Two-Body Off-Shell T and S Matrices

Abstract

The two-body- off-shell T matrix elements with complex energy parameter for uncoupled and coupled states of the orbital angular momenta are investigated in a numerical method. The off-shell T matrix has a branch cut from the origin along the positive energy axis on the complex energy plane. If a two-body potential. has several bound states, the poles of -, the off-shell T matrix appear only on the negative energy axis and correspond to the, bound states of the potential. Redundant poles never appear. Once the off-shell T matrix is de­ termined, an off-shell S matrix with two complex energy parameters can be made of the off-shell T matrix. The scattering amplitude on the energy shell can be derived from the on-shell limit of the off-shell S matrix. The above approach can be extended to the case of the Coulomb plus nuclear potential. In this case, the Coulomb potential must be included into the unperturbed Hamiltonian, because of the long range property of the potential. Previously one of the present authors (T.T.) gave an exact expression/), 2) referred to as T1 and T2 for Refs. 1) and 2), respectively, of the off-shell T matrix element for a finite range local potential with hard core. The expression can be used for both cases of uncoupled and coupled states of the orbital ang1,dar momenta. He has really shown that the on-shell limit of the off-shell T matrix for the Ramada-Johnston potential 3) (HJ) is in agreement with the usual well­ known T matrix1) on the energy shell which is closely related with the two-body scattering amplitude. The expression of the off-shell element can still be applic­ able to a potential without hard core if the hard core radius can be brought to a negligibly small value. W e 4) have confirmed this fact for the off-shell element with negative energy parameters, using the Reid soft core potential 5) (RSC). It has also been shown that the off-shell element with negative energy parameter has a pole at the position corresponding to the bound state if the potential has one or more bound states. The bound state energy for the triplet coupled state of the total angular momentum J = 1 was - 2.269 MeV for HJ potential and -2.2245 MeV for RSC potential. In those cases, a pair of the bound state wave functions is represented only by the solutions of coupled Schri:idinger equation. Although the previous works1), 2), 4) were done only within the framework- of real energy parameters, the generalization to complex energy parameters is easily done. In fact, for the three-body problem, such a kind of off-shell T matrix with ~ complex energy parameter appears in the Fadd~ev equation6) as the kernel