Nonuniqueness of solutions to the Lippmann-Schwinger equation in a soluble three-body model.

Abstract

It is generally accepted that with suitable boundary conditions the Lippmann-Schwinger (LS) integral equation is equivalent to the Schr\"odinger equation, even in systems of more than two particles. It also is generally accepted that although in two-particle systems the scattering solutions to the LS equation are uniquely defined by that equation without need for boundary conditions, the same cannot be said of systems containing more than two particles; in such many-particle systems, it is generally agreed, unique scattering solutions to the LS equation are not obtained unless boundary conditions are imposed. However, these assertions have not been verified heretofore in any system of three or more particles, because exact closed-form solutions of the Schr\"odinger or LS equation for such systems are so difficult to achieve. We have examined these questions in a one-dimensional three-body model first discussed by McGuire, wherein three equal-mass particles interact via equal and finite strength attractive \ensuremath{\delta}-function potentials. With this model, the scattering solutions to the Schr\"odinger equation can be written exactly, in closed form. Thereby we are able to demonstrate explicitly that in this model the scattering solutions to the Schr\"odinger equation do satisfy the LS equation; we also demonstrate explicitly that the LS equation's scattering solutions really are nonunique unless boundary conditions are imposed. This latter result strongly suggests that, in three-particle systems at any rate, recent criticisms of the aforementioned generally accepted nonuniqueness thesis are not well taken.