On Approximate Asymptotic Solution of Integral Equations of Collision Theory

Abstract

It is well known that multi-particle integral equations of collision theory, in general, are not compact. At the same time it has been shown that the motion of three and four particles is described with consistent integral equations. In particular, by using identical transformations of the kernel of the Lipman-Schwinger equation for certain classes of potentials Faddeev obtained Fredholm type integral equations for three-particle problems $[1]$. The motion of for bodies is described by equations of Yakubovsky and Alt-Grassberger-Sandhas-Khelashvili $[2.3]$, which are obtained as a result of two subsequent transpormations of the kernel of Lipman-Schwinger equation. in the case of $N>4$ the compactness of multi-particle equations has not been proven yet. In turn out that for sufficiently high energies the $N$-particle $\left( {N \ge 3} \right)$ dynamic equations have correct asymptotic solutions satisfying unitary condition $[4]$. In present paper by using the Heitler formalism we obtain the results briefly summarized in Ref. [4]. In particular, on the bases of Heitler's equation [5] a unitary asymptotic solution of the system of $N$-particle scattering integral equations is found, which represents a generalization to any number of particles of the result of Ref. $[6]$ obtained for three particles.