Upper Bounds on Scattering Lengths When Composite Bound States Exist

In the case of the zero-energy scattering of one compound system by another, where one real scattering length completely characterizes the problem (e.g., the reaction A+B→C+D, in addition to A+B→A+B, cannot take place) it has previously been shown that the Kohn-Hulth\'en variational principle provides an upper bound on the scattering length if no composite bound states exist. The extension of this result to the case where one or more composite bound states do exist is presented here. The inclusion of tensor forces, exchange forces, and Coulomb forces is allowed. Several methods are given for obtaining a rigorous upper bound on the scattering length, which involve the addition of certain positive terms to the Kohn-Hulth\'en variational expression. The approximate information about the composite bound states which is required to construct these additional terms can be found by standard methods. As a consequence of one of the results obtained, it is shown that under certain circumstances some ordinary variational calculations give a bound. Thus, an analysis of a previous calculation in the light of the present results leads, without further calculations, to a rigorous upper bound on the singlet electron-hydrogen scattering length.