Off the Energy ShellTMatrix. II

Toshinori Takemiya
1973 Progress of theoretical physics  
The coupled equations of the off-shell Lippman-Schwinger equation for triplet states are solved. From the outgoing wave parts of the solutions, the expressions of the off-shell T matrices for triplet-state interactions are derived. It is shown that the present T matrices on the energy shell agree with the usual ones and the off-shell 'T matrices have the symmetry property, tf.\<,+> (p, k; z) =tf,~1+l (k,p; z). § I. hbroduction The importance of the investigation of the off-shell T matrix
more » ... s in the fact that the elements of the T matrix are measurable quantities and if all of them will be known in the future from the experimental data, it will be useful to construct the interaction potential. It is only the on-shell elements that can be obtained from the analysis of the free two-body scattering data, but the offshell elements must be obtained from the analysis of the P-P bremsstrahlung, nuclear matter, three-body bound and scattering problems, etc. Unfortunately, the knowledge of the off-shell T matrix has been scarecely obtained from the experimental data up to the present day. Therefore it seems to be the second best that the off-shell T matrix can be obtained from the realistic potential which reproduces the free two-body scattering data. Once a realistic potential is assumed, the T matrix should be uniquely determined, because it is defined through the integral equation T=V+ V(E-H 0 +ie)-1 T. Wong and ZambottPl solved the itpove integral equation for the Yukawa potential by use of the matrix inversion n'iethod. However this method is of no use for the potential with hard cores. We have to look for the method to obtain the off-shell T matrix, which is valid for the potential with or without hard cores. Previously, we presented a simple method 2 l to obtain the off-shell T matrix for the singlet interaction potentials with hard cores. Although the method was applied to the singlet-state interaction of the Ramada-Johnston potentia1, 3 l it is considerably general and is applicable to any finite-range local potential with or without hard cores even if the potential may have energy dependences.
doi:10.1143/ptp.49.1602 fatcat:gvejv63tbbednnpgxqbcdrwq2q