Completeness relations and resonant state expansions

The completeness properties of the discrete set of bound states, virtual states, and resonant states characterizing the system of a single nonrelativistic particle moving in a central cutoff potential are investigated. We do not limit ourselves to the restricted form of completeness that can be obtained from Mittag-Leffler theory in this case. Instead we will make use of the information contained in the asymptotic behavior of the discrete states to get a new approach to the question of eventual overcompleteness. Using the theory of analytic functions we derive a number of completeness relations in terms of discrete states and complex continuum states and give some criteria for how to use them to form resonant state expansions of functions, matrix elements, and Green’s functions. In cases where the integral contribution vanishes, the discrete part of the expansions is of the same form as that given by Mittag-Leffler theory but with regularized inner products. We also consider the possibility of using the discrete states as basis in a matrix representation.