Unified treatment of bound-state and scattering problems.

Abstract

The iteration-subtraction method for the unified treatment of bound-state and scattering problems is compared and contrasted with a similar method for the two-body bound-state problem via nonsingular scattering equations developed recently. We also compare another recent method for solving bound-state problems with the iteration-subtraction method. A recently proposed iteration-subtraction method' for the unified treatment of bound state and scattering problems in the momentum space suggests that solutions to both these problems can essentially be constructed using the iterative solution of an auxiliary integral equation of the Fredholm type. This auxiliary integral equation has a kernel which is free from singularities and is also sufBciently weak in order to allow a convergent iterative solution for a wide class of potentials. The solution of the original Lippmann-Schwinger scattering equation in the momentum space is then related to that of the auxiliary equation. At the bound state energy the solution of the auxiliary equation satisfies a certain condition which is used to pick up the binding energy, and the bound-state wave function is readily constructed using the solution of the auxiliary equation at this energy. This method has been used to define a class of Jostlike functions and has been studied in the case of nonlocal potentials. This method has also been extended to the case of multichannel scattering problems and applied to three-body scattering problems. The approach has been demonstrated by Dzhibuti and Tsiklauri to be very useful to solve three- and four-body bound state problems using the hyperspherical harmonics method in momentum space. The method has easily been extended to study virtual states and resonances. Sasakawa proposed an eKcient method for computing phase shifts using a wave-function description of scattering. Later it was shown that the Sasakawa method can be reformulated to yield a practical method for computing half-on-shell t matrix elements using the momentum-space Lippmann-Schwinger equations. Also, a method by Kowalski and Noyes' has the advantage of the methods of Refs. 1 — 7 in that it treats the scattering problem through the solution of an auxiliary equation. The above mentioned iteration-subtraction method' should be considered a generalization of methods of Refs. 8-10 to calculate fully-off-shell t matrix elements, binding energies, and bound state wave functions. More recently, and specially after the completion of