Green’s function and propagator for the one-dimensionalδ-function potential

A particle in a one-dimensional \ensuremath{\delta}-function potential possesses both discrete and continuum solutions. The configuration-space Green's function and propagator for this problem are derived by explicit summation over the spectrum of eigenstates. The momentum-space Green's function is also obtained. The propagator does not contain the classical action function in any simple way, in contrast to the usual structure in Feynman's path-integral formalism. Various analogies between the \ensuremath{\delta}-function and Coulomb problems are discussed.