Properties of Resonance Wave Functions

Resonances of quantum systems are associated with poles of the Green's function which occur at complex energies. The wave functions corresponding to such poles increase exponentially at large distances and so are very badly divergent. Nevertheless, these resonance wave functions have useful properties which can be exploited in cases where only their behavior at small distances is relevant. In this paper we construct and study such resonance wave functions for several illustrative quantum systems of theoretical interest. It is shown that the wave functions may be considered renormalized in a sense analogous to that of quantum field theory. However, the renormalization which occurs here is entirely automatic and the theory has neither ad hoc procedures nor infinite quantities. In addition to other results, we obtain a representation of the Green's function in terms of the resonance wave functions; this representation appears likely to be useful because it has an energy dependence that is especially simple.