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

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... potential problems with δ- and δ'-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.