Low Energy Inverse Problems in Three-body Scattering

Scattering theory is the analysis of the motion of several interacting particles. Inverse problems in scattering theory seek the answer to the question: can one determine the interactions between particles by a scattering experiment, so, for instance, does the scattering matrix, or a part of the scattering matrix, determine the interaction? The answer, and the difficulty, greatly depends on the part of the scattering data one wishes to use. Before explaining the various settings, we remark that the inverse problems are highly non-linear since the scattering data do not depend linearly on the interactions. Hence, one of the usual methods in the field is to transfer the problem to an asymptotically linear one, which is then easier to analyze. The simplest setting is the study of high energy asymptotics of the scattering matrix, for then the potentials behave like small perturbations of the Laplacian. In addition, the leading term in the asymptotics depends linearly on them. This problem was studied, under various assumptions, by Enss and Weder [3, 4], Novikov [13] and Wang [20]. Here we are interested in finite energy problems, i.e. where the scattering data are known either only at a fixed energy, or in a fixed bounded interval of energies. Thus, the problems are not immediately equivalent to a linear perturbation problem. The flavor of the problem greatly depends on the part of the scattering matrix one wishes to use. In some situations a principal symbol calculation for an S-matrix allows one to use the 2-body inverse results. An example of this is free-to-free scattering: as shown in [17], in three-body scattering the singularities of the free-to-free S-matrix at energy λ > 0 determine the S-matrices in all proper subsystems at all energies in (0, λ), which then determine the pair interactions by two-body results. More precisely, the principal symbol of the part of the free-to-free S-matrix corresponding to a single collision is essentially given by the subsystem S-matrix at the energies in (0, λ). Slightly more involved arguments using the results of [19] are expected to work in the many-body setting to show that the free-to-free S-matrix determines all pair interactions. However, one may wish to study inverse problems where the parts of the S-matrix that are known do not have any singularities, so the previous method cannot be