Narrow Results

A general recipe is given how to use the method of complex scaling in calculation of scattering observables. The physical quantity has to be expressed in an explicit form in terms of the total Green operator and this operator have to be replaced with the continuum discretized complex scaled Green-operator. Contrary to earlier expectations the two-body scattering problem in the presence of the Coulomb interaction can be solved using the standard uniform complex scaling. It is sketched how to apply the procedure in the case of three-body scattering. The great advantage of the use of the complex scaling is that the explicit construction of the scattering boundary condition can be avoided.

Energy resonance in scattering is usually investigated either directly in the complex energy plane (E-plane) or indirectly in the complex angular momentum plane (ℓ-plane). Another formulation complementing these two approaches was introduced recently. It is an indirect algebraic method that studies resonances in a complex charge plane (Z-plane). This latter approach will be generalized to provide a unified algebraic treatment of resonances in the complex E-, ℓ-, and Z-planes. The complex scaling (rotation) method will be used in the development of this approach. The resolvent operators (Green's functions) are formally defined in these three spaces. Bound states spectrum and resonance energies in the E-plane are mapped onto a discrete set of poles of the respective resolvent operator on the real line of the ℓ- and Z-planes. These poles move along trajectories as the energy is varied. A finite L² basis is used in the numerical implementation of this approach. Stability of poles and trajectories against variation in all computational parameters is demonstrated. Resonance energies for a given potential are calculated and compared with those obtained by other studies.

Charmed dibaryon states with the spin-parity $J^{\pi }=0^{+},1^{+}$, and $2^{+}$ are predicted for the two-body $Y_{c}N$ ($={\mathrm{\Lambda }}_c$, ${\mathrm{\Sigma }}_c$, or ${\mathrm{\Sigma }}_c^{\ast }$) systems. We employ the complex scaling method for the coupled channel Hamiltonian with the $Y_{c}N$-CTNN potentials, which were proposed in our previous study. We find four sharp resonance states near the ${\mathrm{\Sigma }}_{c}N$ and ${\mathrm{\Sigma }}_c^{\ast }N$ thresholds. From the analysis of the binding energies of partial channel systems, we conclude that these resonance states are Feshbach resonances. We compare the results with the $Y_{c}N$ resonance states in the heavy quark limit, where the ${\mathrm{\Sigma }}_{c}N$ and ${\mathrm{\Sigma }}_c^{\ast }N$ thresholds are degenerate, and find that they form two pairs of the heavy-quark doublets in agreement with the heavy quark spin symmetry.