A non-relativistic model of two-particle decay IV. Relation to the scattering theory,spectral concentration,and bound states

The present paper which represents the fourth part of the series devoted to analysis of a simple Lee-type model of two-particle decay deals with three problems. The first one concerns relation of the model to the scattering theory. We prove asymptotic completeness for the elastic scattering of the two light particles and show that for a sufficiently weak coupling this system has just one resonance whose position is the same as that of the pole which yield the main contribution to the decay law. The second problem concerns spectral concentration; we prove its occurrence for families of intervals aroundE that shrink slower than quadratically ing. Finally, necessary and sufficient conditions for the existence of bound states are discussed.The authors are indebted to Dr. M. Sotona for Remark 5.7.