The existing time-dependent formal theory of scattering is valid only for the simplest case. The theory is extended to the general case of rearrangement collisions including the case where all or some particles are indistinguishable.

The outstanding feature of the scattering process is the existence of asymptotically constant observables. The formal expression of this fact is the existence of a set of basis functions ${\Phi }_n$ such that, if the Schrödinger state is written as a linear combination $\int c(n, t)\exp (-i{E}_{n}t){\Phi }_n\mathrm{dn}$, the coefficients $c$ have time-independent limits $c_{\pm }\mathrm{}\left(n\right)\mathrm{}$ for $t=\pm \infty$, the squares of which are observed probabilities. While in the simplest case the ${\Phi }_n\text{'}\mathrm{s}$ are eigenfunctions of the unperturbed Hamiltonian, they form in general a nonorthogonal and linearly not-independent set which can be explicitly given, but there exists no linear operator of which they are eigenfunctions. As a consequence, it is impossible to define an interaction representation in which the states have asymptotic limits, and no linear $S$-operator exists. An $S$-matrix is defined, and it is shown to be connected with cross sections in the usual manner. An expression for the $S$-matrix in terms of time-independent solutions is given and shown to reduce to the usual one in the simplest case, but not in general. An integral equation for the scattering amplitude is given which is nonlinear. It has, however, the advantage of exhibiting directly the contribution of bound states in addition to Born's approximation. Unitarity and reciprocity relations for the general case are derived.

• Received 15 August 1955

DOI:https://doi.org/10.1103/PhysRev.101.880