Outgoing Boundary Condition in Rearrangement Collisions

Abstract

The boundary condition on the solution to the nonrelativistic time-independent Schrödinger equation for arbitrarily complicated rearrangements of spinless particles is carefully examined. For real energies $E$ it is shown that the outgoing boundary condition on the scattered wave $\varphi$ need not imply that $\varphi$ is "every-where outgoing." This and similar considerations make apparent the fact remarked by Foldy and Tobocman, namely that the Lippmann-Schwinger integral equation need not have a unique solution for real energies. The relationship of this result to the added fact that solutions to the Lippmann-Schwinger integral equation are unique for complex energies $E+i\epsilon$, $\epsilon >0\mathrm{}$ is discussed, as is also the relationship of the usual operator manipulations to the outgoing boundary condition.