Collapse to the Center and Ambiguity in the Asymptotic Behavior of the Off-Shell Scattering Amplitude in Singular Three-Body Problems

Some examples of equations for the there-body problem where there is an oscillating high-momentum behavior are discussed. Specifically, these are (i) the equation in the fixed-center approximation; (ii) the unitarized equation in the fixed-center approximation; (iii) the Skornyakov–Ter-Martirosyan equation; and (iv) equations involving operators that are used in effective field theory—that is, those that are expandable in positive-power series in momentum. It is shown that, in such problems, there arises a situation analogous to a collapse to the center—that is, an infinite number of bound states. The energy of these states is unbounded from below. In this sense, the situation in the models being considered is close to a collapse to the center in the two-body problem.