Wave operators for multi-channel long-range scattering

Abstract

We consider a multichannel scattering system composed of N ⩾ 2 spinless, distinguishable, nonrelativistic particles, each with configuration space ℝ³ and interacting pairwise by local potentials V _ij (1 ⩽ i < j ⩽ N) consisting of long-range and short-range parts. Each V _ij can be chosen, roughly, with the same degree of generality as in ALSHOLM [3], when the configuration space in the latter reference is taken as ℝ³. Using techniques similar to those of ALSHOLM [2,3], we have proved for the class of potentials considered that suitable modified wave operators μ ^α_± . exist and have a generalized intertwining property for each channel α such that the corresponding bound states have a mild decay property of infinity. For the present class of V _ij 's, this property is known to be possessed by those bound states corresponding to eigenvalues of the discrete spectrum of the pertinent cluster Hamiltonian, or even to arbitrary nonthreshold eigenvalues if in addition the V _ij 's are dilatation analytic. We have also proved that the usual range-orthogonality property of the μ ^α_± holds under the conditions stated below. The results of this paper can be readily generalized to the case when the single-particle configuration space is ℝ^v(v ⩾ 1).