Resonant state expansion of the resolvent.

Abstract

An analytic method of generating resonant state expansions from the standard completeness relation of nonrelativistic quantum mechanics is described and shown to reproduce the generalized completeness relations, earlier derived, involving resonant states. The method is then applied to the expansion of the resolvent (the complete Green's function), the symmetry properties of which seem to be destroyed if a conventional application of the completeness relations is made. These forms of expansions have a continuum term which contains symmetry-restoring contributions and can therefore never vanish identically, nor can it be neglected. The symmetry-conserving form of the expansion has a set of discrete terms which are identical in form to those of the Mittag-Leffler series for the resolvent. In addition, it contains a continuum contribution which in some cases vanishes identically, but in general does not. We illustrate these findings with numerical applictions in which the potential (a square well) is chosen so as to permit analytic evaluation of practically all functions and quantities involved.