Variational principle for the bloch unified reaction theory

Abstract

The unified reaction theory formulated by Bloch uses a boundary value operator to write the Schr\"odinger equation for a scattering state as an inhomogeneous equation over the interaction region. As suggested by Lane and Robson, this equation can be solved by using a matrix representation on any set which is complete over the interaction volume. Lane and Robson have proposed, however, that a variational form of the Bloch equation can be used to obtain a "best" value for the $S$ matrix when a finite subset of this basis is used. We first consider the variational principle suggested by Lane and Robson, which gives a many-channel $S$ matrix different from the matrix solution on a finite basis, and show that the difference results from the fact that their variational principle is not, in fact, equivalent to the Bloch equation. We then present a variational principle which is fully equivalent to the Bloch form of the Schr\"odinger equation and show that the resulting $S$ matrix is the same as that obtained from the matrix solution of this equation.NUCLEAR REACTIONS Variational principle for the Bloch equation.