Upper Bounds on Scattering Lengths for Compound Systems:n−DQuartet Scattering

Abstract

In the zero-energy scattering of a particle by a compound system under the conditions that (1) only one exit channel is open (elastic scattering) and (2) no composite bound state exists for the particle and the scattering system in the state of given total angular momentum, the Kohn variational principle gives an upper bound on the scattering length. This is one of several results given previously for the case of scattering by a center of force which may be taken over directly, provided conditions (1) and (2) are satisfied. As a particular application of these results, several previous calculations of the $n-D$ quartet scattering length, $A_Q$, based on the Kohn principle (the method of Verde and the static approximation of Buckingham and Massey are included) are reanalyzed using the rigorous criterion that the best result is the one giving the lowest value. Further, some calculations of $A_Q$ based on the Rubinow formulation, which do not necessarily provide a bound, are converted to the Kohn form, thereby obtaining, in addition to a bound, an improved approximation to the scattering length. Some limitations and possible extensions of the method are discussed.