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

Abstract

We discuss some examples of equations of the three-body problem with the oscillating asymptotics at large momentum: (i) the fixed-center approximation, (ii) the unitarized equation in the fixed-center approximation, (iii) Skornyakov--Ter-Martirosyan equation, and (iv) equations with operators used in the effective field theory, i.e., which can be expanded in power-series with positive powers of momentum. We show that in the aforementioned three-body problems the situation analogous to the falling down to the center in the two-body problem takes place -- there appears an infinite number of bound states. The energy of these states is not bounded from below. In that sense the situation is close to the falling down to the center in the two-body problem.