Explicit Representations for the T-Matrix on Unphysical Energy Sheets and Resonances in Two- and Three-Body Systems

We discuss the structure of the two- and three-body T-matrices, scattering matrices, and resolvents continued to the unphysical energy sheets. Our conclusions arise due to the representations that have been found for analytically continued momentum-space kernels of the T-operators. These representations are explicitly written only in terms of the physical-sheet kernels of the T-matrix itself. One of advantages of the representations in the three-body case is that they show which portions of the physical-sheet three-body scattering matrix are ``responsible'' for the resonances associated with a particular unphysical sheet. A resonance appears to be the energy where the correspondingly truncated scattering matrix (taken on the physical sheet) has eigenvalue zero. We also mention applications of this approach to some specific three-body systems, based on the Faddeev differential equations.Comment: Based on a lecture given at the International Workshop ``Critical Stability of Few-Body Quantum Systems'' (Dresden, October 17--22, 2005

Representations for Three-Body T-Matrix on Unphysical Sheets: Proofs

A proof is given for the explicit representations which have been formulated in the author's previous work (nucl-th/9505028) for the Faddeev components of three-body T-matrix continued analytically on unphysical sheets of the energy Riemann surface. Also, the analogous representations for analytical continuation of the three-body scattering matrices and resolvent are proved. An algorithm to search for the three-body resonances on the base of the Faddeev differential equations is discussed.Comment: 98 Kb; LaTeX; Journal-ref was added (the title changed in the journal

Representations for Three-Body T-Matrix on Unphysical Sheets

Explicit representations are formulated for the Faddeev components of three-body T-matrix continued analytically on unphysical sheets of the energy Riemann surface. According to the representations, the T-matrix on unphysical sheets is obviously expressed in terms of its components taken on the physical sheet only. The representations for T-matrix are used then to construct similar representations for analytical continuation of three-body scattering matrices and resolvent. Domains on unphysical sheets are described where the representations obtained can be applied.Comment: 123 Kb; LaTeX; Journal-ref was added (the title changed in the journal

Scattering and resonances in the ^4He three-atomic system

A mechanism of disappearance and formation of the Efimov levels of the helium ^4He_3 trimer is studied when the force of interatomic interaction is changed. The resonances including virtual levels are calculated by the method based on the solving the boundary value problem, at complex energies, for the Faddeev differential equations describing the (2+1 --> 2+1; 1+1+1) scattering processes.Comment: RevTeX, 5 pages; Contribution to Proceedings of the First International Conference on Modern Trends in Computational Physics, June 15-20, 1998, Dubna (Russia