Bounds on Multichannel Scattering Parameters

Abstract

Using the projection techniques recently developed in the formal theory of reactions, it is shown that a bound on the exact reactance matrix $K$ is provided by the close-coupling reactance-matrix approximation $K^P$; that is, $K-K^P$ is, in a sense that can be made precise, a positive definite operator. This is of more than formal interest since the numerical solution of the finite number of coupled equations which arise when we allow the target system to be excited to only a restricted number of virtual states, and the determination of $K^P$ is feasible for a variety of three-body problems which includes, of course, three-body model problems. Furthermore, $K^P$ improves monotonically as one includes more and more virtual states. The recognition of this monotonicity property is useful in self-consistency analyses during the course of numerical calculations, and provides a more precise meaning for the numerical results obtained. Choosing a particular representation, the bound on $K$ generates bounds on the appropriately defined eigenphase shifts. The question of the absolute definition of phase shifts and of eigenphase shifts is discussed in some detail and it is shown that the presently used definition has serious deficiencies.