Crossing, unitarity, and the impulse approximation in pion-nucleus scattering

We present a theory of pion-nucleus scattering which incorporates crossing symmetry. The resulting nonlinear $T$ matrix equation is a generalization of the Low equation to the case of complex targets. If one neglects crossing symmetry in the formalism, the theory reduces to the linear theory of pion-nucleus scattering discussed previously. Our method requires the introduction of a non-Hermitian effective Hamiltonian (for the description of elastic scattering) whose definition is based on a projection operator technique. It is pointed out that one great advantage of our theory over other approaches which incorporate crossing symmetry lies in the fact that (in the simplest approximation) the driving term in our equation may be identified with the "triangle diagram," that is, the leading term in a covariant multiple scattering series for the relativistic optical potential. The approximations which are necessary to obtain familiar expressions for the driving term are made explicit in the context of this theory.