Noyes-Kowalski Approximation for the Off-ShellT-Matrix and a Separable Potential

Abstract

> 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<qNIV' (3) where <qNI and lqN) are N-dimensional column and row vectors made up of a set of N trial functions li), and AN is the inverse of the matrix which has the element <iiVI j). It was shown that the given potential Vis approximated by the separable potential UN, if appropriate functions are chosen as the trial functions 3 > • 4 >. A single trial function is taken in Eq. (3) (i.e., N =1). At first, the wave function of a free particle I Xk) is selected as the trial function. The separable potential U/ is then given byfrom Eq. (3). The on-shell T-matrix for U/ yields exactly that for V which is given by choosing a free particle wave function as the trial function in Schwinger's variational expression for the on-shell T-matrix. 5 > Next, the trial function is chosen to be the partial wave projection I <,bk <+ >) of the scattering wave function which is governed by the potential V and is asymptotically equal to the sum of an incident plane wave and an outgoing scattering wave. Then Eq. (3) yields the separable potential at UPVA on June 16, 2015 http://ptp.oxfordjournals.org/ Downloaded from