Variational Approach to the On- and Off-ShellTMatrix

Abstract

A variational procedure for calculating the two-body T matrix T, (p, p';s) is proposed, and studied numerically for the case of the Reid So soft core potential. The method is based on a variational principle of the Schwinger type, in which the trial functions are themselves offenergy-shell T matrices with fixed s andp (or fixed s andp'), which are expressed as linear combinations of a convenient basis set. The variationally calculated T matrix turns out to have the interesting form T =V+VAN'V, where 6' is a finite-rank approximation to the full Green's function, of rank equal to the number of basis functions. It also turns out that for potentials of finite rank the approximation is exact, provided that the space spanned by the basis functions includes the form factors of the potential. Numerical results are given for the Reid potential at energies from -50 to 300 MeV, and show good convergence for both onand off-shell T matrix elements. The nonvariational estimates obtained directly from the trial functions also converge quite well, but less rapidly than the variational results.

Scattering Theory of Waves and Particles

“…(11.68) and (11.70). Verde (1949); Hulthen and Olsson (1950); T. Kato (I950b); Massey and Moiseiwitch (1951); J. L. Jackson (1951); Troesch and Verde (1951); H. E. Moses (1953b and; Borowitz and Friedman (1953); Turner and Makinson (1953); Boyet and Borowitz (1954); Moe and Saxon (1958); c. Joachain (1965a and b); C. Schwartz (1966); Y. Hahn (I968b); H. Morawitz (1970); Sloan and Brady (1972); Rabitz and Conn (1973); Carew and Rosenberg (1973); Oberoi and Nesbeth (1973); L. ; D. G. (11.69) is due to B.…”

Numerical quasiparticle representation of the nucleon-nucleonTmatrix

“…(3.2), we obtain It is worthwhile to note that one can look at the expressions Ts ', Ts", and Ts ' from quite a different point of view: These expressions are indeed separable representations for the operators T, T -V, and T -V -VGOV, respectively. The different accuracy which one obtains for the separable representation of T and T -V has already been emphasized in the literature [17], which in turn depends on the difference between the various variational principles [17,18] which underlie the separable representations.…”

Finite-Rank Multivariate-Basis Expansions of the Resolvent Operator as a Means of Solving the Multivariable Lippmann–Schwinger Equation for Two-Particle Scattering

“…18 would follow from using G A 0 , with A = O, L , R or I , directly in the T-LS equation. We also note that the IP-approximation G I is equivalent to the solution of the G-LS equation via a Schwinger-type variational principle [21][22][23] (which in this context is sometimes referred to as the Newton variational principle [24,26]), while LP-approximation G L corresponds to solution of the G-LS equation via the Galerkin method [11].…”

“…We find that, for a given basis (and the P associated with it), inner and Boolean projections provide the most promising computational schemes. We note in passing that some of these projection schemes are intimately related to Schwinger-type variational methods for the resolvent [14,[21][22][23][24][25][26]. All previous applications of these methods, however, have been on single-variable scattering equations that result from expansions over internal states and partial wave analysis.…”