Multiple scattering by two centers of force

A solvable model for scattering by two centers of force is studied. The force at each center is assumed to be a separable force acting on the S wave only. A force law of this kind enables one to find the scattering amplitude for overlapping as well as nonoverlapping potentials analytically. The resulting scattering amplitudes can be used to compare the range of validity of the Born and other high energy approximations and also to test the accuracy of the result obtained by retaining the leading terms in the multiple scattering series. By examining the relation between the exact scattering amplitude for the two centers and the scattering amplitude of one center, it is shown that in the case of nonoverlapping potentials, the former amplitude can be expressed completely in terms of the latter. Hence no additional information about the nature of force results from the determination of the two-center scattering amplitude. However, for overlapping potentials, one can discriminate among equivalent potentials by finding the scattering amplitude for the two-center-of-force problem. Finally to justify the use of a potential which acts only in one partial wave (l = 0), the following observation has been made. If one calculates the scattering length for two nonoverlapping hard sphere interactions, using S-wave scattering by each center, and then repeats the calculation for the same interactions but considers S and P waves and their interference, the difference between the two results will be quite small.