Two-particle resonance poles and the energy spectra of three- and four-particle systems

Abstract

The position of the two-particle resonance poles is shown to be the dominant factor in a number of interesting phenomena appearing in the energy spectra of bound three-and four-particle systems with separable two-particle interactions. The validity of the boundstate pole dominance assumption in few-particle systems is shown to depend on the location of the two-particle resonance poles. These results generalize the ground state collapse obtained earlier for the Tabakin rank-one separable interactions in the triton. However, our results also indicate that the collapse does not necessarily occur for more general nonlocal interactions.

Three-particle system for nonlocal interactions with continuum bound states

“…This result generalizes the ground state collapse [2,3] previously found for the rank-one separable interaction of Tabakin [4] having a continuum bound state (CBS), or equivalently a resonance of zero width. It was also found that the ground state collapse occurred sooner for the four-particle than for the three-particle system due to its higher density [1]. Another even more surprising result obtained was the fact that the binding energy of the collapsed state tends to increase when the resonance pole is shifted to higher energies [1], (e.g.…”

“…It was also found that the ground state collapse occurred sooner for the four-particle than for the three-particle system due to its higher density [1]. Another even more surprising result obtained was the fact that the binding energy of the collapsed state tends to increase when the resonance pole is shifted to higher energies [1], (e.g. Elab~2,000 MeV instead of Elab~330 MeV), which indicates that a CBS or a resonance pole near enough to the real axis is even more "dangerous" at very high energies than at lower energies.…”

“…Sofianos et al [1] previously showed that a ground state collapse, producing, a deeply bound ground state, occurs in the three particle system for separable two-body potentials of rank-one having a resonance pole sufficiently close to the real axis. This result generalizes the ground state collapse [2,3] previously found for the rank-one separable interaction of Tabakin [4] having a continuum bound state (CBS), or equivalently a resonance of zero width.…”

“…They show a similar trend, as a function of the position of the resonance pole, as the actual three particle binding energy. A similar indication for the occurrence of a ground state collapse for separable potentials of rank one is given by the three particle bound state propagator [1] and can be related to the sharp increase in the two-particle phase shift 6o(k ) (defined as the phase of Dr(k), where D*(k) is the corresponding Fredholm determinant [63.) However the picture for separable potentials of ranktwo was less clear.…”

“…However the picture for separable potentials of ranktwo was less clear. Sofianos et al [1] constructed several separable potentials of rank two all having the same two-nucleon bound state wave function. The rank one separable potential with the same bound state wave function, i.e.…”