorganization of perturbation theory information to yield high-order nonperturbative results. The size of the matrices that enter the decomposition formulation is controlled by the states included in the projection operator. Coupling between these states is included to all orders and is not restricted to transitions between states coupled directly by the potential. The remaining degrees of freedom are treated in a renormalized Born approximation. The formulations can be developed for K or T, either fully off-shell, half off-shell, of fully on-shell, and the final approximations are rational in the potential strength. The approximations to the K matrix are Hermitian so that the resulting S matrix is unitary.