> decomposed the off-shell two-body T-matrix for an arbitrary potential into a single separable and a nonseparable term. The off-shell T-matrix is approximately represented with the separable form by neglecting the non-separable term in the decomposition.ll,S> Such an approximation for representing the off-shell T-matrix is hereafter referred to as the NK approximation. In calculating the onshell T-matrix, the approximation of setting the non-separable term equal to zero is exactly equivalent to choosing a plane wave function as the trial function in Schwinger's variational expression for the on-shell T-matrix. 2 > The non-separable term vanishes exactly for a separable potential of the rank one. The NK approximation is reduced to a separable approximation for a given potential, provided that the given potential is approximated by a separable potential of the rank one and the off-shell T-matrix for that separable potential is equal to the separable term in the decomposition of the off-shell T-matrix for the given potential. This makes it possible to investigate properties of the NK approximation on the basis of such a separable potential. In this note, it is shown that the NK approximation is exactly equivalent to approximating the potential by a separable potential of the rank one.A two-body potential is denoted by V. In the momentum representation, the offshell two-body T-matrix T 1 (p,p;k 2 ) for the l-th partial wave satisfies the Lippmann-
Schwinger equationT 1 (p, q; k 2 ) = V 1 (p, q) +;:: Xi"" V, (p, p') Tz (p'' q; k2) p'2 dp' (1) 0 k 2 -P"+i~£ .Here hp, hp', hq are momenta, p. is the reduced mass and h 2 k2 /2p. is the kinetic energy in the c.m. system. The partial wave function of a free particle is given bywhere j 1 (x) is the spherical Bessel function of the l-th order. The partial wave index will be omitted from now on. Sugar and Blankenbecler 3 > gave the separable potential of the rank N UN= VlqN)AN
A high·energy scattering approximation which expresses the wave function as a sum of the second Born form and correction terms is developed. It is shown for example of a Gaussian potential that the approximation well gives the scattering amplitude for the coupling parameter larger as well as smaller than unity.In previous work,l),2) an approximation method was developed to describe the scattering of a high-energy Schrodinger particle from a central potential of finite range. Under the conditions that the typical strength of the potential is much smaller than the incident energy and its range is greater than the reduced wavelength, the method can well give the scattering amplitude in the weak and intermediate coupling cases far beyond the conventional angular range of validity of Glauber's formula. 3 ) It . is applicable to the scattering from an optical potential by analytic continuation,I) andits relativistic extension is possible within the frawework of the non-relativistic Schrodinger equation by modifying the wave number and the potentia1. 2 ) Its advantage is that the scattering amplitude as well as the wave function is expressed in terms' of integrals evaluated along a straight line parallel to the incident direction independent of a scattering angle. In this respect, it contrasts with the eikonal approach 4H ) which has a mathematical problem of changing the integration path after having calculated the scattering amplitude from the wave function and ambiguity on selecting an integration path in evaluating the off-shell amplitude. On the other hand, however, the method has a physically important problem of extending an allowed range of the coupling parameter. In the present paper, an attempt is reported to improve the method by introducing manageable correction. The notations in what follows are the same as those in Refs. 1) and 2). The scattering of a particle with an incident wave number ko from a potential V(r) is described by the integral equation W(r)=exp{iko(so' r)}+ j G(r, r') U(r') W(r')dr'(1)References 1) and 2) adopted its approximate solution W(r)=exp{iko(so'r)}+ jG(r, r')U(r') 1fft(r') dr' ,where 1fft(r) is the internal wave function for the scattering from the square well
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