  The path decomposition expansion represents the propagator of the
irreversible reaction as a convolution of the first-passage, last-passage and
rebinding time probability densities. Using path integral technique, we give an
elementary, yet rigorous, proof of the path decomposition expansion of the
Green's functions describing the non-reactive case and the irreversible
reaction of an isolated pair of molecules. To this end, we exploit the
connection between boundary value problems and interaction potential problems
with $\delta$- and $\delta'$-function perturbation. In particular, we employ a
known exact summation of a perturbation series to derive exact relations
between the Green's functions of the perturbed and unperturbed problem. Along
the way, we are able to derive a number of additional exact identities that
relate the propagators describing the free-space, the non-reactive as well as
the completely and partially reactive case.
